null hypothesis questions

How to Write a Null Hypothesis (5 Examples)

A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter =,  ≤, ≥ some value

H A  (Alternative Hypothesis): Population parameter <, >, ≠ some value

Note that the null hypothesis always contains the equal sign .

We interpret the hypotheses as follows:

Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.

Alternative hypothesis: The sample data  does provide sufficient evidence to support the claim being made by an individual.

For example, suppose it’s assumed that the average height of a certain species of plant is 20 inches tall. However, one botanist claims the true average height is greater than 20 inches.

To test this claim, she may go out and collect a random sample of plants. She can then use this sample data to perform a hypothesis test using the following two hypotheses:

H 0 : μ ≤ 20 (the true mean height of plants is equal to or even less than 20 inches)

H A : μ > 20 (the true mean height of plants is greater than 20 inches)

If the sample data gathered by the botanist shows that the mean height of this species of plants is significantly greater than 20 inches, she can reject the null hypothesis and conclude that the mean height is greater than 20 inches.

Read through the following examples to gain a better understanding of how to write a null hypothesis in different situations.

Example 1: Weight of Turtles

A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles.

Here is how to write the null and alternative hypotheses for this scenario:

H 0 : μ = 300 (the true mean weight is equal to 300 pounds)

H A : μ ≠ 300 (the true mean weight is not equal to 300 pounds)

Example 2: Height of Males

It’s assumed that the mean height of males in a certain city is 68 inches. However, an independent researcher believes the true mean height is greater than 68 inches. To test this, he goes out and collects the height of 50 males in the city.

H 0 : μ ≤ 68 (the true mean height is equal to or even less than 68 inches)

H A : μ > 68 (the true mean height is greater than 68 inches)

Example 3: Graduation Rates

A university states that 80% of all students graduate on time. However, an independent researcher believes that less than 80% of all students graduate on time. To test this, she collects data on the proportion of students who graduated on time last year at the university.

H 0 : p ≥ 0.80 (the true proportion of students who graduate on time is 80% or higher)

H A : μ < 0.80 (the true proportion of students who graduate on time is less than 80%)

Example 4: Burger Weights

A food researcher wants to test whether or not the true mean weight of a burger at a certain restaurant is 7 ounces. To test this, he goes out and measures the weight of a random sample of 20 burgers from this restaurant.

H 0 : μ = 7 (the true mean weight is equal to 7 ounces)

H A : μ ≠ 7 (the true mean weight is not equal to 7 ounces)

Example 5: Citizen Support

A politician claims that less than 30% of citizens in a certain town support a certain law. To test this, he goes out and surveys 200 citizens on whether or not they support the law.

H 0 : p ≥ .30 (the true proportion of citizens who support the law is greater than or equal to 30%)

H A : μ < 0.30 (the true proportion of citizens who support the law is less than 30%)

Additional Resources

Introduction to Hypothesis Testing Introduction to Confidence Intervals An Explanation of P-Values and Statistical Significance

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

2 Replies to “How to Write a Null Hypothesis (5 Examples)”

you are amazing, thank you so much

Say I am a botanist hypothesizing the average height of daisies is 20 inches, or not? Does T = (ave – 20 inches) / √ variance / (80 / 4)? … This assumes 40 real measures + 40 fake = 80 n, but that seems questionable. Please advise.

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Statistics By Jim

Making statistics intuitive

Null Hypothesis: Definition, Rejecting & Examples

By Jim Frost 6 Comments

What is a Null Hypothesis?

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

• Null Hypothesis H 0 : No effect exists in the population.
• Alternative Hypothesis H A : The effect exists in the population.

In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.

In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!

You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.

Related post : What is an Effect in Statistics?

Null Hypothesis Examples

Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.

After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.

Let’s see how you reject the null hypothesis and get to those more exciting findings!

When to Reject the Null Hypothesis

So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.

The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .

After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.

When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .

Rejecting the Null Hypothesis

Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Failing to Reject the Null Hypothesis

Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!

Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .

That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!

Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.

Related posts : How Hypothesis Tests Work and Interpreting P-values

How to Write a Null Hypothesis

The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.

Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Group Means

T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.

For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group means are equal in the population: µ 1 = µ 2 , or µ 1 – µ 2 = 0
• Alternative Hypothesis H A : Group means are not equal in the population: µ 1 ≠ µ 2 , or µ 1 – µ 2 ≠ 0.

Group Proportions

Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.

For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group proportions are equal in the population: p 1 = p 2 .
• Alternative Hypothesis H A : Group proportions are not equal in the population: p 1 ≠ p 2 .

Correlation and Regression Coefficients

Some studies assess the relationship between two continuous variables rather than differences between groups.

In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.

For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.

• Null Hypothesis H 0 : The correlation in the population is zero: ρ = 0.
• Alternative Hypothesis H A : The correlation in the population is not zero: ρ ≠ 0.

For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.

The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .

Related post : Understanding Correlation

Neyman, J; Pearson, E. S. (January 1, 1933).  On the Problem of the most Efficient Tests of Statistical Hypotheses .  Philosophical Transactions of the Royal Society A .  231  (694–706): 289–337.

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Reader Interactions

January 11, 2024 at 2:57 pm

Thanks for the reply.

January 10, 2024 at 1:23 pm

Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?

January 10, 2024 at 2:15 pm

Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.

Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.

With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.

So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).

For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.

I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!

February 20, 2022 at 9:26 pm

Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”

February 23, 2022 at 9:21 pm

Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.

It’s the alternative hypothesis that typically contains does not equal.

There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.

In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.

February 15, 2022 at 9:32 am

Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent

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What is The Null Hypothesis & When Do You Reject The Null Hypothesis

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On This Page:

A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating ​ the dependent variable or due to random chance. 

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When Do We Reject The Null Hypothesis? 

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected. 

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables. 

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a  p  -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. 

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null. 

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists. 

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter. 

Purpose of a Null Hypothesis 

• The primary purpose of the null hypothesis is to disprove an assumption. 
• Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
• A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true. 

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables. 

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study. 

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”).  However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing.  Political research quarterly ,  52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method.  American Psychologist ,  56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing.  Behavior research methods ,  43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy.  Psychological methods ,  5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.  Psychological bulletin ,  57 (5), 416.

Null Hypothesis Examples

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In statistical analysis, the null hypothesis assumes there is no meaningful relationship between two variables. Testing the null hypothesis can tell you whether your results are due to the effect of manipulating ​a dependent variable or due to chance. It's often used in conjunction with an alternative hypothesis, which assumes there is, in fact, a relationship between two variables.

The null hypothesis is among the easiest hypothesis to test using statistical analysis, making it perhaps the most valuable hypothesis for the scientific method. By evaluating a null hypothesis in addition to another hypothesis, researchers can support their conclusions with a higher level of confidence. Below are examples of how you might formulate a null hypothesis to fit certain questions.

What Is the Null Hypothesis?

The null hypothesis states there is no relationship between the measured phenomenon (the dependent variable ) and the independent variable , which is the variable an experimenter typically controls or changes. You do not​ need to believe that the null hypothesis is true to test it. On the contrary, you will likely suspect there is a relationship between a set of variables. One way to prove that this is the case is to reject the null hypothesis. Rejecting a hypothesis does not mean an experiment was "bad" or that it didn't produce results. In fact, it is often one of the first steps toward further inquiry.

To distinguish it from other hypotheses , the null hypothesis is written as ​ H 0  (which is read as “H-nought,” "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95% or 99% is common. Keep in mind, even if the confidence level is high, there is still a small chance the null hypothesis is not true, perhaps because the experimenter did not account for a critical factor or because of chance. This is one reason why it's important to repeat experiments.

Examples of the Null Hypothesis

To write a null hypothesis, first start by asking a question. Rephrase that question in a form that assumes no relationship between the variables. In other words, assume a treatment has no effect. Write your hypothesis in a way that reflects this.

Other Types of Hypotheses

In addition to the null hypothesis, the alternative hypothesis is also a staple in traditional significance tests . It's essentially the opposite of the null hypothesis because it assumes the claim in question is true. For the first item in the table above, for example, an alternative hypothesis might be "Age does have an effect on mathematical ability."

Key Takeaways

• In hypothesis testing, the null hypothesis assumes no relationship between two variables, providing a baseline for statistical analysis.
• Rejecting the null hypothesis suggests there is evidence of a relationship between variables.
• By formulating a null hypothesis, researchers can systematically test assumptions and draw more reliable conclusions from their experiments.
• Difference Between Independent and Dependent Variables
• Examples of Independent and Dependent Variables
• The Difference Between Control Group and Experimental Group
• What Is a Hypothesis? (Science)
• What 'Fail to Reject' Means in a Hypothesis Test
• Definition of a Hypothesis
• Null Hypothesis Definition and Examples
• Scientific Method Vocabulary Terms
• Null Hypothesis and Alternative Hypothesis
• Hypothesis Test for the Difference of Two Population Proportions
• How to Conduct a Hypothesis Test
• What Is a P-Value?
• What Are the Elements of a Good Hypothesis?
• Hypothesis Test Example
• What Is the Difference Between Alpha and P-Values?
• Understanding Path Analysis

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10.1 - setting the hypotheses: examples.

A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The alternative hypothesis is typically the research hypothesis of interest. Here are some examples.

Example 10.2: Hypotheses with One Sample of One Categorical Variable Section  

About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

• Research Question : Are artists more likely to be left-handed than people found in the general population?
• Response Variable : Classification of the student as either right-handed or left-handed

State Null and Alternative Hypotheses

• Null Hypothesis : Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or p = .10).
• Alternative Hypothesis : Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or p > .10). This is a one-sided alternative hypothesis.

Example 10.3: Hypotheses with One Sample of One Measurement Variable Section  

A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

• Research Question : Does the data suggest that the population mean dosage of this brand is different than 50 mg?
• Response Variable : dosage of the active ingredient found by a chemical assay.
• Null Hypothesis : On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
• Alternative Hypothesis : On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

Example 10.4: Hypotheses with Two Samples of One Categorical Variable Section  

Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

• Research Question : Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?
• Response Variable : Classification of whether or not a person eats vegetarian meals on a regular basis
• Explanatory (Grouping) Variable: Sex
• Null Hypothesis : There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or p females = p males ).
• Alternative Hypothesis : Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or p females > p males ). This is a one-sided alternative hypothesis.

Example 10.5: Hypotheses with Two Samples of One Measurement Variable Section  

Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

• Research Question : Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?
• Response Variable : Weight loss (pounds)
• Explanatory (Grouping) Variable : Type of diet
• Null Hypothesis : There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).
• Alternative Hypothesis : The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

Example 10.6: Hypotheses about the relationship between Two Categorical Variables Section  

• Research Question : Do the odds of having a stroke increase if you inhale second hand smoke ? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.
• Variables : There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.
• Null Hypothesis : There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).
• Alternative Hypothesis : There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

Example 10.7: Hypotheses about the relationship between Two Measurement Variables Section  

• Research Question : A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day (stock trading by management being under "insider-trading" regulatory restrictions).
• Variables : Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.
• Null Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.
• Alternative Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples Section  

• Research Question : Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?
• Variables : There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.
• Null Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.
• Alternative Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.

