## Hypothesis Testing with the Binomial Distribution

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

## Hypothesis Testing

## Worked Example

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

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## Binomial Hypothesis Test

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## Types of hypotheses

There are two main types of hypotheses:

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.

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

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

STEP 3 – Write out our binomial distribution .

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

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

## Two-tailed test example

## Critical values and critical regions

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

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

## 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.

## Frequently Asked Questions about Binomial Hypothesis Test

--> how many samples do you need for the binomial hypothesis test.

## --> 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.

## Final Binomial Hypothesis Test Quiz

Binomial hypothesis test quiz - teste dein wissen.

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.

What is an alternative hypothesis?

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

A critical value is the value where we start to reject the null hypothesis.

of the users don't pass the Binomial Hypothesis Test quiz! Will you pass the quiz?

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## Hypothesis Testing for Binomial Distribution

## One-sided Test

H 0 : π ≤ 1/6; i.e. the die is not biased towards the number three H 1 : π > 1/6

Using a significance level of α = .05, we have

P ( x ≥ 4) = 1–BINOM.DIST(3, 10, 1/6, TRUE) = 0.069728 > 0.05 = α .

For a 95% confidence level, α = .05, and so

BINOM.INV( n, p , 1– α ) = BINOM.INV(9, .5, .95) = 7

We use a one-tailed test with null and alternative hypotheses:

p-value = 1–BINOM.DIST(12, 24, .35, TRUE) = .04225 < .05 = α

and so conclude with 95% confidence that the new process shows a significant improvement.

## Two-sided Test

## 88 thoughts on “Hypothesis Testing for Binomial Distribution”

Charles, Took a bit to figure out how binom.test evaluates p-value.

Excel notation below produces same p-value provided by binom.test(x, n, p) in R.

A4 is an array, so need to hit ctrl+shift+enter.

Correction A3: p should be A3: x

Thanks, Muzaffar. Do you know how many people are in the dementia home? Charles

I’m a bit confused as to which test we would use… I assumed we use Lower-tailed test

Sammy, What are your null and alternative hypotheses? Charles

I had H0: p=.10 And H1: p<.10 (This one I'm not so sure about)

This is a two-tailed test. Can you please tell me how to do it?

I have a question about Example 3:

In example 3, where did the number 12 come from? Is it 13-1?

What is the theory behind subtracting 1? Why don’t we use 13 instead of 12?

Hello Mok Wai Ming, This problem is very similar to Example 1. Charles

This is an assignment but I am completely lost. Any help

What if the test value isn’t given and you have to guess and find the critical region?

Sorry, but I don’t understand your question. Charles

i appreciate the good job of this site

______________________________ Binomial with n = 100 and p = 0,03

x P( X <= x ) x P( X <= x ) 2 0,419775 3 0,647249 ______________________________

LEFT ONE TAIL TEST xBI Result of the BINOM.INV function xLC Lef tail critical value

x xLC Non rejection interval xBI >= xLC

If F(xBI) = alpha THEN xBI = xLC Do not subtract 1 from xBI

Function xlBinom_CV(n As Integer, p, alpha, nTails, pTail)

‘ ATTENTION – This function is not fully tested

For the binomial distribution, use the BINOM.DIST or BINOMDIST function. Charles

Probably so. I plan to revise all these functions along the lines that you have suggested. Charles

k_crit = min{k : P(X>=k) <= alpha}

but what the Excel function returns is

Sorry, don’t know what happed with formulas in my previous post above. The second one should read

k_excel = min{k : P(X= 1- alpha} = min{k : 1- P(X<=k) k) <= alpha }

Erik, If you want the pdf instead of the cdf, change the last argument from TRUE to FALSE. Charles

How will we know how many number of heads?

## Leave a Comment Cancel reply

## Hypothesis Testing Using the Binomial Distribution

An intuitive introduction and the 5-step-process..

Let’s play a game and throw a coin. Heads, I win. Tails, you win.

We start throwing the coin. First up is heads. I win.

Because, if the coin was fair, you probably wouldn’t have rolled 4 heads in a row.

The probability of getting four heads in a row is thus given by

## How does hypothesis testing work? A brief explanation.

## General Procedure in Hypothesis Testing

Let’s summarize the steps of hypothesis testing again.

Step 1: Figure out the new belief that you would like to prove (H₁) and define its opposite (H₀).

Step 3: Based upon the hypothesis, the error level, and the experiment, determine the decision rule.

Step 4: Conduct your experiment.

Step 5: Conclude according to the decision rule determined in step 3.

## Coin Example — Step by Step

Let’s get back to the coin example and go through the steps.

## Step 1: Determining H₀ and H₁

We would like to prove that the coin is rigged.

