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Binomial Test Calculator

This binomial test calculator determines the probability of a particular outcome (K) across a certain number of trials ( n ), where there are precisely two possible outcomes.

To use the calculator, enter the values of n , K and p into the table below ( q will be calculated automatically), where n is the number of trials or observations, K is number of occasions the actual (or stipulated) outcome occurred, and p is the probability the outcome will occur on any particular occasion.

Things to remember: (a) the binomial test is appropriate only when you've got just two possible outcomes (or categories, etc.); (b) n and K will be frequencies; and (c) the value for p will fall somewhere between 0 and 1 - it's a proportion.

Binomial Test Calculator

Calculate Binomial Test with this free online tool

How to Calculate Binomial Test

• Set values for Trials, Successes, P(success), alpha, and alternative.
• View the result.
• If there is an error in red text, then try changing the inputs. If it persists, then click the Feedback button at the top of the page.

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Descriptive Statistics

Hypothesis test, binomial test calculator.

The domain of statistics is vast, with numerous tests and tools designed to decode the complexities of data. Among these, the binomial test stands out as a powerful technique to test the probability of observed outcomes. Whether you're examining the success rate of a new drug, the likelihood of customers choosing a product, or any scenario involving two potential outcomes, the Binomial Test Calculator is your analytical companion. This post delves into the significance of the binomial test and how our calculator simplifies its execution.

Understanding the Binomial Test

At its core, the binomial test determines whether the observed proportion of successes in a sample matches a pre-defined expectation. It's especially handy when you're dealing with yes/no, success/failure, or any other dichotomous outcomes.

Key Features of the Binomial Test Calculator

• User-Friendly Interface: Regardless of your statistical prowess, our calculator is designed for ease of use, ensuring every user can quickly input data and derive results.
• Quick Analysis: The need for manual calculations or complex software setups is eliminated. Input your data, and obtain results promptly.
• Detailed Outputs: Along with the p-value, the calculator provides clear interpretations, helping you gauge whether the observed outcomes significantly deviate from expectations.
• Graphical Illustration: Visual learners can benefit from the distribution graphs, making it easier to understand the data's significance in a graphical context.

How to Navigate the Binomial Test Calculator

• Data Input: Begin by inputting the number of trials, the number of observed successes, and the expected probability of success.
• Calculation: After feeding the data, click 'Calculate'. The calculator promptly processes the data and provides the results.
• Interpreting Results: A crucial metric provided is the p-value. Typically, a p-value less than 0.05 indicates that the observed data is significantly different from what was expected under the null hypothesis.

Practical Applications

The binomial test and our calculator find usage in various fields:

• Medical Research: To analyze the effectiveness of a treatment when compared to a known success rate.
• Marketing Studies: To determine if a new advertisement has a significant impact on product choices.
• Quality Control: To assess if the defect rate in a production batch deviates from acceptable standards.

The Binomial Test Calculator transforms these analytical tasks into smooth operations, ensuring accuracy and efficiency.

In Conclusion

The realm of statistics offers profound insights, provided we have the right tools in our arsenal. The Binomial Test Calculator serves as a bridge between complex statistical principles and actionable insights. Whether you're a researcher, a student, or a professional looking to make data-driven decisions, our calculator ensures you do so with confidence and clarity. Dive into the fascinating world of binomial tests and let our calculator guide your explorations!

Binomial test / Exact Binomial Test

Binomial Theorem > Binomial test

What is a binomial test?

The binomial test is used when an experiment has two possible outcomes (i.e. success/failure) and you have an idea about what the probability of success is. A binomial test is run to see if observed test results differ from what was expected. Example : you theorize that 75% of physics students are male. You survey a random sample of 12 physics students and find that 7 are male. Do your results significantly differ from the expected results?

Solution : Use the binomial formula to find the probability of getting your results. The null hypothesis for this test is that your results do not differ significantly from what is expected.

• p, the observed proportion, is 7/12, or .47.
• q, (1 – p) is 1 – .47.
• X is your expected number of successes (75% of 12 is 9).