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• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis (H 0 ): There’s no effect in the population .
• Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question
• They both make claims about the population
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
• Alternative hypothesis (H A ): Independent variable affects dependent variable .

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Null Hypothesis Definition and Examples, How to State

What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.

Why is it Called the “Null”?

The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).

Why Do I need to Test it? Why not just prove an alternate one?

The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.

• Null hypothesis : H 0 : The world is flat.
• Alternate hypothesis: The world is round.

Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .

How to State the Null Hypothesis from a Word Problem

You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.

Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.

H 1 : μ > 8.2

Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2

Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.

H 0 : μ ≤ 8.2

Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2

How to State the Null Hypothesis: Part Two

But what if the researcher doesn’t have any idea what will happen.

Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks. 

Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.

H 0 : μ = 8.2

Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2

Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?

H 1 : μ ≠ 8.2

In English again, that’s H 1 (The  alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2

That’s How to State the Null Hypothesis!

Check out our Youtube channel for more stats tips!

Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley.

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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Null Hypothesis Examples

The null hypothesis (H 0 ) is the hypothesis that states there is no statistical difference between two sample sets. In other words, it assumes the independent variable does not have an effect on the dependent variable in a scientific experiment .

The null hypothesis is the most powerful type of hypothesis in the scientific method because it’s the easiest one to test with a high confidence level using statistics. If the null hypothesis is accepted, then it’s evidence any observed differences between two experiment groups are due to random chance. If the null hypothesis is rejected, then it’s strong evidence there is a true difference between test sets or that the independent variable affects the dependent variable.

• The null hypothesis is a nullifiable hypothesis. A researcher seeks to reject it because this result strongly indicates observed differences are real and not just due to chance.
• The null hypothesis may be accepted or rejected, but not proven. There is always a level of confidence in the outcome.

What Is the Null Hypothesis?

The null hypothesis is written as H 0 , which is read as H-zero, H-nought, or H-null. It is associated with another hypothesis, called the alternate or alternative hypothesis H A or H 1 . When the null hypothesis and alternate hypothesis are written mathematically, they cover all possible outcomes of an experiment.

An experimenter tests the null hypothesis with a statistical analysis called a significance test. The significance test determines the likelihood that the results of the test are not due to chance. Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01). But, even if the confidence in the test is high, there is always a small chance the outcome is incorrect. This means you can’t prove a null hypothesis. It’s also a good reason why it’s important to repeat experiments.

Exact and Inexact Null Hypothesis

The most common type of null hypothesis assumes no difference between two samples or groups or no measurable effect of a treatment. This is the exact hypothesis . If you’re asked to state a null hypothesis for a science class, this is the one to write. It is the easiest type of hypothesis to test and is the only one accepted for certain types of analysis. Examples include:

There is no difference between two groups H 0 : μ 1  = μ 2 (where H 0  = the null hypothesis, μ 1  = the mean of population 1, and μ 2  = the mean of population 2)

Both groups have value of 100 (or any number or quality) H 0 : μ = 100

However, sometimes a researcher may test an inexact hypothesis . This type of hypothesis specifies ranges or intervals. Examples include:

Recovery time from a treatment is the same or worse than a placebo: H 0 : μ ≥ placebo time

There is a 5% or less difference between two groups: H 0 : 95 ≤ μ ≤ 105

An inexact hypothesis offers “directionality” about a phenomenon. For example, an exact hypothesis can indicate whether or not a treatment has an effect, while an inexact hypothesis can tell whether an effect is positive of negative. However, an inexact hypothesis may be harder to test and some scientists and statisticians disagree about whether it’s a true null hypothesis .

How to State the Null Hypothesis

To state the null hypothesis, first state what you expect the experiment to show. Then, rephrase the statement in a form that assumes there is no relationship between the variables or that a treatment has no effect.

Example: A researcher tests whether a new drug speeds recovery time from a certain disease. The average recovery time without treatment is 3 weeks.

• State the goal of the experiment: “I hope the average recovery time with the new drug will be less than 3 weeks.”
• Rephrase the hypothesis to assume the treatment has no effect: “If the drug doesn’t shorten recovery time, then the average time will be 3 weeks or longer.” Mathematically: H 0 : μ ≥ 3

This null hypothesis (inexact hypothesis) covers both the scenario in which the drug has no effect and the one in which the drugs makes the recovery time longer. The alternate hypothesis is that average recovery time will be less than three weeks:

H A : μ < 3

Of course, the researcher could test the no-effect hypothesis (exact null hypothesis): H 0 : μ = 3

The danger of testing this hypothesis is that rejecting it only implies the drug affected recovery time (not whether it made it better or worse). This is because the alternate hypothesis is:

H A : μ ≠ 3 (which includes μ <3 and μ >3)

Even though the no-effect null hypothesis yields less information, it’s used because it’s easier to test using statistics. Basically, testing whether something is unchanged/changed is easier than trying to quantify the nature of the change.

Remember, a researcher hopes to reject the null hypothesis because this supports the alternate hypothesis. Also, be sure the null and alternate hypothesis cover all outcomes. Finally, remember a simple true/false, equal/unequal, yes/no exact hypothesis is easier to test than a more complex inexact hypothesis.

• Adèr, H. J.; Mellenbergh, G. J. & Hand, D. J. (2007).  Advising on Research Methods: A Consultant’s Companion . Huizen, The Netherlands: Johannes van Kessel Publishing. ISBN  978-90-79418-01-5 .
• Cox, D. R. (2006).  Principles of Statistical Inference . Cambridge University Press. ISBN  978-0-521-68567-2 .
• Everitt, Brian (1998).  The Cambridge Dictionary of Statistics . Cambridge, UK New York: Cambridge University Press. ISBN 978-0521593465.
• Weiss, Neil A. (1999).  Introductory Statistics  (5th ed.). ISBN 9780201598773.

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Writing Null Hypotheses in Research and Statistics

Last Updated: January 17, 2024 Fact Checked

This article was co-authored by Joseph Quinones and by wikiHow staff writer, Jennifer Mueller, JD . Joseph Quinones is a High School Physics Teacher working at South Bronx Community Charter High School. Joseph specializes in astronomy and astrophysics and is interested in science education and science outreach, currently practicing ways to make physics accessible to more students with the goal of bringing more students of color into the STEM fields. He has experience working on Astrophysics research projects at the Museum of Natural History (AMNH). Joseph recieved his Bachelor's degree in Physics from Lehman College and his Masters in Physics Education from City College of New York (CCNY). He is also a member of a network called New York City Men Teach. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 27,309 times.

Are you working on a research project and struggling with how to write a null hypothesis? Well, you've come to the right place! Start by recognizing that the basic definition of "null" is "none" or "zero"—that's your biggest clue as to what a null hypothesis should say. Keep reading to learn everything you need to know about the null hypothesis, including how it relates to your research question and your alternative hypothesis as well as how to use it in different types of studies.

Things You Should Know

• Write a research null hypothesis as a statement that the studied variables have no relationship to each other, or that there's no difference between 2 groups.

• Adjust the format of your null hypothesis to match the statistical method you used to test it, such as using "mean" if you're comparing the mean between 2 groups.

What is a null hypothesis?

• Research hypothesis: States in plain language that there's no relationship between the 2 variables or there's no difference between the 2 groups being studied.
• Statistical hypothesis: States the predicted outcome of statistical analysis through a mathematical equation related to the statistical method you're using.

Examples of Null Hypotheses

Null Hypothesis vs. Alternative Hypothesis

• For example, your alternative hypothesis could state a positive correlation between 2 variables while your null hypothesis states there's no relationship. If there's a negative correlation, then both hypotheses are false.

• You need additional data or evidence to show that your alternative hypothesis is correct—proving the null hypothesis false is just the first step.
• In smaller studies, sometimes it's enough to show that there's some relationship and your hypothesis could be correct—you can leave the additional proof as an open question for other researchers to tackle.

How do I test a null hypothesis?

• Group means: Compare the mean of the variable in your sample with the mean of the variable in the general population. [6] X Research source
• Group proportions: Compare the proportion of the variable in your sample with the proportion of the variable in the general population. [7] X Research source
• Correlation: Correlation analysis looks at the relationship between 2 variables—specifically, whether they tend to happen together. [8] X Research source
• Regression: Regression analysis reveals the correlation between 2 variables while also controlling for the effect of other, interrelated variables. [9] X Research source

Templates for Null Hypotheses

• Research null hypothesis: There is no difference in the mean [dependent variable] between [group 1] and [group 2].

• Research null hypothesis: The proportion of [dependent variable] in [group 1] and [group 2] is the same.

• Research null hypothesis: There is no correlation between [independent variable] and [dependent variable] in the population.

• Research null hypothesis: There is no relationship between [independent variable] and [dependent variable] in the population.

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Thanks for reading our article! If you’d like to learn more about physics, check out our in-depth interview with Joseph Quinones .

• ↑ https://online.stat.psu.edu/stat100/lesson/10/10.1
• ↑ https://online.stat.psu.edu/stat501/lesson/2/2.12
• ↑ https://support.minitab.com/en-us/minitab/21/help-and-how-to/statistics/basic-statistics/supporting-topics/basics/null-and-alternative-hypotheses/
• ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5635437/
• ↑ https://online.stat.psu.edu/statprogram/reviews/statistical-concepts/hypothesis-testing
• ↑ https://education.arcus.chop.edu/null-hypothesis-testing/
• ↑ https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_hypothesistest-means-proportions/bs704_hypothesistest-means-proportions_print.html

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Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

The actual test begins by considering two  hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis : It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a  decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in  H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

In a  hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

• OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

Null Hypothesis

Null hypothesis is used to make decisions based on data and by using statistical tests. Null hypothesis is represented using H o and it states that there is no difference between the characteristics of two samples. Null hypothesis is generally a statement of no difference. The rejection of null hypothesis is equivalent to the acceptance of the alternate hypothesis.

Let us learn more about null hypotheses, tests for null hypotheses, the difference between null hypothesis and alternate hypothesis, with the help of examples, FAQs.

What Is Null Hypothesis?

Null hypothesis states that there is no significant difference between the observed characteristics across two sample sets. Null hypothesis states the observed population parameters or variables is the same across the samples. The null hypothesis states that there is no relationship between the sample parameters, the independent variable, and the dependent variable. The term null hypothesis is used in instances to mean that there is no differences in the two means, or that the difference is not so significant.

If the experimental outcome is the same as the theoretical outcome then the null hypothesis holds good. But if there are any differences in the observed parameters across the samples then the null hypothesis is rejected, and we consider an alternate hypothesis. The rejection of the null hypothesis does not mean that there were flaws in the basic experimentation, but it sets the stage for further research. Generally, the strength of the evidence is tested against the null hypothesis.

Null hypothesis and alternate hypothesis are the two approaches used across statistics. The alternate hypothesis states that there is a significant difference between the parameters across the samples. The alternate hypothesis is the inverse of null hypothesis. An important reason to reject the null hypothesis and consider the alternate hypothesis is due to experimental or sampling errors.

Tests For Null Hypothesis

The two important approaches of statistical interference of null hypothesis are significance testing and hypothesis testing. The null hypothesis is a theoretical hypothesis and is based on insufficient evidence, which requires further testing to prove if it is true or false.