## Step 2: Error Level and Experiment

Therefore, let’s throw the coin 10 times and decide upon α = 0.05.

## Step 3: Decision Rule

Now, the question is, given that H₀ is true, what is the probability to observe certain results?

(If you want a deeper explanation of the binomial distribution, check this story here).

Therefore, the decision rule should be:

If 8 or more heads are observed, reject H₀. At a significance level of 95 % (an error level of 5 %), we can assume that the coin is rigged. If 7 or less heads are observed, we cannot reject H₀ and we cannot statistically prove that the coin is rigged.

## Step 4: Conduct the Experiment

Now comes the moment to actually throw the coin 10 times:

Heads, heads, tails, heads, tails, heads, tails, heads, heads, heads.

## Step 5: Experiment Conclusion

Does it mean that the coin is actually fair? No.

At the end of the day (or the research paper) hypothesis testing always follows the same 5 steps.

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## What is the Binomial test?

## What are the hypotheses of the binomial test?

The hypotheses for the Binomial test are as follows:

- The null hypothesis (H0) is that the population proportion of one outcome equals a specific hypothesized value (this can be denoted as π = π o ).
- The alternative hypothesis (H1) is that the population proportion of one outcome does not equal a specific hypothesized value (π ≠ π o ).

## When to use the Binomial test

## Binomial test assumptions

- B – the variable of interest should be a binary outcome meaning it can take only one of two values (e.g. a coin toss (heads/tails), presence of a disease (yes/no), morality (dead/alive)). This is sometimes also referred to as a dichotomous variable.
- I – observations should be independent , meaning that one observation should not have any bearing on the probability of another.
- N – the experiment should have a fixed sample size denoted n .
- S – all independent observations should have the same probability of having the outcome. This is similar to the independence assumption and can be achieved through random sampling.

## Binomial test example

- The null hypothesis (H0) is that the proportion of survey participants (30%) with HSV is equal to 20% (0.2).
- The alternative hypothesis (H1) is that the proportion of survey participants (30%) with HSV is not equal to 20% (0.2).

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## Example Questions

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Make sure you are happy with the following topics before continuing.

## Hypothesis Testing

## When to do a Binomial Hypothesis Test

## How to do a Binomial Hypothesis Test

- Define the parameter in the context of the question – for a binomial hypothesis test the parameter is p which is always the probability of something.
- Write down the null hypothesis and the alternate hypothesis .
- Define the test statistic X in the context of the question .
- Write down the distribution of X under the null hypothesis .
- State the significance level \alpha – even though you are likely given it in the question, not stating it risks losing a mark .
- Test for significance or find the critical region .
- Write a concluding sentence , linking the acceptance or rejection of H_{0} to the context .

## Critical Region and Actual Significance Level

## Example: Binomial Hypothesis Test

p is the probability of the sandwiches being in stock on a given day

Test statistic: X is the number of sandwiches in stock over five days.

Significance level: \alpha=0.05

Do not reject H_{0} . Insufficient evidence to suggest Edith is correct.

p is the probability of having the disease.

Test statistic X is the number of people in Hammerton who have the disease.

Under H_{0}: X\sim B(200,0.06)

Two tailed test so we are looking for a probability smaller than \dfrac{0.05}{2}=0.025

i) Find the critical region for a 5\% level one tail test on:

i) \mathbb{P}(X=2)=0.0621>0.05

So 2 is not in the critical region while 1 is.

Hence, the critical region is X=0,1

ii) They reach different conclusions.

p is the probability of a bacterium splitting.

Test statistic X is the number of bacteria that split.

Significance level: \alpha=0.01

Do not reject H_{0} . Insufficient evidence to suggest the probability of splitting is too high.

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To hypothesis test with the binomial distribution, we must calculate the probability, p p , of the observed event and any more extreme event happening.

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

Both One-sample Proportion Test Tool and R's function prop.test(x, n, p0) give the same results, where x is # of Successes, n is Sample size and p0 is Hyp

Now the basic idea is as follows: To prove H₁, we first assume that the opposite is true. The opposite is called the null hypothesis and

Hypothesis testing for the binomial distribution. In this video, I'll show you how to conduct a Hypothesis test for Binomial

Example question on hypothesis testing for the binomial distribution.YOUTUBE CHANNEL at https://www.youtube.com/ExamSolutionsEXAMSOLUTIONS

In statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations

Binomial Distribution: Hypothesis Testing, Confidence. Intervals (CI), and Reliability with Implementation in. S-PLUS by Joseph C. Collins. ARL-TR-5214.

Binomial test example · The null hypothesis (H0) is that the proportion of survey participants (30%) with HSV is equal to 20% (0.2). · The

Since binomial hypothesis tests test a probability parameter, words like probability, proportion and percentage are all clues that you should use a binomial