Plugging those values into the formula and solving, you get: 0.158, which is the probability of 7 or fewer males out of 12. Doubling this (for a two tailed test ), gives 0.315. These are your p-values . With very few exceptions, you’ll always use the doubled value.

As the p-value of 0.315 is large (I’m assuming a 5% alpha level here, which would mean p-values of less than 5% would be significant), you cannot reject the null hypothesis that the results are expected. In other words, 7 is not outside of the range of what you would expect.

If, on the other hand, you had run the test with 4 males (p=.333 and q=.666), the doubled p-value would have been .006, which means you would have rejected the null.

Another option : use an online calculator, like this one: VassarStats binomial calculator.

Assumptions for the Binomial Test

• Items are dichotomous (i.e. there are two of them) and nominal.
• The sample size is significantly less than the population size.
• The sample is a fair representation of the population.
• Sample items are independent(one item has no bearing on the probability of another).

Binomial Hypothesis Testing ( Edexcel A Level Maths: Statistics )

Revision note.

Binomial Hypothesis Testing

How is a hypothesis test carried out with the binomial distribution.

• The population parameter being tested will be the probability, p   in a binomial distribution B(n , p)
• A hypothesis test is used when the assumed probability is questioned
• Make sure you clearly define p before writing the hypotheses
• The null hypothesis will always be H 0  : p = ...
• The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
• The alternative hypothesis, H 1 will be H 1 : p > ...  or   H 1 : p < ...  
• The alternative hypothesis,  H 1 will be H 1 : p ≠ ...  
• To carry out a hypothesis test with the binomial distribution, the test statistic will be the number of successes in a defined number of trials
• When defining the test statistic, remember that the value of p   is being tested, so this should be written as p in the original definition, followed by the null hypothesis stating the assumed value of p
• The binomial distribution will be used to calculate the probability of the test statistic taking the observed value or a more extreme value
• either calculating the probability of the test statistic taking the observed or a more extreme value ( p – value ) and comparing this with the significance level
• Finding the critical region can be more useful for considering more than one observed value or for further testing

How is the critical value found in a hypothesis test with the binomial distribution?

• The binomial distribution is a discrete distribution so the probability of the observed value being within the critical region, given a true null hypothesis may be less than the significance level
• This is the actual significance level and is the probability of incorrectly rejecting the null hypothesis
• Use the tables of the binomial distribution or your calculator to find the first value for which the probability of that or a more extreme value is less than half of the given significance level in both the upper and lower tails
• Often one of the critical regions will be much bigger than the other

What steps should I follow when carrying out a hypothesis test with the binomial distribution?

• Step 1. Define the probability, p
• Step 2. Write the null and alternative hypotheses clearly using the form

H 0 : p = ...

H 1 : p = ...

• Step 4. Calculate either the critical value(s) or the p – value for the test
• Step 5. Compare the observed value of the test statistic with the critical value(s) or the p - value with the significance level
• Step 6. Decide whether there is enough evidence to reject H 0 or whether it has to be accepted
• Step 7. Write a conclusion in context

Worked example

Jacques, a breadmaker, claims that more than 80% of people that shop in a particular supermarket buy his brand of bread.  Jacques takes a random sample of 100 customers that have purchased bread and asks them which brand of bread they have purchased.  He records that 86 of them had purchased his brand of bread.  Test, at the 10% level of significance, whether Jacques’ claim is justified.

• Make sure you are careful with the inequalities when finding critical values from the binomial tables or from your calculator.

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Binomial Probability Calculator

Use the Binomial Calculator to compute individual and cumulative binomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems .

To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution .

• Enter a value in each of the first three text boxes (the unshaded boxes).
• Click the Calculate button to compute binomial and cumulative probabilities.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What is a binomial experiment?

A binomial experiment has the following characteristics:

• The experiment involves repeated trials.
• Each trial has only two possible outcomes - a success or a failure.
• The probability that any trial will result in success is constant.
• All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution . It refers to the probabilities associated with the number of successes in a binomial experiment .

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution , as shown below.

What is the number of trials?