Significance Testing

The aim of significance testing is to provide evidence to reject the null hypothesis. If the difference is strong enough then reject the null hypothesis and accept the alternate hypothesis. The testing is designed to test the strength of the evidence against the hypothesis. The four important steps of significance testing are as follows.

• First state the null and alternate hypotheses.
• Calculate the test statistics.
• Find the p-value.
• Test the p-value with the α and decide if the null hypothesis should be rejected or accepted.

If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.

• Hypothesis Testing

Hypothesis testing takes the parameters from the sample and makes a derivation about the population. A hypothesis is an educated guess about a sample, which can be tested either through an experiment or an observation. Initially, a tentative assumption is made about the sample in the form of a null hypothesis.

There are four steps to perform hypothesis testing. They are:

• Identify the null hypothesis.
• Define the null hypothesis statement.
• Choose the test to be performed.
• Accept the null hypothesis or the alternate hypothesis.

There are often errors in the process of testing the hypothesis. The two important errors observed in hypothesis testing is as follows.

• Type - I error is rejecting the null hypothesis when the null hypothesis is actually true.
• Type - II error is accepting the null hypothesis when the null hypothesis is actually false.

Difference Between Null Hypothesis And Alternate Hypothesis

The difference between null hypothesis and alternate hypothesis can be understood through the following points.

• The opposite of the null hypothesis is the alternate hypothesis and it is the claim which is being proved by research to be true.
• The null hypothesis states that the two samples of the population are the same, and the alternate hypothesis states that there is a significant difference between the two samples of the population.
• The null hypothesis is designated as H o and the alternate hypothesis is designated as H a .
• For the null hypothesis, the same means are assumed to be equal, and we have H 0 : µ 1 = µ 2. And for the alternate hypothesis, the sample means are unequal, and we have H a : µ 1 ≠ µ 2.
• The observed population parameters and variables are the same across the samples, for a null hypothesis, but in an alternate hypothesis, there is a significant difference between the observed parameters and variables across the samples.

☛ Related Topics

The following topics help in a better understanding of the null hypothesis.

• Probability and Statistics
• Basic Statistics Formula
• Sample Space

Examples on Null Hypothesis

Example 1: A medical experiment and trial is conducted to check if a particular drug can serve as the vaccine for Covid-19, and can prevent from occurrence of Corona. Write the null hypothesis and the alternate hypothesis for this situation.

The given situation refers to a possible new drug and its effectiveness of being a vaccine for Covid-19 or not. The null hypothesis (H o ) and alternate hypothesis (H a ) for this medical experiment is as follows.

• H 0 : The use of the new drug is not helpful for the prevention of Covid-19.
• H a : The use of the new drug serves as a vaccine and helps for the prevention of Covid-19.

Example 2: The teacher has prepared a set of important questions and informs the student that preparing these questions helps in scoring more than 60% marks in the board exams. Write the null hypothesis and the alternate hypothesis for this situation.

The given situation refers to the teacher who has claimed that her important questions helps to score more than 60% marks in the board exams. The null hypothesis(H o ) and alternate hypothesis(H a ) for this situation is as follows.

• H o : The important questions given by the teacher does not really help the students to get a score of more than 60% in the board exams.
• H a : The important questions given by the teacher is helpful for the students to score more than 60% marks in the board exams.

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Practice Questions on Null Hypothesis

Faqs on null hypothesis, what is null hypothesis in maths.

Null hypothesis is used in statistics and it states if there is any significant difference between the two samples. The acceptance of null hypothesis mean that there is no significant difference between the two samples. And the rejection of null hypothesis means that the two samples are different, and we need to accept the alternate hypothesis. The null hypothesis statement is represented as H 0 and the alternate hypothesis is represented as H a .

How Do You Test Null Hypothesis?

The null hypothesis is broadly tested using two methods. The null hypothesis can be tested using significance testing and hypothesis testing.Broadly the test for null hypothesis is performed across four stages. First the null hypothesis is identified, secondly the null hypothesis is defined. Next a suitable test is used to test the hypothesis, and finally either the null hypothesis or the alternate hypothesis is accepted.

How To Accept or Reject Null Hypothesis?

The null hypothesis is accepted or rejected based on the result of the hypothesis testing. The p value is found and the significance level is defined. If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.

What Is the Difference Between Null Hypothesis And Alternate Hypothesis?

The null hypothesis states that there is no significant difference between the two samples, and the alternate hypothesis states that there is a significant difference between the two samples. The null hypothesis is referred using H o and the alternate hypothesis is referred using H a . As per null hypothesis the observed variables and parameters are the same across the samples, but as per alternate hypothesis there is a significant difference between the observed variables and parameters across the samples.

What Is Null Hypothesis Example?

A few quick examples of null hypothesis are as follows.

• The salary of a person is independent of his profession, is an example of null hypothesis. And the salary is dependent on the profession of a person, is an alternate hypothesis.
• The performance of the students in Maths from two different classes is a null hypothesis. And the performance of the students from each of the classes is different, is an example of alternate hypothesis.
• The nutrient content of mango and a mango milk shake is equal and it can be taken as a null hypothesis. The test to prove the different nutrient content of the two is referred to as alternate hypothesis.

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Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.

• Null Hypothesis (H 0 ) – This can be thought of as the implied hypothesis. “Null” meaning “nothing.”  This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.
• Alternative Hypothesis (H a ) – This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.

Null Hypothesis:   H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis :  H a : Male factory workers have a higher salary than female factory workers.

Null Hypothesis :  H 0 : There is no relationship between height and shoe size. Alternative Hypothesis :  H a : There is a positive relationship between height and shoe size.

Null Hypothesis :  H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis :  H a : The quality of a brick mason’s work is influenced by on-the-job experience.

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• Math Article

Null Hypothesis

In mathematics, Statistics deals with the study of research and surveys on the numerical data. For taking surveys, we have to define the hypothesis. Generally, there are two types of hypothesis. One is a null hypothesis, and another is an alternative hypothesis .

In probability and statistics, the null hypothesis is a comprehensive statement or default status that there is zero happening or nothing happening. For example, there is no connection among groups or no association between two measured events. It is generally assumed here that the hypothesis is true until any other proof has been brought into the light to deny the hypothesis. Let us learn more here with definition, symbol, principle, types and example, in this article.

Table of contents:

• Comparison with Alternative Hypothesis

Null Hypothesis Definition

The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data. This hypothesis is either rejected or not rejected based on the viability of the given population or sample . In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0 .

Null Hypothesis Symbol

In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero), such that H 0 . It is pronounced as H-null or H-zero or H-nought. At the same time, the alternative hypothesis expresses the observations determined by the non-random cause. It is represented by H 1 or H a .

Null Hypothesis Principle

The principle followed for null hypothesis testing is, collecting the data and determining the chances of a given set of data during the study on some random sample, assuming that the null hypothesis is true. In case if the given data does not face the expected null hypothesis, then the outcome will be quite weaker, and they conclude by saying that the given set of data does not provide strong evidence against the null hypothesis because of insufficient evidence. Finally, the researchers tend to reject that.

Null Hypothesis Formula

Here, the hypothesis test formulas are given below for reference.

The formula for the null hypothesis is:

H 0 :  p = p 0

The formula for the alternative hypothesis is:

H a = p >p 0 , < p 0 ≠ p 0

The formula for the test static is:

Remember that,  p 0  is the null hypothesis and p – hat is the sample proportion.

Also, read:

Types of Null Hypothesis

There are different types of hypothesis. They are:

Simple Hypothesis

It completely specifies the population distribution. In this method, the sampling distribution is the function of the sample size.

Composite Hypothesis

The composite hypothesis is one that does not completely specify the population distribution.

Exact Hypothesis

Exact hypothesis defines the exact value of the parameter. For example μ= 50

Inexact Hypothesis

This type of hypothesis does not define the exact value of the parameter. But it denotes a specific range or interval. For example 45< μ <60

Null Hypothesis Rejection

Sometimes the null hypothesis is rejected too. If this hypothesis is rejected means, that research could be invalid. Many researchers will neglect this hypothesis as it is merely opposite to the alternate hypothesis. It is a better practice to create a hypothesis and test it. The goal of researchers is not to reject the hypothesis. But it is evident that a perfect statistical model is always associated with the failure to reject the null hypothesis.

How do you Find the Null Hypothesis?

The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don’t have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent).

When is Null Hypothesis Rejected?

The null hypothesis is rejected using the P-value approach. If the P-value is less than or equal to the α, there should be a rejection of the null hypothesis in favour of the alternate hypothesis. In case, if P-value is greater than α, the null hypothesis is not rejected.

Null Hypothesis and Alternative Hypothesis

Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.

Null Hypothesis Examples

Here, some of the examples of the null hypothesis are given below. Go through the below ones to understand the concept of the null hypothesis in a better way.

If a medicine reduces the risk of cardiac stroke, then the null hypothesis should be “the medicine does not reduce the chance of cardiac stroke”. This testing can be performed by the administration of a drug to a certain group of people in a controlled way. If the survey shows that there is a significant change in the people, then the hypothesis is rejected.

Few more examples are:

1). Are there is 100% chance of getting affected by dengue?

Ans: There could be chances of getting affected by dengue but not 100%.

2). Do teenagers are using mobile phones more than grown-ups to access the internet?

Ans: Age has no limit on using mobile phones to access the internet.

3). Does having apple daily will not cause fever?

Ans: Having apple daily does not assure of not having fever, but increases the immunity to fight against such diseases.

4). Do the children more good in doing mathematical calculations than grown-ups?

Ans: Age has no effect on Mathematical skills.

In many common applications, the choice of the null hypothesis is not automated, but the testing and calculations may be automated. Also, the choice of the null hypothesis is completely based on previous experiences and inconsistent advice. The choice can be more complicated and based on the variety of applications and the diversity of the objectives. 

The main limitation for the choice of the null hypothesis is that the hypothesis suggested by the data is based on the reasoning which proves nothing. It means that if some hypothesis provides a summary of the data set, then there would be no value in the testing of the hypothesis on the particular set of data. 

Frequently Asked Questions on Null Hypothesis

What is meant by the null hypothesis.

In Statistics, a null hypothesis is a type of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.

What are the benefits of hypothesis testing?

Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.

When a null hypothesis is accepted and rejected?

The null hypothesis is either accepted or rejected in terms of the given data. If P-value is less than α, then the null hypothesis is rejected in favor of the alternative hypothesis, and if the P-value is greater than α, then the null hypothesis is accepted in favor of the alternative hypothesis.

Why is the null hypothesis important?

The importance of the null hypothesis is that it provides an approximate description of the phenomena of the given data. It allows the investigators to directly test the relational statement in a research study.

How to accept or reject the null hypothesis in the chi-square test?

If the result of the chi-square test is bigger than the critical value in the table, then the data does not fit the model, which represents the rejection of the null hypothesis.

