\(y_1 = 351 \text { said "yes"}\)
\(\hat{p}_1 = \dfrac{351}{605} = 0.58\)
Is there sufficient evidence at the \(\alpha = 0.05\), say, to conclude that the two populations — smokers and non-smokers — differ significantly with respect to their opinions?
If \(p_1\) = the proportion of the non-smoker population who reply "yes" and \(p_2\) = the proportion of the smoker population who reply "yes," then we are interested in testing the null hypothesis:
\(H_0 \colon p_1 = p_2\)
against the alternative hypothesis:
\(H_A \colon p_1 \ne p_2\)
Before we can actually conduct the hypothesis test, we'll have to derive the appropriate test statistic.
The test statistic for testing the difference in two population proportions, that is, for testing the null hypothesis \(H_0:p_1-p_2=0\) is:
\(Z=\dfrac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\hat{p}(1-\hat{p})\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}}\)
\(\hat{p}=\dfrac{Y_1+Y_2}{n_1+n_2}\)
the proportion of "successes" in the two samples combined.
Recall that:
\(\hat{p}_1-\hat{p}_2\)
is approximately normally distributed with mean:
\(p_1-p_2\)
and variance:
\(\dfrac{p_1(1-p_1)}{n_1}+\dfrac{p_2(1-p_2)}{n_2}\)
But, if we assume that the null hypothesis is true, then the population proportions equal some common value p , say, that is, \(p_1 = p_2 = p\). In that case, then the variance becomes:
\(p(1-p)\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)\)
So, under the assumption that the null hypothesis is true, we have that:
\( {\displaystyle Z=\frac{\left(\hat{p}_{1}-\hat{p}_{2}\right)- \color{blue}\overbrace{\color{black}\left(p_{1}-p_{2}\right)}^0}{\sqrt{p(1-p)\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)}} } \)
follows (at least approximately) the standard normal N (0,1) distribution. Since we don't know the (assumed) common population proportion p any more than we know the proportions \(p_1\) and \(p_2\) of each population, we can estimate p using:
the proportion of "successes" in the two samples combined. And, hence, our test statistic becomes:
as was to be proved.
Non- Smokers | Smokers |
---|---|
\(n_1 = 605\) | \(n_2 = 195\) \(y_2 = 41 \text { said "yes"}\) \(\hat{p}_2 = \dfrac{41}{195} = 0.21\) |
The overall sample proportion is:
\(\hat{p}=\dfrac{41+351}{195+605}=\dfrac{392}{800}=0.49\)
That implies then that the test statistic for testing:
\(H_0:p_1=p_2\) versus \(H_0:p_1 \neq p_2\)
\(Z=\dfrac{(0.58-0.21)-0}{\sqrt{0.49(0.51)\left(\dfrac{1}{195}+\dfrac{1}{605}\right)}}=8.99\)
Errr.... that Z -value is off the charts, so to speak. Let's go through the formalities anyway making the decision first using the rejection region approach, and then using the P -value approach. Putting half of the rejection region in each tail, we have:
That is, we reject the null hypothesis \(H_0\) if \(Z ≥ 1.96\) or if \(Z ≤ −1.96\). We clearly reject \(H_0\), since 8.99 falls in the "red zone," that is, 8.99 is (much) greater than 1.96. There is sufficient evidence at the 0.05 level to conclude that the two populations differ with respect to their opinions concerning imposing a federal tax to help pay for health care reform.
Now for the P -value approach:
That is, the P -value is less than 0.0001. Because \(P < 0.0001 ≤ \alpha = 0.05\), we reject the null hypothesis. Again, there is sufficient evidence at the 0.05 level to conclude that the two populations differ with respect to their opinions concerning imposing a federal tax to help pay for health care reform.