The number of trials refers to the number of replications in a binomial experiment.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. We classify Heads as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

Note: Each trial results in a success or a failure. So the number of trials in a binomial experiment is equal to the number of successes plus the number of failures.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY r successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

In a binomial experiment, the probability that the experiment results in exactly x successes is indicated by the following notation: P(X=x);

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X<2); the probability of getting AT MOST 2 successes is indicated by P(X≤2); the probability of getting AT LEAST 2 successes is indicated by P(X≥2); the probability of getting MORE THAN 2 successes is indicated by P(X>2).

Sample Problem

• The probability of success (i.e., getting a Head) on any single trial is 0.5.
• The number of trials is 12.
• The number of successes is 7 (since we define getting a Head as success).

Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button.

The calculator reports that the binomial probability is 0.193. That is the probability of getting EXACTLY 7 Heads in 12 coin tosses. (The calculator also reports the cumulative probabilities. For example, the probability of getting AT MOST 7 heads in 12 coin tosses is a cumulative probability equal to 0.806.)

• The probability of success for any individual student is 0.6.
• The number of trials is 3 (because we have 3 students).
• The number of successes is 2.

The calculator reports that the probability that two or fewer of these three students will graduate is 0.784.

Oops! Something went wrong.

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Sign and binomial test

Use the binomial test when there are two possible outcomes. You know how many of each kind of outcome (traditionally called "success" and "failure") occurred in your experiment. You also have a hypothesis for what the true overall probability of "success" is. The binomial test answers this question: If the true probability of "success" is what your theory predicts, then how likely is it to find results that deviate as far, or further, from the prediction.

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

Binomial test

The binomial test is a hypothesis test used when there is a categorical variable with two expressions, e.g., gender with "male" and "female". The binomial test can then check whether the frequency distribution of the variable corresponds to an expected distribution, e.g.:

• Men and women are equally represented.
• The proportion of women is 54%.

This is a special case when you want to test whether the frequency distribution of the variables is random or not. In this case, the probability of occurrence is set to 50%.

The binomial test can therefore be used to test whether or not the frequency distribution of a sample is the same as that of the population.

The binomial test checks whether the frequency distribution of a variable with two values/categories in the sample corresponds to the distribution in the population.

Hypotheses in binomial test

The hypothesis of the binomial test results in the one tailed case to

• Null hypothesis: The frequency distribution of the sample corresponds to that of the population.
• Alternative hypothesis: The frequency distribution of the sample does not correspond to that of the population.

Thus, the non-directional hypothesis only tests whether there is a difference or not, but not in which direction this difference goes.

In the two sided case, the aim is to investigate whether the probability of occurrence of an expression in the sample is greater or less than a given or true percentage.

In this case, an expression is defined as "success" and it is checked whether the true "probability of success" is smaller or larger than that in the sample.

The alternative hypothesis then results in:

• Alternative hypothesis: True probability of success is smaller/larger than specified value

Binomial test calculation

To calculate a binomial test you need the sample size, the number of cases that are positive of it, and the probability of occurrence in the population.

Binomial test example

A possible example for a binomial test would be the question whether the gender ratio in the specialization marketing at the university XY differs significantly from that of all business students at the university XY (population).

Listed below are the students majoring in marketing; women make up 55% of the total business degree program.

Binomial test with DATAtab:

Calculate the example in the statistics calculator. Simply add the upper table including the first row into the hypothesis test calculator .

DATAtab gives you the following result for this example data:

Interpretation of a Binomial Test

With an expected test value of 55%, the p-value is 0.528. This means that the p-value is above the signification level of 5% and the result is therefore not significant. Consequently, the null hypothesis must not be rejected. In terms of content, this means that the gender ratio of the marketing specialization (=sample) does not differ significantly from that of all business administration students at XY University (=population).

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Hypothesis Testing with the Binomial Distribution

Contents Toggle Main Menu 1 Hypothesis Testing 2 Worked Example 3 See Also

Hypothesis Testing

To hypothesis test with the binomial distribution, we must calculate the probability, $p$, of the observed event and any more extreme event happening. We compare this to the level of significance $\alpha$. If $p>\alpha$ then we do not reject the null hypothesis. If $p<\alpha$ we accept the alternative hypothesis.