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing
• Examples of null and alternative hypotheses

Writing null and alternative hypotheses

• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations
• Using P-values to make conclusions

• (Choice A)   H 0 : p = 0.1 H a : p ≠ 0.1 ‍   A H 0 : p = 0.1 H a : p ≠ 0.1 ‍  
• (Choice B)   H 0 : p ≠ 0.1 H a : p = 0.1 ‍   B H 0 : p ≠ 0.1 H a : p = 0.1 ‍  
• (Choice C)   H 0 : p = 0.1 H a : p > 0.1 ‍   C H 0 : p = 0.1 H a : p > 0.1 ‍  
• (Choice D)   H 0 : p = 0.1 H a : p < 0.1 ‍   D H 0 : p = 0.1 H a : p < 0.1 ‍  

Create a null and alternative hypothesis with a rationale of what you’ll be testing. Describe how you collected your data, what calculations or statistics you ran, and what the dependent and independent variables are. Identifying Types of Tissues in Slides Chicken Nugget Necropsy Step 1 Title: Write a descriptive title that tells the reader what the research objective is and what the results are in a succinct manner. Step 2 Introduction: Write the Introduction paragraph(s). This should include some background research on the topic which will have in-text citations in APA format. State the research question and objective in your own words (use the objective and questions below, just reword them in your own words). Then create a null and alternative hypotheses with a rationale of what you’ll be testing. Objective The purpose of this lab is to use your knowledge of tissues to determine the composition of three processed meat products. Research Question Which of these processed meat products has the most meat (skeletal muscle) and least fat (adipose tissue)? Burger King McDonalds Health Food Store Brand Chicken nuggets are a popular food among children therefore choosing the healthier option will provide for nutrients. Chicken nuggets are a great source of protein and low in calories compared to meatless options for the same amount of protein. Is store bought chicken nuggets healthier than fast food chicken nuggets? Samples from three different chicken nuggets (Burger King, Mcdonalds and Health Food Store). By taking three samples from the three different chicken nuggets and examining them under a microscope we can find the percentage of skeletal tissue, adipose tissue and other tissues per sample In order to determine which chicken nugget is healthier we need to measure the amount of meat (skeletal muscle) to the amount of fat (adipose tissue). Ideal the chicken nugget with the most skeletal muscle and least amount of adipose tissue would be the healthiest chicken nugget. Step 3 Methodology: Look at the images here. Classify the tissues under each intersection of lines as Skeletal muscle (SM), Adipose tissue (AP), or "Other" (other includes fibrous connective tissue, nervous tissue, epithelium, etc.). If a point falls on open space (i.e., not on the sample), do not count that point. Determine the relative abundance of each category by dividing the total number of points which contained the tissue divided by the total number of points which fell over the sample (see below). Do this for EACH of the nine samples (three for each meat). Then calculate the AVERAGE percentage of each tissue in each of the three lunch meats. You will then summarize how you collected your data, what calculations or statistics you ran, and what the dependent and independent variables are.

Null hypothesis (H0): There is no significant difference in the composition of skeletal muscle and adipose tissue among the three processed meat products (Burger King, McDonald's, and Health Food Store Brand).

Alternative hypothesis (Ha): There is a significant difference in the composition of skeletal muscle and adipose tissue among the three processed meat products, indicating that one product has the highest percentage of skeletal muscle and the lowest percentage of adipose tissue.

Rationale: The null hypothesis assumes that there is no difference in the composition of skeletal muscle and adipose tissue among the meat products. The alternative hypothesis suggests that there is a difference, which aligns with the objective of determining the meat content and fat content in the processed meat products.

To collect the data, nine samples will be taken, with three samples from each of the three meat products (Burger King, McDonald's, and Health Food Store Brand). Each sample will be examined under a microscope, and the tissues will be classified as skeletal muscle, adipose tissue, or "other" categories (including fibrous connective tissue, nervous tissue, epithelium, etc.).

The calculations involved will include determining the relative abundance of each tissue category by dividing the total number of points containing the tissue by the total number of points falling over the sample. This will be done for each of the nine samples. The average percentage of each tissue category (skeletal muscle, adipose tissue, and other) will then be calculated for each of the three lunch meats.

In this study, the dependent variable is the composition of tissues (percentage of skeletal muscle, adipose tissue, and other), while the independent variable is the type of processed meat product (Burger King, McDonald's, and Health Food Store Brand). The objective is to examine if the composition of tissues varies significantly among the different meat products and identify which product has the highest percentage of skeletal muscle and lowest percentage of adipose tissue, indicating a potentially healthier option.

Learn more about Rationale here:

brainly.com/question/17366042

Related Questions

​​​​​​​ 4) Prove that if \( A \subset \mathbb{R} \) bounded above, then \[ \sup A \in \bar{A}=A \cup A^{\prime} \text {. } \]

To prove that if (A) is a subset of (\mathbb{R}) bounded above, then (\sup A) belongs to the closure of (A), which is defined as (\bar{A} = A \cup A'), where (A') denotes the set of limit points of (A), we need to show two things:

(\sup A \in A) or (\sup A) is an element of (A).

(\sup A \in A') or (\sup A) is a limit point of (A).

Let's prove these two statements:

To show that (\sup A) is an element of (A), we consider two cases:

a) If (\sup A \in A), then it is trivially in (A).

b) If (\sup A \notin A), then there must exist some element (x) in (A) such that (x > \sup A). Since (A) is bounded above, (\sup A) serves as an upper bound for (A). However, (x) is greater than this upper bound, which contradicts the assumption . Hence, this case is not possible, and we conclude that (\sup A) must be in (A).

To demonstrate that (\sup A) is a limit point of (A), we need to show that for any neighborhood of (\sup A), there exists a point in (A) (distinct from (\sup A)) that lies within the neighborhood.

Let (U) be a neighborhood of (\sup A). We can consider two cases:

a) If (\sup A) is an isolated point of (A), meaning there exists some (\epsilon > 0) such that (N(\sup A, \epsilon) \cap A = {\sup A}), where (N(\sup A, \epsilon)) is the (\epsilon)-neighborhood of (\sup A), then there are no points in (A) other than (\sup A) within the neighborhood. In this case, (\sup A) is not a limit point.

b) If (\sup A) is not an isolated point of (A), it is a limit point. For any (\epsilon > 0), the (\epsilon)-neighborhood (N(\sup A, \epsilon)) contains infinitely many elements of (A). This is because any interval around (\sup A) will contain points from (A) since (\sup A) is the least upper bound of (A). Hence, we can always find a point distinct from (\sup A) within the neighborhood, satisfying the definition of a limit point.

Since we have shown that (\sup A) belongs to both (A) and (A'), we can conclude that (\sup A) is an element of the closure of (A) ((\sup A \in \bar{A} = A \cup A')).

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Assume a continuous random variable X follows a uniform distribution on [1, 4]. So the probability density function (pdf) of X can be written as, f(x) = 1/k, 1 ≤ x ≤ 4. (Give answers with 3 digits after decimal) a) [1pt] Find the value of k. Submit Answer Tries 0/99 b) [1pt] What is the probability P(X ≥ 3.5)? Submit Answer Tries 0/99 c) [2pts] What is the expected value of X? Submit Answer Tries 0/99 d) [2pts] What is the variance of X? Submit Answer Tries 0/99

the answers are:a) k = 3b) P(X ≥ 3.5) ≈ 0.167c) E(X) = 2.5d) Var(X) ≈ 0.778

a) Calculation of k:For the uniform distribution on [a, b], the probability density function is given as:f(x) = 1/(b − a) for a ≤ x ≤ bHere, a = 1, b = 4Thus, f(x) = 1/(4 − 1) = 1/3Therefore, the value of k = 3.

b) Calculation of P(X ≥ 3.5):P(X ≥ 3.5) = ∫[3.5,4] f(x) dx∫[3.5,4] 1/3 dx = [x/3]3.5 to 4 = (4/3 − 7/6) = 1/6 ≈ 0.167

Calculation of the expected value of X:

We know that the expected value of X is given as:E(X) = ∫[1,4] xf(x) dx∫[1,4] x(1/3) dx = [x^2/6]1 to 4 = (16/6 − 1/6) = 5/2 = 2.5d)

Calculation of the variance of X:We know that the variance of X is given as:

Var(X) = ∫[1,4] (x − E(X))^2f(x) dx= ∫[1,4] (x − 2.5)^2(1/3) dx= [x^3/9 − 5x^2/6 + 25x/18]1 to 4= (64/9 − 40/3 + 100/18 − 1/9)= 7/9 ≈ 0.778Thus, the answers are:a) k = 3b) P(X ≥ 3.5) ≈ 0.167c) E(X) = 2.5d) Var(X) ≈ 0.778

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If n(A∪B)=99 and n(A)=n(B)=67, find n(A∩B)

(A∩B) is 35.

n (A∪B) =99,

n(A)=n(B)=67.

We have to find the value of n(A∩B). To find the value of n(A∩B), we will use the below formula,

n(A∪B) = n(A) + n(B) - n(A∩B).

We know that n(A∪B) = 99n(A) = 67n(B) = 67. Putting these values in the above formula,

n(A∪B) = n(A) + n(B) - n(A∩B)99 = 67 + 67 - n(A∩B)99 = 134 - n(A∩B)n(A∩B) = 134 - 99n(A∩B) = 35.

Hence, the value of n (A∩B) is 35.

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Consider a Poisson distribution with = 9. (Round your answers to four decimal places.) (a)Write the appropriate Poisson probability function. f(x) = (b) Compute f(2). f(2) = (c) Compute f(1). f(1) = (d) Compute P(x ≥ 2). P(x ≥ 2) =

In a Poisson distribution with a mean of 9, the appropriate Poisson probability function is used to calculate the probabilities of different outcomes. The function is denoted as f(x), where x represents the number of events.

(a) The appropriate Poisson probability function is given by:

f(x) = (e^(-λ) * λ^x) / x!

Here, λ represents the mean of the Poisson distribution, which is 9.

(b) To compute f(2), we substitute x = 2 into the probability function:

f(2) = (e^(-9) * 9^2) / 2!

(c) Similarly, to compute f(1), we substitute x = 1 into the probability function:

f(1) = (e^(-9) * 9^1) / 1!

(d) To compute P(x ≥ 2), we need to calculate the sum of probabilities for x = 2, 3, 4, and so on, up to infinity. Since summing infinite terms is not feasible, we often approximate it by calculating 1 minus the cumulative probability for x less than 2:

P(x ≥ 2) = 1 - P(x < 2)

The calculation of P(x < 2) involves summing the probabilities for x = 0 and x = 1.

In summary, the appropriate Poisson probability function is used to calculate probabilities for different values of x in a Poisson distribution with a mean of 9. These probabilities can be computed by substituting the values of x into the probability function.

Additionally, the probability of x being greater than or equal to a specific value can be calculated by subtracting the cumulative probability for x less than that value from 1.

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Bob walks 200 m south, then jogs 400 m northwest, then walks 200 m in a 30º direction. south east. a. Draw a graph of Bob's movements. Use a ruler and protractor. (14 points) b. Use graphical and analytical methods to find the total displacement that Bob traveled. (Magnitude and direction) (20 pts) c. Compare the results obtained by the graphical and analytical method. (Percent of difference). (6 points) 2. Knowing that α = 35, determine (graph and analytically) the resultant of the forces that are show in the figure. Compare your results calculating the percent difference. (Analytically it must be by components rectangular

The total displacement that Bob traveled is approximately 4.11 units at an angle of -30.96º. The percent difference between the graphical and analytical methods is 0%.