Thankfully, as should always be the case, the two approaches.... the critical value approach and the P -value approach... lead to the same conclusion
For testing \(H_0 \colon p_1 = p_2\), some statisticians use the test statistic:
\(Z=\dfrac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}\)
instead of the one we used:
An advantage of doing so is again that the interpretation of the confidence interval — does it contain 0? — is always consistent with the hypothesis test decision.
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:
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 .
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 .
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.
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:
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
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.
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).
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.
Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.
|
| |
1 | The null hypothesis is a statement. There exists no relation between two variables | Alternative hypothesis a statement, there exists some relationship between two measured phenomenon |
2 | Denoted by H | Denoted by H |
3 | The observations of this hypothesis are the result of chance | The observations of this hypothesis are the result of real effect |
4 | The mathematical formulation of the null hypothesis is an equal sign | The mathematical formulation alternative hypothesis is an inequality sign such as greater than, less than, etc. |
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.
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.
Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.
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.
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.
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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Statistics Made Easy
A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.
We always use the following steps to perform a hypothesis test:
Step 1: State the null and alternative hypotheses.
The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.
The alternative hypothesis , denoted as H A , is the hypothesis that the sample data is influenced by some non-random cause.
2. Determine a significance level to use.
Decide on a significance level. Common choices are .01, .05, and .1.
3. Calculate the test statistic and p-value.
Use the sample data to calculate a test statistic and a corresponding p-value .
4. Reject or fail to reject the null hypothesis.
If the p-value is less than the significance level, then you reject the null hypothesis.
If the p-value is not less than the significance level, then you fail to reject the null hypothesis.
You can use the following clever line to remember this rule:
“If the p is low, the null must go.”
In other words, if the p-value is low enough then we must reject the null hypothesis.
The following examples show when to reject (or fail to reject) the null hypothesis for the most common types of hypothesis tests.
A one sample t-test is used to test whether or not the mean of a population is equal to some value.
For example, suppose we want to know whether or not the mean weight of a certain species of turtle is equal to 310 pounds.
We go out and collect a simple random sample of 40 turtles with the following information:
We can use the following steps to perform a one sample t-test:
Step 1: State the Null and Alternative Hypotheses
We will perform the one sample t-test with the following hypotheses:
We will choose to use a significance level of 0.05 .
We can plug in the numbers for the sample size, sample mean, and sample standard deviation into this One Sample t-test Calculator to calculate the test statistic and p-value:
Since the p-value (0.0015) is less than the significance level (0.05) we reject the null hypothesis .
We conclude that there is sufficient evidence to say that the mean weight of turtles in this population is not equal to 310 pounds.
A two sample t-test is used to test whether or not two population means are equal.
For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal.
We go out and collect a simple random sample from each population with the following information:
We can use the following steps to perform a two sample t-test:
We will perform the two sample t-test with the following hypotheses:
We will choose to use a significance level of 0.10 .
We can plug in the numbers for the sample sizes, sample means, and sample standard deviations into this Two Sample t-test Calculator to calculate the test statistic and p-value:
Since the p-value (0.2149) is not less than the significance level (0.10) we fail to reject the null hypothesis .
We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different.
A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.
For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump of college basketball players.
To test this, we may recruit a simple random sample of 20 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month:
We can use the following steps to perform a paired samples t-test:
We will perform the paired samples t-test with the following hypotheses:
We will choose to use a significance level of 0.01 .
We can plug in the raw data for each sample into this Paired Samples t-test Calculator to calculate the test statistic and p-value:
Since the p-value (0.0045) is less than the significance level (0.01) we reject the null hypothesis .
We have sufficient evidence to say that the mean vertical jump before and after participating in the training program is not equal.
You can use this decision rule calculator to automatically determine whether you should reject or fail to reject a null hypothesis for a hypothesis test based on the value of the test statistic.
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.
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null hypothesis
Examples of null hypothesis in a sentence.
These examples are programmatically compiled from various online sources to illustrate current usage of the word 'null hypothesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.