Worked Example

A coin is tossed twenty times, landing on heads six times. Perform a hypothesis test at a $5$% significance level to see if the coin is biased.

First, we need to write down the null and alternative hypotheses. In this case

The important thing to note here is that we only need a one-tailed test as the alternative hypothesis says “in favour of tails”. A two-tailed test would be the result of an alternative hypothesis saying “The coin is biased”.

We need to calculate more than just the probability that it lands on heads $6$ times. If it landed on heads fewer than $6$ times, that would be even more evidence that the coin is biased in favour of tails. Consequently we need to add up the probability of it landing on heads $1$ time, $2$ times, $\ldots$ all the way up to $6$ times. Although a calculation is possible, it is much quicker to use the cumulative binomial distribution table. This gives $\mathrm{P}[X\leq 6] = 0.058$.

We are asked to perform the test at a $5$% significance level. This means, if there is less than $5$% chance of getting less than or equal to $6$ heads then it is so unlikely that we have sufficient evidence to claim the coin is biased in favour of tails. Now note that our $p$-value $0.058>0.05$ so we do not reject the null hypothesis. We don't have sufficient evidence to claim the coin is biased.

But what if the coin had landed on heads just $5$ times? Again we need to read from the cumulative tables for the binomial distribution which shows $\mathrm{P}[X\leq 5] = 0.021$, so we would have had to reject the null hypothesis and accept the alternative hypothesis. So the point at which we switch from accepting the null hypothesis to rejecting it is when we obtain $5$ heads. This means that $5$ is the critical value .

Selecting a Hypothesis Test

Binomial Distribution: Find Critical Values

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How to Do Hypothesis Testing with Binomial Distribution

A hypothesis test has the objective of testing different results against each other. You use them to check a result against something you already believe is true. In a hypothesis test, you’re checking if the new alternative hypothesis  H A would challenge and replace the already existing null hypothesis  H 0 .

Hypothesis tests are either one-sided or two-sided. In a one-sided test, the alternative hypothesis is left-sided with p < p 0 or right-sided with p > p 0 . In a two-sided test, the alternative hypothesis is p ≠ p 0 . In all three cases, p 0 is the pre-existing probability of what you’re comparing, and p is the probability you are going to find.

Note! In hypothesis testing, you calculate the alternative hypothesis to say something about the null hypothesis.

Hypothesis Testing Binomial Distribution

For example, you would have a reason to believe that a high observed value of p , makes the alternative hypothesis H a : p > p 0 seem reasonable.

There is a drug on the market that you know cures 8 5 % of all patients. A company has come up with a new drug they believe is better than what is already on the market. This new drug has cured 92 of 103 patients in tests. Determine if the new drug is really better than the old one.

This is a classic case of hypothesis testing by binomial distribution. You now follow the recipe above to answer the task and select 5 % level of significance since it is not a question of medication for a serious illness.

The alternative hypothesis in this case is that the new drug is better. The reason for this is that you only need to know if you are going to approve for sale and thus the new drug must be better:

This result indicates that there is a 1 3 . 6 % chance that more than 92 patients would be cured with the old medicine.

so H 0 cannot be rejected. The new drug does not enter the market.

If the p value had been less than the level of significance, that would mean that the new drug represented by the alternative hypothesis is better, and that you are sure of this with statistical significance.

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• Binomial Hypothesis Test

When calculating probabilities using binomial expansions, we can calculate these probabilities  for an individual value (\(P(x = a)\))  or a cumulative value \(P(x<a), \space P(x\leq a), \space P(x\geq a)\) .

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What is a hypothesis test?

What is a null hypothesis?

What is an alternative hypothesis?

What is a one-tailed test?

What is a two-tailed test?

What is a significance level?

What is a critical value?

What is a critical region?

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In hypothesis testing , we are testing as to whether or not these calculated probabilities can lead us to accept or reject a hypothesis.

We will be focusing on regions of binomial distribution ; therefore, we are looking at cumulative values.