To draw a graph of Bob's movements, we can use a ruler and protractor to accurately represent the distances and directions. Let's assume that each unit on the graph represents 100 meters.

1. Bob walks 200 m south:

Starting from the origin (0, 0), we move down 2 units to represent 200 m south.

2. Bob jogs 400 m northwest :

From the endpoint of the previous step, we move 4 units to the left and 4 units up to represent 400 m northwest.

3. Bob walks 200 m in a 30º southeast direction:

From the endpoint of the previous step, we move 2 units down and 3.46 units to the right (since cos(30º) ≈ 0.866 and sin(30º) ≈ 0.5) to represent 200 m in a 30º southeast direction.

  |          ○    (3.46, -2)

  |  ○    (0, -2)

  |______________________ x

  0    1    2    3    4

To find the total displacement, we need to calculate the magnitude and direction of the displacement.

Analytical Method

To find the total displacement analytically, we can add up the displacements in the x and y directions separately.

Displacement in the x-direction :

The graph shows that Bob's displacement in the x-direction is approximately 3.46 units to the right.

Displacement in the y-direction:

The graph shows that Bob's displacement in the y-direction is approximately 2 units down.

The magnitude of the Total Displacement:

Using the Pythagorean theorem, we can find the magnitude of the total displacement:

magnitude = √((displacement in x)^2 + (displacement in y)^2)

          = √((3.46)^2 + (-2)^2)

          ≈ 4.11 units

The direction of the Total Displacement:

To find the direction of the total displacement, we can use trigonometry:

tan(θ) = (displacement in y) / (displacement in x)

θ = atan((displacement in y) / (displacement in x))

θ = atan((-2) / 3.46)

θ ≈ -30.96º (measured counterclockwise from the positive x-axis)

Therefore, the total displacement that Bob traveled is approximately 4.11 units at an angle of -30.96º.

We can calculate the percent difference between the magnitudes obtained to compare the results obtained by the graphical and analytical methods.

Percent Difference = |(graphical result - analytical result) / analytical result| * 100%

Percent Difference = |(4.11 - 4.11) / 4.11| * 100%

                 = 0%

The percent difference between the graphical and analytical methods is 0%.

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Sampling bias. One way of checking for the effects of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known demographic facts about the population. The 2010 census found that 13.9%, or 32,576,000, of the 235,016,000 adults (aged 18 and older) in the United States identified themselves as being of Hispanic origin. Is the value 13.9% a parameter or a statistic? Explain your answer.

In summary, the value 13.9% is a parameter, not a statistic .

A parameter is a characteristic or measure that describes a population, while a statistic is a characteristic or measure that describes a sample.

In this case, the value of 13.9% represents the proportion of adults in the entire United States population who identified themselves as being of Hispanic origin , as determined by the 2010 census. It is a fixed value that describes the population as a whole and is based on complete information from the census.

On the other hand, a statistic would be obtained from a sample , which is a subset of the population. It is an estimate or measurement calculated from the data collected in the sample and is used to make inferences about the population parameter.

In this context, a statistic could be the proportion of adults of Hispanic origin based on a sample survey.

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Let X and Y have joint density f(x,y)={ cx, 0, ​ when 0 2 x ​ <1 otherwise. ​ Determine the distribution of XY. 26. Suppose that X and Y are random variables with a joint density f(x,y)={ y 1 ​ e −x/y e −y , 0, ​ when 0 otherwise. ​ Show that X/Y and Y are independent standard exponential random variables and exploit this fact in order to compute EX and VarX. 27. Let X and Y have joint density f(x,y)={ cx, 0, ​ when 0 3 x ​ <1 otherwise. ​ Determine the distribution of XY.

In the first part, we determine the distribution of XY by integrating the joint density function. In the second part, we verify the independence and exponential distribution to compute EX and VarX.

The joint density function \(f(x, y)\) describes the distribution of two random variables, X and Y. To determine the distribution of XY, we need to find the cumulative distribution function (CDF) of XY.

To do this, we integrate the joint density function over the appropriate region. In this case, we integrate \(f(x, y)\) over the region where \(XY\) takes on a specific value.

Once we have the CDF of XY, we can differentiate it to obtain the probability density function ( PDF ) of XY. This will give us the distribution of XY.

Regarding the second part of the question, we are given the joint density function of X and Y. To show that X/Y and Y are independent standard exponential random variables , we need to verify two conditions:

1. Independence: We need to show that the joint density function of X/Y and Y can be expressed as the product of their individual density functions.

2. Exponential distribution: We need to show that the individual density functions of X/Y and Y follow the standard exponential distribution.

Once we establish the independence and exponential distribution, we can use these properties to compute the expected value (EX) and variance (VarX) of X.

In summary, the first part involves finding the distribution of XY by integrating the joint density function, while the second part involves verifying the independence and exponential distribution to compute EX and VarX.

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(6) Solving triangle A B C with c=25, a=15 , and B=60^{\circ} . Round each answer to the nearest tenth.

The sides of the triangle are a = 15, b = 12.9, c = 25. The angles of the triangle are A = 60°, B = 60°, C = 60°.

We are given the following information: c=25, a=15 and B=60°.

Using this information, we can solve for the remaining sides and angles of the triangle using the Law of Sines and the fact that the sum of angles in a triangle is 180°.

Let's begin by finding angle `C`. We know that the sum of angles in a triangle is 180°, so we can use this fact to find angle C.  

A + B + C = 180  

C = 180 - A - B

C = 180 - 60 - A  

C = 120 - A

Now, we can use the Law of Sines to find `B` and `c`.

The Law of Sines states that:

(sin A)/a = (sin B)/b = (sin C)/c

We know a, b, and A. Let's find b.

(sin A)/a = (sin B)/b

(sin 60)/15 = (sin B)/b

sqrt(3)/15 = (sin B)/b

b = (sin B)(15/sqrt(3))

b = (sin B)(5sqrt(3))

Now, we can find `c` using the Law of Sines.

(sin C)/c = (sin B)/b

(sin C)/25 = (sin 60)/(5sqrt(3))

sin C = (25 sin 60)/(5sqrt(3))

sin C = (5sqrt(3))/2

C = sin^-1((5sqrt(3))/2)

Now we can find angle `A`.

A = 180 - B - C

A = 180 - 60 - 60

Finally, we can use the Law of Sines to find `c` using `A` and `a`.

(sin A)/a = (sin C)/c

(sin 60)/15 = (sin 60)/c

So the sides of the triangle are a = 15, b = 12.9, c = 25. The angles of the triangle are A = 60°, B = 60°, C = 60°.

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figure A is a scale copy of figure B

The value of x is 42.

To determine the value of x, we need to analyze the given information regarding the scale factor between Figure A and Figure B.

The scale factor is expressed as the ratio of the corresponding side lengths or dimensions of the two figures.

Let's assume that the length of a side in Figure B is represented by 'x'. According to the given information, Figure A is a scale copy of Figure B with a scale factor of 2/7. This means that the corresponding side length in Figure A is 2/7 times the length of the corresponding side in Figure B.

Applying this scale factor to the length of side x in Figure B, we can express the length of the corresponding side in Figure A as (2/7)x.

Given that the length of side x in Figure B is 12, we can substitute it into the equation:

(2/7)x = 12

To solve for x, we can multiply both sides of the equation by 7/2:

x = (12 * 7) / 2

Simplifying the expression:

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Consider the equation sin(z)=cos(z) You know, visually from right triangles, that z=π/4 and z=3π/4 are solutions (up to multiples of 2π ). Are there any other (complex) solutions? Solve the equation to address this question. ( 20 points)

The solutions include

z = π/4 + π = 5π/4

z = π/4 + 2π = 9π/4

z = π/4 + 3π = 13π/4

These solutions cover all the possible complex solutions for the equation sin(z) = cos(z).

To solve the equation sin(z) = cos(z), we can use the trigonometric identity sin(z) = cos(z) when z = π/4 and z = 3π/4. However, we need to check if there are any other complex solutions as well.

Let's solve the equation algebraically to find all possible solutions:

sin(z) = cos(z)

Divide both sides by cos(z):

sin(z) / cos(z) = 1

Using the identity tan(z) = sin(z) / cos(z), we have:

To find the solutions, we can take the inverse tangent (arctan) of both sides:

z = arctan(1)

The principal value of arctan (1) is π/4, which corresponds to one of the known solutions.

Now, let's consider the periodicity of the tangent function. The tangent function has a period of π, so we can add or subtract any multiple of π to the solution.

Therefore, the general solution is:

z = π/4 + nπ

where n is an integer representing any multiple of π.

So, in addition to z = π/4, the solutions include:

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Find the sum: \( -5+2+9+\ldots+44 \) Answer:

The sum of given series is 198.

The given series is -5+2+9+.....+44.In this series, the first term (a) is -5, the common difference (d) is 7 and the last term (l) is 44.

We can find the last term using the formula:[tex]\[l = a + (n-1)d \]where n is the number of terms. Therefore, \[44 = -5 + (n-1)7 \] .[/tex]

Simplifying the equation, we get \[n = 9\]Therefore, there are 9 terms in this series.

Now, we can find the sum of this series using the formula:[tex]\[S_n = \frac{n}{2} (a + l) \].[/tex]

Substituting the values, we get: [tex]\[S_9 = \frac{9}{2} (-5 + 44) = 198\].[/tex]

Hence, the main answer is 198. We can write the conclusion as:Therefore, the sum of the given series -5+2+9+.....+44 is 198.

The series has a total of 9 terms. We used the formula for the sum of an arithmetic series to find the main answer.

The formula is[tex]\[S_n = \frac{n}{2} (a + l) \][/tex]where n is the number of terms, a is the first term, l is the last term and d is the common difference.

We substituted the values of a, l and n to find the sum. We also found that there are 9 terms in the series. Therefore, the  answer is 198.

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in the Polya system if the number is 64 with lower number being 1/2 of second

The Polya system , if the number is 64 and the lower number is half of the second number,

it implies that the lower number can be any value , and the second number is twice that value.

In the Polya system, numbers are represented using a notation where the number 64 is written as [tex]2^6[/tex],

Indicating that it is 2 raised to the power of 6.

According to the statement, the lower number is half of the second number.

Let's represent the lower number as "x" and the second number as "2x" (since it is twice the value of the lower number).

Given that x is half of 2x, we have the equation:

x = (1/2) × 2x

Simplifying this equation, we get:

This equation indicates that x can take any value since both sides are equal.

Question: Simplify [tex]64^{1/2}[/tex] using Polya system and state the system.

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Find a so that f(x) = ax^2 + 8x + 2 has two real roots. (Enter your answer using interval notation.)

Given the function [tex]f(x) = ax² + 8x + 2[/tex] to have two real roots . Then, the discriminant, [tex]b² - 4ac > 0[/tex] We know that the quadratic formula is used to solve quadratic equations. Therefore, the value of a < 8 so that [tex]f(x) = ax² + 8x + 2[/tex] has two real roots in interval notation is[tex]:(-∞, 8).[/tex]

The quadratic formula is

[tex]x = (-b ± √(b² - 4ac))/2a[/tex]

The discriminant, [tex]b² - 4ac[/tex], determines the number of real roots. If the discriminant is greater than 0, the quadratic function has two real roots.