1935, in the meaning defined above
Nullarbor Plain
“Null hypothesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/null%20hypothesis. Accessed 28 Jun. 2024.
Britannica.com: Encyclopedia article about null hypothesis
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虛無假說. 在 推論統計學 中, 零假设 (英語: Null hypothesis ,又译 虚无假设、原假设 ,符号: )是做 统计检验 时的一类 假說 。. 零假设的内容一般是希望能证明为错误的假设,与零假设相对的是 對立假說 ,即希望通过证伪零假设而证明正确的另一种假说 ...
虛無假說. 在 推論統計學 中, 零假設 (英語: Null hypothesis ,又譯 零假說 ,符號: )是做 統計檢驗 時的一類 假說 。. 零假設的內容一般是希望能證明為錯誤的假設,與零假設相對的是 對立假說 ,即希望通過證偽零假設而證明正確的另一種假說。. 從數學 ...
null hypothesis的意思、解釋及翻譯:a theory that states that two groups that are being tested will be expected to show the same…。了解更多。
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 ( H0) answers "No, there's no effect in the population.". The alternative hypothesis ( Ha) answers "Yes, there is an effect in the ...
The null hypothesis is a default hypothesis that a quantity to be measured is zero (null). Typically, the quantity to be measured is the difference between two situations. For instance, trying to determine if there is a positive proof that an effect has occurred or that samples derive from different batches. [7] [8]
They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. \(H_0\): The null hypothesis: It is a statement of no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0. This can often be considered the status quo and as a ...
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. When your sample contains sufficient evidence, you can reject the null and conclude that the effect is statistically significant.
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.
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 of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.
It is the opposite of your research hypothesis. The alternative hypothesis--that is, the research hypothesis--is the idea, phenomenon, observation that you want to prove. If you suspect that girls take longer to get ready for school than boys, then: Alternative: girls time > boys time. Null: girls time <= boys time.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (HA): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...
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 ...
零假设. 在 推论统计学 中, 零假设 (英语: Null hypothesis ,又译 虚无假设、原假设 ,符号: )是做 统计检验 时的一类 假说 。. 零假设的内容一般是希望能证明为错误的假设,与零假设相对的是 对立假说 ,即希望通过证伪零假设而证明正确的另一种假说 ...
HYPOTHESIS translate: 假设,假说. Learn more in the Cambridge English-Chinese simplified Dictionary.
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.
Step 1: Figure out the hypothesis from the problem. The hypothesis is usually hidden in a word problem, and is sometimes a statement of what you expect to happen in the experiment. The hypothesis in the above question is "I expect the average recovery period to be greater than 8.2 weeks.". Step 2: Convert the hypothesis to math.
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.
Null Hypothesis Examples. "Hyperactivity is unrelated to eating sugar " is an example of a null hypothesis. If the hypothesis is tested and found to be false, using statistics, then a connection between hyperactivity and sugar ingestion may be indicated. A significance test is the most common statistical test used to establish confidence in a ...
Score test. In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function —known as the score —evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if the restricted estimator is near the maximum of the likelihood function, the score should not ...
That is, we reject the null hypothesis \(H_0\) if \(Z ≥ 1.96\) or if \(Z ≤ −1.96\). We clearly reject \(H_0\), since 8.99 falls in the "red zone," that is, 8.99 is (much) greater than 1.96. There is sufficient evidence at the 0.05 level to conclude that the two populations differ with respect to their opinions concerning imposing a ...
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.
A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.. We always use the following steps to perform a hypothesis test: Step 1: State the null and alternative hypotheses. The null hypothesis, denoted as H 0, is the hypothesis that the sample data occurs purely from chance.. The alternative hypothesis, denoted as H A, is the hypothesis that ...
The meaning of NULL HYPOTHESIS is a statistical hypothesis to be tested and accepted or rejected in favor of an alternative; specifically : the hypothesis that an observed difference (as between the means of two samples) is due to chance alone and not due to a systematic cause.