Types of hypotheses

There are two main types of hypotheses:

The null hypothesis (H 0 ) is the hypothesis we assume happens, and it assumes there is no difference between certain characteristics of a population. Any difference is purely down to chance.

The alternative hypothesis (H 1 ) is the hypothesis we can try to prove using the data we have been given.

We can either:

Accept the null hypothesis OR

Reject the null hypothesis and accept the alternative hypothesis .

What are the steps to undertake a hypothesis test?

There are some key terms we need to understand before we look at the steps of hypothesis testing :

Critical value – this is the value where we go from accepting to rejecting the null hypothesis.

Critical region – the region where we are rejecting the null hypothesis.

Significance Level – a significance level is the level of accuracy we are measuring, and it is given as a percentage . When we find the probability of the critical value, it should be as close to the significance level as possible.

One-tailed test – the probability of the alternative hypothesis is either greater than or less than the probability of the null hypothesis.

Two-tailed test – the probability of the alternative hypothesis is just not equal to the probability of the null hypothesis.

So when we undertake a hypothesis test, generally speaking, these are the steps we use:

STEP 1 – Establish a null and alternative hypothesis, with relevant probabilities which will be stated in the question.

STEP 2 – Assign probabilities to our null and alternative hypotheses.

STEP 3 – Write out our binomial distribution .

STEP 4 – Calculate probabilities using binomial distribution. (Hint: To calculate our probabilities, we do not need to use our long-winded formula, but in the Casio Classwiz calculator, we can go to Menu -> Distribution -> Binomial CD and enter n as our number in the sample, p as our probability, and X as what we are trying to calculate).

STEP 5 – Check against significance level (whether this is greater than or less than the significance level).

STEP 6 – Accept or reject the null hypothesis.

Let's look at a few examples to explain what we are doing.

One-tailed test example

As stated above a one-tailed hypothesis test is one where the probability of the alternative hypothesis is either greater than or less than the null hypothesis.

A researcher is investigating whether people can identify the difference between Diet Coke and full-fat coke. He suspects that people are guessing. 20 people are selected at random, and 14 make a correct identification. He carries out a hypothesis test.

a) Briefly explain why the null hypothesis should be H 0 , with the probability p = 0.5 suggesting they have made the correct identification.

b) Complete the test at the 5% significance level.

Two-tailed test example

In a two-tailed test, the probability of our alternative hypothesis is just not equal to the probability of the null hypothesis.

A coffee shop provides free espresso refills. The probability that a randomly chosen customer uses these refills is stated to be 0.35. A random sample of 20 customers is chosen, and 9 of them have used the free refills.

Carry out a hypothesis test to a 5% significance level to see if the probability that a randomly chosen customer uses the refills is different to 0.35.

So our key difference with two-tailed tests is that we compare the value to half the significance level rather than the actual significance level.

Critical values and critical regions

Remember from earlier critical values are the values in which we move from accepting to rejecting the null hypothesis. A binomial distribution is a discrete distribution; therefore, our value has to be an integer.

You have a large number of statistical tables in the formula booklet that can help us find these; however, these are inaccurate as they give us exact values not values for the discrete distribution.

Therefore the best way to find critical values and critical regions is to use a calculator with trial and error till we find an acceptable value:

STEP 1 - Plug in some random values until we get to a point where for two consecutive values, one probability is above the significance level, and one probability is below.

STEP 2 - The one with the probability below the significance level is the critical value.

STEP 3 - The critical region, is the region greater than or less than the critical value.

Let's look at this through a few examples.

Worked examples for critical values and critical regions

A mechanic is checking to see how many faulty bolts he has. He is told that 30% of the bolts are faulty. He has a sample of 25 bolts. He believes that less than 30% are faulty. Calculate the critical value and the critical region.

Let's use the above steps to help us out.

A teacher believes that 40% of the students watch TV for two hours a day. A student disagrees and believes that students watch either more or less than two hours. In a sample of 30 students, calculate the critical regions.