[tex]b² - 4ac > 0[/tex]

We are given

[tex]f(x) = ax² + 8x + 2[/tex]

Substituting the values into the above inequality , we get:

[tex]$$64 - 8a > 0$$[/tex]

Solving the above inequality, we get:

[tex]$$\begin{aligned} 64 - 8a &> 0 \\ 64 &> 8a \\ a &< 8 \end{aligned}$$[/tex]

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Given the following functions, use function composition to determine if f(x) and g(x) are inverse fucntions. f(x)=x+7 and g(x)=x−7 (a) (f∘g)(x)= (b) (g∘f)(x)= (c) Thus g(x) an inverse function of f(x)

a) (f∘g)(x)=x

b) (g∘f)(x)=x

c)g(x) is an inverse function of f(x).

Given the following functions, use function composition to determine if f(x) and g(x) are inverse functions, f(x)=x+7 and g(x)=x−7.

(a) (f∘g)(x)=f(g(x))=f(x−7)=(x−7)+7=x, therefore (f∘g)(x)=x

(b) (g∘f)(x)=g(f(x))=g(x+7)=(x+7)−7=x, therefore (g∘f)(x)=x

(c) Thus, g(x) is an inverse function of f(x).

In function composition, one function is substituted into another function.

The notation (f∘g)(x) represents f(g(x)) or the function f with the output of the function g replaced with the variable x.

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Given μ=50 and σ=6.25 : (a) Find the bounds which represent a lower bound of 93.75% of information. (b) Find the bounds which represent a lower bound of 89% of information.

(a) The lower bound that represents 93.75% of the information is

    approximately 42.8125.

(b) The lower bound that represents 89% of the information is     approximately 42.3125.

To find the bounds that represent a lower percentage of information, we need to calculate the corresponding z-scores and then use them to find the values that fall within those bounds.

(a) Finding the bounds for 93.75% of information:

Step 1: Find the z-score corresponding to the desired percentage. Since we want to find the lower bound, we need to find the z-score that leaves 6.25% of the data in the tail.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to the lower tail of 6.25% is approximately -1.15.

Step 2: Calculate the lower bound using the z-score formula:

Lower Bound = μ + (z-score * σ)

Lower Bound = 50 + (-1.15 * 6.25)

Lower Bound ≈ 50 - 7.1875

Lower Bound ≈ 42.8125

So, the lower bound that represents 93.75% of the information is approximately 42.8125.

(b) Finding the bounds for 89% of information:

Step 1: Find the z-score corresponding to the desired percentage. Since we want to find the lower bound, we need to find the z-score that leaves 11% of the data in the tail (100% - 89%).

Using a standard normal distribution table or a calculator , we find that the z-score corresponding to the lower tail of 11% is approximately -1.23.

Lower Bound = 50 + (-1.23 * 6.25)

Lower Bound ≈ 50 - 7.6875

Lower Bound ≈ 42.3125

So, the lower bound that represents 89% of the information is approximately 42.3125.

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26 packets are sampled. If you have a sample mean of $24.70 and a standard deviation of $5.47. Estimate the average cost of all packages at the 99 level of confidence.

The estimated average cost of all packages at the 99% confidence level is $24.70.

To estimate the average cost of all packages at the 99% confidence level, we can use the formula for the confidence interval of the mean:

Confidence interval = sample mean ± (critical value * standard deviation / √sample size)

First, we need to find the critical value corresponding to a 99% confidence level. Since the sample size is relatively small (26 packets), we'll use the t-distribution instead of the normal distribution.

The degrees of freedom for the t-distribution is equal to the sample size minus 1 (df = 26 - 1 = 25). Looking up the critical value for a 99% confidence level and 25 degrees of freedom in a t-table, we find that the critical value is approximately 2.796.

Now, we can calculate the confidence interval:

Confidence interval = $24.70 ± (2.796 * $5.47 / √26)

Confidence interval = $24.70 ± (2.796 * $5.47 / 5.099)

Confidence interval = $24.70 ± (2.796 * $1.072)

Confidence interval = $24.70 ± $2.994

This means that we can be 99% confident that the true average cost of all packages lies within the range of $21.706 to $27.694.

Therefore, the estimated average cost of all packages at the 99% confidence level is $24.70.

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1. Determine the volume of one balloon (sphere with radius r ) using the following equation (volume of a sphere) to find V: V= 3 4 ​ πr 3 2. One cubic centimeter of helium will lift about 0.0009958736 grams, so 9.958736 ∗ 10 ∧ −7 ∗ V gives us how many kg one balloon can lift. 3. We want to find how many balloons are needed, so given an object's weight M, we get the number of balloons needed by the following equation: =N=1.0+M/(V∗9.958736 ∗ 10 ∧ −7) - *Note: You need to add 1.0 because we are calculating full balloons, not partial balloons. Make a program that calculates how many helium balloons are needed to lift an object (i.e. calculate N). The program asks users to enter two numbers: 1. The average radius the balloons r (in cm ) 2. The weight of the object being lifted M (in kg ). After calculating N, display the calculated volume V for one balloon ( cm/3 ), followed by the number of balloons needed to lift the object (N). Finally display the total volume of all the balloons (N ∗ V). You may need to use the cmath library to properly represent the equations. Note that there are many tools available through the cmath library including π,sin,cos, pow, and sqrt. Note: we will use test cases to assist in grading the homework. Please ensure that you follow the format below to make sure that the grading scripts pass.

The program calculates the number of helium balloons needed to lift an object based on its weight by formulas for the volume of a sphere, the lifting capacity of helium, and the number of balloons required.

To calculate the number of helium balloons needed to lift an object, we can use the given equations. First, we find the volume of one balloon using the formula for the volume of a sphere: V = (3/4)πr^3, where r is the average radius of the balloons. Next, we determine how many kilograms one balloon can lift by multiplying the volume (V) by the conversion factor 9.958736 * 10^-7.

To find the number of balloons needed (N) to lift an object with weight M, we use the equation N = 1.0 + M / (V * 9.958736 * 10^-7). It is important to add 1.0 to account for the calculation of full balloons, rather than partial balloons.

In summary, the program will prompt the user to enter the average radius of the balloons (in cm) and the weight of the object being lifted (in kg). It will then calculate the volume of one balloon (V) in cm^3 using the sphere volume formula. The program will display V and the number of balloons needed (N) to lift the object. Finally, it will show the total volume of all the balloons by multiplying N and V.

The program utilizes the given formulas to determine the number of helium balloons required to lift a given object based on its weight , providing the necessary output for each step of the calculation.

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The level of lead in the blood was determined for a sample of 152 male hazardous-waste workers age 20-30 and also for a sample of 86 female workers, resulting in a mean ± standard error of 5.8 ±0.3 for the men and 3.5 ± 0.2 for the women. Calculate an estimate of the difference between true average blood lead levels for male and female workers in a way that provides information about reliability and precision. (Use a 95% confidence interval. Round your answers to two decimal places.) Suppose 1 and 2 are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. The data follows: m = 6, x = 114.7, s1 = 5.01, n = 6,y = 129.8, and s2 = 5.33. Calculate a 95% CI for the difference between true average stopping distances for cars equipped with system 1 and cars equipped with system 2. (Round your answers to two decimal places.)

the estimate of the difference between true average stopping distances for cars equipped with system 1 and cars equipped with system 2 is 15.1 with a 95% confidence interval of (8.60, 21.60).

Confidence interval estimate :

We are to calculate an estimate of the difference between true average blood lead levels for male and female workers in a way that provides information about reliability and precision . We can use the following formula to calculate the confidence interval estimate:

Confidence interval = (X1 - X2) ± t(α/2) x SE(X1 - X2)

where, X1 - X2 = 5.8 - 3.5 = 2.3α = 0.05 for 95% confidence interval

df = (n1 + n2 - 2) = (152 + 86 - 2) = 236

t(α/2) = t(0.025) = 1.97 (from the t-distribution table)

SE(X1 - X2) = sqrt( [(s1^2 / n1) + (s2^2 / n2)] ) = sqrt( [(0.3^2 / 152) + (0.2^2 / 86)] )= 0.049

So, substituting the values, we get the 95% confidence interval estimate as follows:

Confidence interval = (2.3) ± (1.97 x 0.049)= (2.3) ± (0.09653)= 2.20 to 2.40

Hence, the estimate of the difference between true average blood lead levels for male and female workers is 2.3 with a 95% confidence interval of (2.20, 2.40).

Stopping distances:

We are to calculate a 95% CI for the difference between true average stopping distances for cars equipped with system 1 and cars equipped with system 2. We can use the following formula to calculate the confidence interval estimate:

where, X1 - X2 = 129.8 - 114.7 = 15.1α = 0.05 for 95% confidence interval

df = (n1 + n2 - 2) = (6 + 6 - 2) = 10

t(α/2) = t(0.025) = 2.228 (from the t-distribution table)

SE(X1 - X2) = sqrt[ (s1^2 / n1) + (s2^2 / n2) ] = sqrt[ (5.01^2 / 6) + (5.33^2 / 6) ]= 2.921

Confidence interval = (15.1) ± (2.228 x 2.921)= (15.1) ± (6.50)= 8.60 to 21.60

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. Let X1, X2,... be a sequence of independent uniform [0, 1] random variables. For a fixed constant CE [0, 1], define the random variable N by N = min{n: X,, > c}. (a) Explain in just a few words how N relates to the X's and c. (b) Is N independent of XN? Give an intuitive explanation as well as a rigorous one.

N relates to X's and c in a manner that N is the minimum number n for which the nth random variable Xn is greater than the constant c. This suggests that N is determined by the first n Xn values that are less than or equal to c, since we are taking the minimum of the sequence .

The random variable N and XN are not independent. Intuitively, if XN is less than c, then N cannot be equal to N. We have two cases: if XN < c, then N = N, while if XN > c, then N < N. This means that knowing XN gives information about N, which means they are not independent.

Furthermore, we can prove this rigorously by using conditional probability .

The random variable N is defined as N = min{n : Xn > c}, where X1, X2, X3, ... is a sequence of independent uniform [0, 1] random variables and C is a constant in the range [0, 1]. This implies that N is the minimum index n such that the nth random variable Xn is greater than c.

Since N is the minimum index such that Xn > c, we can say that N is determined by the first n Xn values that are less than or equal to c, as we are taking the minimum of the sequence.

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For the function: y(x) = e^-x + e^x/2 this function is given a special name: "cosh(x)", or "hyperbolic cosine". a) Determine any critical points of this function, and the inflection points (if any exist). b) Compute the second derivative of y(x) (i.e. y" (x)), and compare it to y(x). How are these two functions related? c) The first derivative of above function is given the name sinh(x), or "hyperbolic sine". Use your knowledge of the previous parts to compute, and give your answer in terms of cosh(x) or sinh(x): d^10/dx^10 sinh (x) d) Integrate y(x) over the interval [-1,1], by using the fundamental theorem of calculus. You do not need to use the sinh and cosh definitions for this.

a)  x = i(pi/2) is an inflection point of y(x).

b) ) y''(x) =[tex]e^-x + e^x/2[/tex]

c) sinh(x) = cosh(x)/2

d) ∫(-1 to 1)y(x)dx =[tex]e^-1/2 - e^-1 + e^1/2 - e^1.[/tex]

Given function is[tex]y(x) = e^-x + e^x/2[/tex], which is called hyperbolic cosine or cosh(x).

a) Critical points:

[tex]y(x) = e^-x + e^x/2[/tex]

Critical points can be calculated by finding the derivative of y(x) and then equating it to zero.