As this is a two-tailed test, there are two critical regions, one on the lower end and one on the higher end. Also, remember the probability we are comparing with is that of half the significance level.

Binomial Hypothesis Test - Key takeaways

• Hypothesis testing is the process of using binomial distribution to help us reject or accept null hypotheses.
• A null hypothesis is what we assume to be happening.
• If data disprove a null hypothesis, we must accept an alternative hypothesis.
• We use binomial CD on the calculator to help us shortcut calculating the probability values.
• The critical value is the value where we start rejecting the null hypothesis.
• The critical region is the region either below or above the critical value.
• Two-tailed tests contain two critical regions and critical values.

Flashcards in Binomial Hypothesis Test 8

A hypothesis test is a test to see if a claim holds up, using probability calculations.

A null hypothesis is what we assume to be true before conducting our hypothesis test.

An alternative hypothesis is what we go to accept if we have rejected our null hypothesis.

A one tailed test is a test where the probability of the alternative hypothesis can be either greater than or less than the probability of the null hypothesis.

A two tailed test is a hypothesis test where the probability of the alternative hypothesis can be both greater than and less than the probability of the null hypothesis (simply the probability of the alternative hypothesis is not equal to that of the null hypothesis).

A significance level is the level we are testing to. The smaller the significance level, the more difficult it is to disprove the null hypothesis.

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Frequently Asked Questions about Binomial Hypothesis Test

How many samples do you need for the binomial hypothesis test?

There isn't a fixed number of samples, any sample number you are given you will use as n in X-B(n , p).

What is the null hypothesis for a binomial test?

The null hypothesis is what we assume is true before we conduct our hypothesis test.

What does a binomial test show?

It shows us the probability value is of undertaking a test, with fixed outcomes.

What is the p value in the binomial test?

The p value is the probability value of the null and alternative hypotheses.

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Hypothesis testing using the binomial distribution (2.05a)

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Statistics By Jim

Making statistics intuitive

Binomial Distribution: Uses & Calculator

By Jim Frost 2 Comments

What is the Binomial Distribution?

The binomial distribution is a discrete probability distribution that calculates the likelihood an event will occur a specific number of times in a set number of opportunities. Use this distribution when you have a binomial random variable. These variables count how often an event occurs within a fixed number of trials. They have only two possible outcomes that are mutually exclusive.

For example, the binomial probability distribution can answer the following questions. What is the probability of getting:

• Six heads when you toss the coin ten times?
• 12 women in a sample size of 20?
• Three defective items in a batch of 100?
• Two flu infections over 20 years?

This distribution is an example of a Probability Mass Function (PMF) because it calculates likelihoods for discrete random variables. It is an extension of the Bernoulli distribution that can model only 1 trial.

In this post, learn how to use the binomial distribution, its cumulative form, and when you can use it. I also include a binomial calculator that you can use with what you learn.

Note that this post focuses on how to use and graph the binomial distribution. If you want to learn how to calculate the probabilities by hand, please read Binomial Distribution Formula: Probability, Standard Deviation & Mean .

Binomial Probabilities

I’ll start by using statistical software to calculate the binomial probabilities and create distribution plots. This process will help you understand what you can learn from it.

Suppose you’re playing a game where rolling sixes on a die is really good. You want to know the probability of rolling exactly three sixes in ten die rolls. In this example, the number of events is 3 (X), the number of trials is 10 (N), and the probability (p) is 1/6 = 0.1667.

My software tells me that the likelihood is:

The binomial probability distribution calculates a likelihood of 0.155095 for rolling precisely three sixes in ten rolls.

That’s interesting but perhaps not so helpful by itself. We’re also interested in the chances for rolling other numbers of sixes. Seeing the distribution of probabilities for different numbers of sixes is much more helpful.

Binomial Distribution Graph

The binomial distribution graph is useful because it displays the probability of differing numbers of successes (Xs) out of the total number of trials (N). In the graph below, the distribution plot finds the likelihood of rolling exactly no sixes, 1 six, 2 sixes, 3 sixes, . . ., and up to 10 sixes in the ten die rolls. Using this approach, the binomial distribution graph covers the complete range of possible successes up to the total number of trials.