[tex]y'(x) = -e^-x + (1/2)e^x[/tex]

Solving the above equation for x, we get x = ln(2).

Therefore, x = ln(2) is a critical point of y(x).

Inflection points: To find the inflection points, we need to find the second derivative of y(x).

[tex]y'(x) = -e^-x + (1/2)e^x[/tex] . . . . . . (1)

[tex]y''(x) = e^-x + (1/2)e^x/2[/tex] . . . . . . (2)

Now equate the equation (2) to zero.

[tex]e^-x + (1/2)e^x/2 = 0[/tex]

On solving the above equation, we get

[tex]e^x/2 = -e^-x/2[/tex]

x = i(pi/2) is an inflection point of y(x).

b) [tex]y''(x) = e^-x + (1/2)e^x/2y(x) \\= e^-x + e^x/2[/tex]

Comparing equation (1) and equation (2), we can see that the second derivative of y(x) is the sum of y(x) and y(x) multiplied by a constant.

c) The first derivative of y(x) is given by sinh(x).

[tex]sinh(x) = (1/2)(e^x - e^-x)[/tex]

From equation (1), we can write the value of e^x as

[tex]e^x = 2e^-x[/tex]

Therefore, sinh(x) can be written as sinh(x) = cosh(x)/2

d) The 10th derivative of sinh(x) is given by the following equation:

[tex]d^10/dx^10 sinh(x) = sinh(x) = (1/2)(e^x - e^-x)[/tex]

Therefore,[tex]d^10/dx^10 sinh(x) = cosh(x)/2.[/tex]

Integration of y(x) over the interval [-1,1]:

Using the fundamental theorem of calculus, we have

∫(-1 to 1)y(x)dx =[tex](e^-1 + e^1/2) - (e^1 + e^-1/2)[/tex]

Therefore, ∫(-1 to 1)y(x)dx = [tex]e^-1/2 - e^-1 + e^1/2 - e^1.[/tex]

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calculator display shows results from a test of the claim that less than 8% of treated subjects experienced headaches. Use the normal distribution as an approximation to the 1-PxopzTest prop <0.08 z=1.949033055 p=0.9743542623 p^=0.1123595506 π=267 a. Is the test two-tailed, left-tailed, or right-tailed? Right tailed test Left-tailed test Two-tailed test b. What is the test statistic? z= c. What is the P-value? P-value =

a. The test is a right-tailed test .

b. The test statistic is z = 1.949033055.

c. The P-value is 0.0256457377 (or approximately 0.0256).

a. The test is a right-tailed test because the claim is that less than 8% of treated subjects experienced headaches, indicating a specific direction.

b. The test statistic is given as z = 1.949033055.

c. The P-value is 0.0256457377 (or approximately 0.0256). The P-value represents the probability of obtaining a test statistic as extreme as the observed value (or even more extreme) under the null hypothesis.

In this case, the null hypothesis states that the proportion of treated subjects experiencing headaches is equal to or greater than 8%. The alternative hypothesis, which is the claim being tested, is that the proportion is less than 8%.

To calculate the P-value , we compare the test statistic (z = 1.949033055) to the standard normal distribution. Since this is a right-tailed test, we calculate the area under the curve to the right of the test statistic.

The P-value of 0.0256457377 indicates that the probability of obtaining a test statistic as extreme as 1.949033055 (or even more extreme) under the null hypothesis is approximately 0.0256. This value is smaller than the significance level (usually denoted as α), which is commonly set at 0.05.

Therefore, if we use a significance level of 0.05, we would reject the null hypothesis and conclude that there is evidence to support the claim that less than 8% of treated subjects experienced headaches.

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21. In a between-subjects, two-way ANOVA, MSinteraction \( =842.33 \) and MSwithin \( =3,578.99 \). What is Finteraction? \( 3.25 \) \( 0.24 \) \( 1.24 \) \( 4.25 \)

The correct option is `0.24.`

In a between-subjects, two-way ANOVA , MSinteraction = 842.33 and MSwithin = 3,578.99. We need to determine Finteraction.

Formula for Finteraction is:  `Finteraction = MSinteraction/MSwithin`  ...[1]Putting values in Equation [1], we get:  `Finteraction = 842.33/3,578.99`Simplifying the above expression, we get:  `Finteraction = 0.23527`Approximating to two decimal places, we get:  `Finteraction = 0.24` Hence, the Finteraction is 0.24.

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The total accumulatd cost, C(t), and revenue, R(t), (in hundreds of dollars), respectively, for a Batman Pinball Machine is given by: C′(t)=2 AND R′(t)=4e^−.2t. where t is in time in years. Find the useful life of the maching to the nearest hundreth. Then find the total profit accumulated during the useful life of the machine. Please upload a picture of your work and answers.

To find the useful life of the machine, we need to determine the time at which the accumulated cost equals the accumulated revenue. In other words, we need to find the intersection point of the cost and revenue functions.

[tex]C'(t) = 2R'(t) = 4e^(-0.2t)[/tex]

Integrating both sides of the equations will give us the accumulated cost and revenue functions:

[tex]C(t) = ∫ 2 dt = 2t + C1R(t) = ∫ 4e^(-0.2t) dt = -20e^(-0.2t) + C2[/tex]

Since the cost and revenue are given in hundreds of dollars, we can divide both functions by 100:

[tex]C(t) = 0.02t + C1R(t) = -0.2e^(-0.2t) + C2[/tex]

To find the intersection point, we set C(t) equal to R(t) and solve for t:

[tex]0.02t + C1 = -0.2e^(-0.2t) + C2[/tex]

This equation can't be solved analytically, so we'll need to use numerical methods or graphing techniques to find the approximate solution.

Once we find the value of t where [tex]C(t) = R(t)[/tex], we can calculate the total profit accumulated during the useful life of the machine by subtracting the accumulated cost from the accumulated revenue:

[tex]Profit(t) = R(t) - C(t)[/tex]

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Calculate the steady-state response associated with the following dynamical equation \( 2 x(t)+2 \dot{x}(t)=b \cos (5 t) \), at \( t=15 \), where \( b=62.9 \).

The steady-state response associated with the given dynamical equation at \(t = 15\) is given by [tex]\[x(15) = \frac{62.9}{-75 + 10j}\][/tex]

To calculate the steady-state response of the given dynamical equation, we need to find the value of [tex]\[x(15) = \frac{62.9}{-75 + 10j}\][/tex]Let's start by rewriting the equation in terms of the Laplace transform. The Laplace transform of a derivative is given by \(sX(s) - x(0)\), where \(s\) is the Laplace variable and \(X(s)\) is the Laplace transform of \(x(t)\). Taking the Laplace transform of both sides of the equation, we get: [tex]\[2sX(s) + 2X(s) = \frac{b}{s^2 + 25}\][/tex] Next, we can solve for \(X(s)\) by rearranging the equation: [tex]\[X(s) = \frac{b}{2s^2 + 2s + 25}\][/tex] To find the steady-state response , we need to evaluate \(X(s)\) at \(s = j\omega\), where \(\omega\) is the frequency of the input signal. In this case, the input signal is \(b\cos(5t)\), so \(\omega = 5\). Substituting \(s = j\omega\) into the equation for \(X(s)\), we have: [tex]\[X(j\omega) = \frac{b}{2(j\omega)^2 + 2(j\omega) + 25}\][/tex] Simplifying the equation: [tex]\[X(j\omega) = \frac{b}{-4\omega^2 + 2j\omega + 25}\][/tex] Now, we can evaluate \(X(j\omega)\) at \(\omega = 5\): [tex]\[X(j5) = \frac{b}{-4(5)^2 + 2j(5) + 25}\][/tex] Simplifying further: \[X(j5) = \frac{b}{-100 + 10j + 25}\] \[X(j5) = \frac{b}{-75 + 10j}\][tex]\[X(j5) = \frac{b}{-100 + 10j + 25}\]\[X(j5) = \frac{b}{-75 + 10j}\][/tex] Finally, we substitute the given value of \(b = 62.9\) into the equation: \[X(j5) = \frac{62.9}{-75 + 10j}\] To calculate the steady-state response at \(t = 15\), we need to find the inverse Laplace transform of \(X(j5)\). However, without knowing the initial conditions of the system, we cannot determine the complete response. In summary, the steady-state response associated with the given dynamical equation at \(t = 15\) is given by: [tex]\[x(15) = \frac{62.9}{-75 + 10j}\][/tex]

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Suppose that there are two random variables X and Y. Suppose we know the joint distribution of Y and X. We would like to use X to predict Y. Our prediction is therefore a function of X, denote as m(X). If we restrict m(X) to have the linear form, m(X)=β 1 ​ X Note that there is no intercept in m(X). Now we ask the question "What is the optimal prediction function we can get?" i.e. to find the optimal value of β 1 ​ (denoted by β 1 ∗ ​ ) in m(X)=β 1 ​ X that minimizes the mean squared error β 1 ∗ ​ =argmin β 1 ​ ​ E X,Y ​ [(Y−β 1 ​ X) 2 ]. Prove that the optimal solution is β 1 ∗ ​ = Var(X)+(E(X)) 2 Cov(X,Y)+E(X)E(Y) ​ = E(X 2 ) E(XY) ​ Note that if E(X)=E(Y)=0 then β 1 ∗ ​ =Cov(X,Y)/Var(X)

The optimal solution is β 1 ∗ = Var(X)+(E(X))2Cov(X,Y)+E(X)E(Y)=E(X2)E(XY). Note that if E(X)=E(Y)=0 then β 1 ∗ = Cov(X,Y)/Var(X)

Given that there are two random variables X and Y and the joint distribution of Y and X are known. The prediction of Y using X is a function of X, m(X) and is a linear function defined as, m(X)=β 1  X.

There is no intercept in this function. We want to find the optimal value of β 1 , β 1 ∗ that minimizes the mean squared error β 1 ∗ = argmin β 1 E(X,Y) [(Y−β 1 X)2].

To prove that the optimal solution is β 1 ∗ = Var(X)+(E(X))2Cov(X,Y)+E(X)E(Y)=E(X2)E(XY).

Note: if E(X) = E(Y) = 0, then β 1 ∗ = Cov(X,Y)/Var(X).

We want to find the optimal value of β 1 that minimizes the mean squared error β 1 ∗ = argmin β 1 E(X,Y) [(Y−β 1 X)2]

Substituting m(X) = β 1 X, we haveE(X,Y) [(Y−β 1 X)2] = E(X,Y) [(Y-m(X))2] (1) Expanding the equation (1), we get E(X,Y) [(Y-m(X))2] = E(X,Y) [(Y2 - 2Ym(X) + m(X)2)]

Using the linearity of expectation, we have E(X,Y) [(Y-m(X))2] = E(X,Y) [Y2] - E(X,Y) [2Ym(X)] + E(X,Y) [m(X)2]Now, E(X,Y) [m(X)] = E(X,Y) [β 1 X] = β 1 E(X,Y) [X]

Using this, we getE(X,Y) [(Y-m(X))2] = E(X,Y) [Y2] - 2β 1 E(X,Y) [XY] + β 1 2E(X,Y) [X2] (2) Differentiating the equation (2) with respect to β 1 and equating it to zero, we get-2E(X,Y) [XY] + 2β 1 E(X,Y) [X2] = 0β 1 = E(X,Y) [XY]/E(X,Y) [X2]

Also, β 1 ∗ = argmin β 1 E(X,Y) [(Y-m(X))2] = E(X,Y) [Y-m(X)]2 = E(X,Y) [Y-β 1 X]2

Substituting β 1 = E(X,Y) [XY]/E(X,Y) [X2], we get β 1 ∗ = E(X,Y) [Y]E(X2) - E(XY)2/E(X2) From the above equation, it is clear that the optimal value of β 1 ∗ is obtained when E(Y|X) = β 1 ∗ X = E(X,Y) [Y]E(X2) - E(XY)2/E(X2)

This is the optimal linear predictor of Y using X. Note that, when E(X) = E(Y) = 0, then we get β 1 ∗ = Cov(X,Y)/Var(X).

Therefore, the optimal solution is β 1 ∗ = Var(X)+(E(X))2Cov(X,Y)+E(X)E(Y)=E(X2)E(XY). Note that if E(X)=E(Y)=0 then β 1 ∗ = Cov(X,Y)/Var(X)

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Agree or Disagree with each of the following statements. Remember to justify your reasoning. a) For any function f[x] and numbers a and b, if ∫ a b f[x] x = 0, then f[x] = 0 for all x’s with a < x < b.

False , "For any function f[x] and numbers a and b, if ∫ a b f[x] x = 0, then f[x] = 0 for all x’s with a < x < b"

The antiderivative of a function f[x] that satisfies ∫ a b f[x] x = 0, which is F[x] = ∫ f[x] x, might not be zero. So, it's not accurate to claim that f[x] = 0 for all x’s with a < x < b based on ∫ a b f[x] x = 0.

For any function f[x] and numbers a and b, the statement "if ∫ a b f[x] x = 0, then f[x] = 0 for all x’s with a < x < b" is false. This is because the antiderivative of a function f[x] that satisfies ∫ a b f[x] x = 0, which is F[x] = ∫ f[x] x, may not be zero.

Hence, it's not accurate to conclude that f[x] = 0 for all x’s with a < x < b based on ∫ a b f[x] x = 0. As an example, consider the function f[x] = 1. Even though ∫ a b f[x] x = 0 for a = 0 and b = 1, f[x] = 1 and not zero. As a result, this statement is incorrect.

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Let f(x,y) = 3+xy-2y and let D be the closed triangular region with vertices (1,0), (5,0), (1,4). Note: be careful as you plot these points, it is common to get the x and y coordinates backwards by accident. Find the boundary critical point along the boundary between points (5,0) and (1,4). ( ________, _________ )

Let's find the boundary critical point along the boundary between points (5,0) and (1,4) of the closed triangular region defined by the vertices (1,0), (5,0), and (1,4).

We need to follow these steps:Identify the boundary of the triangular region.Boundary critical points are candidates for the maxima and minima.Find the values of f(x,y) at the critical points and at the corners of the region.Compare the values obtained in step 3 to find the absolute maximum and minimum values of f(x,y) on the region.

Boundary of the region The boundary of the region is formed by the three line segments joining the vertices of the triangle. The segments are as follows:L1: (x, y) = (t, 0) for 1 ≤ t ≤ 5L2: (x, y) = (1, t) for 0 ≤ t ≤ 4L3: (x, y) = (4-t, t) for 0 ≤ t ≤ 4Note that L1 and L2 are parallel to the x-axis and y-axis, respectively. Also, L3 is a line joining (1,0) to (3,4).The boundary of the region is illustrated in the diagram below: Illustration of the triangular regionFind the boundary critical point along L3The point (5,0) is not on the boundary L3. The point (1,4) is on the boundary L3. We need to find the boundary critical point(s) along L3.

Therefore, we use the parameterization of the boundary L3: x = 4 - t, y = t.Substituting into the function f(x,y) = 3 + xy - 2y, we getg(t) = f(4-t, t) = 3 + (4-t)t - 2t = 3 + 2t - t^2We need to find the critical points of g(t) on the interval 0 ≤ t ≤ 4. Critical points are obtained by solving g'(t) = 0 for t. We haveg'(t) = 2 - 2tSetting g'(t) = 0, we obtaint = 1The value of g(t) at the critical point t = 1 isg(1) = 3 + 2(1) - 1^2 = 4Therefore, the boundary critical point along L3 is (3, 1) because x = 4 - t, and y = t, hence (3,1) = (4-t, t) = (1,4)

The given function is f(x, y) = 3 + xy - 2y.We needed to find the boundary critical point along the boundary between points (5, 0) and (1, 4). We identified the boundary of the triangular region and found that the boundary L3 is formed by the line segment joining the points (1, 4) and (5, 0).We used the parameterization of the boundary L3: x = 4 - t, y = t, and substituted it into the function f(x,y) to get g(t) = f(4-t, t) = 3 + (4-t)t - 2t = 3 + 2t - t^2. We found the critical point(s) of g(t) by solving g'(t) = 0 for t. The value of g(t) at the critical point was determined. Therefore, the boundary critical point along L3 is (3, 1).

The boundary critical point along the boundary between points (5,0) and (1,4) of the closed triangular region defined by the vertices (1,0), (5,0), and (1,4) is (3, 1).

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Derive the three-point central formula that approximate the derivative of a function f(x) is a point x 0 ​ . What is the error made using this approximation? QUESTION 3 [3 marks] Determine the derivative of the function f(x)=ln(1−x 2 ) in the point x 0 ​ =−0.5 using three-point central formula with h=0.1

The three-point central difference formula for approximating the derivative of a function f(x) at a point x₀ is given by:

f'(x₀) ≈ (f(x₀ + h) - f(x₀ - h)) / (2h)

where h is the step size or interval between neighboring points.

The error made using this approximation is on the order of O(h²), which means it is proportional to the square of the step size. In other words, as h becomes smaller, the error decreases quadratically . This makes the three-point central difference formula a second-orde r accurate approximation for the derivative.

To determine the derivative of the function f(x) = ln(1 - x²) at x₀ = -0.5 using the three-point central formula with h = 0.1, we can apply the formula as follows:

f'(-0.5) ≈ (f(-0.5 + 0.1) - f(-0.5 - 0.1)) / (2 * 0.1)

f'(-0.5) ≈ (f(-0.4) - f(-0.6)) / 0.2

Substituting the function f(x) = ln(1 - x²):

f'(-0.5) ≈ (ln(1 - (-0.4)²) - ln(1 - (-0.6)²)) / 0.2

f'(-0.5) ≈ (ln(1 - 0.16) - ln(1 - 0.36)) / 0.2

Evaluating the logarithmic terms:

f'(-0.5) ≈ (ln(0.84) - ln(0.64)) / 0.2

Calculating the difference of logarithms and dividing by 0.2 will give the approximate value of the derivative at x₀ = -0.5.

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The Space Shuttle travels at a speed of about 9.31×10 3 m/s. The blink of an astronaut's eye lasts about 118 ms. How many football felds (length =91.4 m ) does the Space Shuttle cover in the blink of an eye?

The Space Shuttle covers approximately 12.9 football fields in the blink of an eye.

To determine the number of football fields covered by the Space Shuttle in the blink of an eye, we need to calculate the distance traveled by the Space Shuttle in that time.

Speed of the Space Shuttle = 9.31×10^3 m/s

Duration of the blink of an eye = 118 ms = 0.118 s

Length of a football field = 91.4 m

First, we can calculate the distance traveled by the Space Shuttle in the blink of an eye using the formula:

Distance = Speed × Time

Distance = 9.31×10^3 m/s × 0.118 s

Distance ≈ 1099.58 m

Now, we can determine the number of football fields covered by dividing the distance by the length of a football field:

Number of football fields = Distance / Length of a football field

Number of football fields = 1099.58 m / 91.4 m

Number of football fields ≈ 12.02

Therefore, the Space Shuttle covers approximately 12.9 football fields in the blink of an eye.

In the blink of an eye, the Space Shuttle, traveling at a speed of about 9.31×10^3 m/s, covers a distance of approximately 1099.58 meters. To put this distance into perspective, we can compare it to the length of a football field, which is 91.4 meters.

By dividing the distance covered by the Space Shuttle (1099.58 meters) by the length of a football field (91.4 meters), we find that the Space Shuttle covers approximately 12.02 football fields in the blink of an eye. This means that within a fraction of a second, the Space Shuttle traverses a distance equivalent to more than 12 football fields.

The calculation highlights the incredible speed at which the Space Shuttle travels, allowing it to cover vast distances in very short periods of time. It also emphasizes the importance of considering the scale and magnitude of distances when dealing with high-speed objects like the Space Shuttle.

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The annual per capita consumption of bottled water was 30.8 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 30.8 and a standard deviation of 12 gallons. a. What is the probability that someone consumed more than 31 gallons of bottled water? b. What is the probability that someone consumed between 25 and 35 gallons of bottled water? c. What is the probability that someone consumed less than 25 gallons of bottled water? d. 97.5% of people consumed less than how many gallons of bottled water?

The answer of the probabilities are a) 49.93% b) 32.6% c) 31.46% d) 54.52 gallons

a. The mean of the distribution is μ = 30.8 gallons, and the standard deviation is σ = 12 gallons. We need to find the probability that someone consumed more than 31 gallons of bottled water. Using the Z-score formula, we have:

z = (x - μ) / σ = (31 - 30.8) / 12 = 0.02 / 12 = 0.0017

P(x > 31) = P(z > 0.0017) = 0.4993

Therefore, the probability that someone consumed more than 31 gallons of bottled water is approximately 0.4993 or 49.93%.

b. We need to find the probability that someone consumed between 25 and 35 gallons of bottled water. Again, using the Z-score formula, we have:

z₁ = (x₁ - μ) / σ = (25 - 30.8) / 12 = -0.48

z₂ = (x₂ - μ) / σ = (35 - 30.8) / 12 = 0.36

P(25 < x < 35) = P(z₁ < z < z₂) = P(z < 0.36) - P(z < -0.48) = 0.6406 - 0.3146 = 0.326

Therefore, the probability that someone consumed between 25 and 35 gallons of bottled water is approximately 0.326 or 32.6%.

c. We need to find the probability that someone consumed less than 25 gallons of bottled water.

z = (x - μ) / σ = (25 - 30.8) / 12 = -0.48

P(x < 25) = P(z < -0.48) = 0.3146

Therefore, the probability that someone consumed less than 25 gallons of bottled water is approximately 0.3146 or 31.46%.

d. We need to find the Z-score that corresponds to the 97.5th percentile of the distribution. Using a Z-score table, we find that this corresponds to a Z-score of 1.96.z = 1.96σ = 12μ = 30.8x = μ + zσ = 30.8 + 1.96(12) = 54.52

Therefore, 97.5% of people consumed less than approximately 54.52 gallons of bottled water.

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