I like these graphs because they emphasize how we’re working with a distribution, and it’s easy to see which values happen more frequently.

In the chart, each bar represents the probability of rolling a specific number of sixes out of ten die rolls. The graph does not show the chances for seven and higher because the likelihoods of that many sixes in just ten rolls are too low to display on the chart.

The binomial distribution graph indicates the probability of rolling no sixes is about 16%. The highest chance is rolling one six (32%). Although, rolling two sixes occurs almost as frequently. Probabilities drop off quickly starting with three sixes. Additionally, the bar for three sixes matches our earlier result of 0.155095.

Related post : Understanding Probability Distributions

Binomial Cumulative Distribution Function

The binomial probability distribution is excellent for understanding the likelihood of obtaining an exact number of events (X) within a certain number of trials (N). However, many times you’re not interested in just one specific value for a binomial random variable. For example, in the die rolling example above, you might know from experience that rolling three or more sixes within ten rolls means you’re doing well. So, you actually want to learn the probability of rolling at least three sixes.

Let me introduce you to the binomial cumulative distribution function.

Technically, the binomial cumulative probability calculates the likelihood of obtaining less than or equal to X events in N trials. If you need to obtain a ≥ probability, use the inverse cumulative distribution. These days, statistical software will generally let you specify the direction of the cumulative function for the binomial distribution from the start. I’ll use the binomial distribution graph again to show you how it works.

For our example, we want to know the chances of rolling ≥ 3 sixes in 10 rolls. Below, the shaded region shows the inverse cumulative probability of rolling at least three sixes in ten die rolls.

The likelihood for rolling three or more sixes in ten rolls is 0.2249, not quite 1 in 4.

For a real-world example, see how I’ve used the binomial distribution to model the number of flu infections (X) for the vaccinated vs. unvaccinated over 20 years (N).

Learn more about Cumulative Distribution Functions: Uses, Graphs & vs PDF .

Binomial Distribution Assumptions and Notation

The binomial distribution models the probabilities for a binomial random variable having exactly X successes occurring in N trials. Your variable must satisfy the following requirements to be a binomial random variable. The binomial distribution is appropriate only for data that fulfill these assumptions.

• There must be only two possible outcomes per trial . For example, defective or not defective, sale or no sale, pass or fail, etc.
• The trials are independent . One trial’s outcome does not affect the subsequent trial. For instance, one coin toss doesn’t affect the result of the following coin toss. Learn more about Independent Events .
• The probability remains constant over time . In some areas, this assumption is true due to the physical characteristics of the process, such as coin tosses and die rolls. However, the probability won’t necessarily remain constant in other contexts. For example, the likelihood that a manufacturing process creates defective parts can change over time. If the probability can change, use the P chart ( a control chart ) to confirm this assumption.

Bernoulli Trials

Typically, you’ll use the binomial distribution when you have Bernoulli Trials, also known as Binomial Experiments. These trials involve binomial random variables that satisfactorily follow the assumptions above. In these trials, analysts label one of the possible outcomes as a success and the other outcome a failure.

A Bernoulli trial contains a set number of trials where the probability of a success is constant. The experiment counts the number of successes (X) out of the total number of trials (N).

You can think of the binomial probability distribution as modeling the number of successes (X) in a sample size of N.

Parameters and Notation

The binomial distribution has two parameters , n and p.

• n : the number of trials.
• p : the event or success probability.

You denote a binomial distribution as b(n,p).

Alternatively, you can write X∼b(n,p), which means that your binomial random variable X follows a binomial probability distribution with n trials and an event probability of p.

The previous examples assess probabilities corresponding with rolling sixes in a series of 10 die rolls. In this scenario, success is rolling a six, while a failure is rolling anything other than a six. The probability of rolling a six is 1/6 = 0.1667.

If rolling sixes is our random variable X, and we roll the die ten times, we can use the following notation for the binomial distribution:

X∼b(10,0.1667)

Binomial Distribution Calculator

Use this binomial distribution calculator to calculate the binomial probabilities and cumulative probabilities. Note that it uses “events” to indicate the number of trials (n).

Next, change exactly r successes to r or more successes . The calculator displays 22.487, matching the results for our example with the binomial inverse cumulative distribution.

Now, try one yourself. Imagine you’re drawing a random sample of 20 from a population where 10% are statisticians. You’re hoping that your study will have 3 or fewer statisticians because they’ll gang up and ask too many pesky questions about your study design. What is the likelihood of obtaining ≤ 3 statisticians?

See the correct answer at the end of this post.

Finally, the binomial and beta distributions are closely related. Click the link to learn more!

For more information about how to use binary data, read my posts, Maximize the Value of Your Binary Data , the Negative Binomial Distribution , the Geometric Distribution , and the Hypergeometric Distribution .

In the calculator example, there is an 86.7% chance of having ≤ 3 statisticians in your sample of 20 people.

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Reader Interactions

August 25, 2022 at 9:55 pm

Is there some way to combine binomial distributions? Here’s an example. Ann, Bob, and Carol are shooting threes on a basketball court. Ann takes 50 shots and has a 30% success rate. Bob takes 30 shots and has a 20% success rate. Carol takes 20 shots and has a 10% success rate. I can use the cumulative binomial distribution to calculate the chance that Ann makes 10 or more shots or that Bob makes 10 or more shots. How do I calculate the probability that the three of them combine to make 20 or more shots?

August 25, 2022 at 6:45 pm

Would binomial distributions be suitable for determining the probability of a prisoner re-offending once released from prison? Thank you.

Comments and Questions Cancel reply

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Unit 12: Significance tests (hypothesis testing)

About this unit.

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The idea of significance tests

• Simple hypothesis testing (Opens a modal)
• Idea behind hypothesis testing (Opens a modal)
• Examples of null and alternative hypotheses (Opens a modal)
• P-values and significance tests (Opens a modal)
• Comparing P-values to different significance levels (Opens a modal)
• Estimating a P-value from a simulation (Opens a modal)
• Using P-values to make conclusions (Opens a modal)
• Simple hypothesis testing Get 3 of 4 questions to level up!
• Writing null and alternative hypotheses Get 3 of 4 questions to level up!
• Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

• Introduction to Type I and Type II errors (Opens a modal)
• Type 1 errors (Opens a modal)
• Examples identifying Type I and Type II errors (Opens a modal)
• Introduction to power in significance tests (Opens a modal)
• Examples thinking about power in significance tests (Opens a modal)
• Consequences of errors and significance (Opens a modal)
• Type I vs Type II error Get 3 of 4 questions to level up!
• Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

• Constructing hypotheses for a significance test about a proportion (Opens a modal)
• Conditions for a z test about a proportion (Opens a modal)
• Reference: Conditions for inference on a proportion (Opens a modal)
• Calculating a z statistic in a test about a proportion (Opens a modal)
• Calculating a P-value given a z statistic (Opens a modal)
• Making conclusions in a test about a proportion (Opens a modal)
• Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
• Conditions for a z test about a proportion Get 3 of 4 questions to level up!
• Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
• Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
• Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

• Writing hypotheses for a significance test about a mean (Opens a modal)
• Conditions for a t test about a mean (Opens a modal)
• Reference: Conditions for inference on a mean (Opens a modal)
• When to use z or t statistics in significance tests (Opens a modal)
• Example calculating t statistic for a test about a mean (Opens a modal)
• Using TI calculator for P-value from t statistic (Opens a modal)
• Using a table to estimate P-value from t statistic (Opens a modal)
• Comparing P-value from t statistic to significance level (Opens a modal)
• Free response example: Significance test for a mean (Opens a modal)
• Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
• Conditions for a t test about a mean Get 3 of 4 questions to level up!
• Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
• Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
• Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

More significance testing videos

• Hypothesis testing and p-values (Opens a modal)
• One-tailed and two-tailed tests (Opens a modal)
• Z-statistics vs. T-statistics (Opens a modal)
• Small sample hypothesis test (Opens a modal)
• Large sample proportion hypothesis testing (Opens a modal)