## What is a 2-Proportion Test?

In a 2-proportion test, two separate proportions are measured at the same time, and their similarity is evaluated. It is also known as a 2-proportion z-test, and a comparison of two different proportions is done in this method.

In the test, we usually consider two different hypotheses, H_{0} and H_{a}. The term H_{0} is used to denote the null hypothesis. If the null hypothesis holds, it means that the two proportions compared are the same. If not, then H_{a} hypothesis holds. This hypothesis is known as the alternate hypothesis, and it can help in ensuring when two proportions compared are not the same.

## When to Use a 2-Proportion Z-Test

Usually, the 2-proportion z-test is used when there are only two parameters upon which we can start the comparison. These two categories or parameters are so classified that their nature is significantly different from each other. In events that can either occur or not using a 2-proportion z-test can be extremely useful. For instance, if we want to categorize a group of girls or boys, or if a person is old or young.

## What is a Test of 2-Proportions?

As stated before, a 2-proportion test uses the two hypotheses H_{0} and H_{a} for finding out the comparison between two proportions. Based on the result, the hypothesis can be selected, which in turn, will determine the course of the events.

### How can we find the Proportion between Two Different Values?

In mathematics, a proportion is simply an expression of finding out whether two separate ratios are equal or not.

## What is a Two Sample Proportion Test?

This is also a form of the z-proportion test. In this test, a single factor is involved with two different samples, and the results are evaluated. Usually, these tests come out in the form of binomial distributions. If only two samples are present, then this test can be performed. For more than two samples, the chi-Square test can be used.

## How does Proportion Testing Work?

In proportion testing, the significance of a value chosen as a sample, and the impact that it can have on the original population or sample space, is calculated.

### Difference of 2-Proportion Test

In a 2-proportion test, the difference between the two possible samples is calculated. We use this result in the formula for the z-test to determine whether the result inclines towards the null hypothesis or the alternate hypothesis.

## Why are Z-Tests Used for Proportions?

The use of z-tests can be extremely useful in proportions. The data of proportion fully supports the z-test as the function of the proportion represents the standard deviation that the proportion undergoes. In a confidence interval, it is important to spot the undergoing deviation in the value. Using a hypothesis test such as the z-test, we can determine whether the conditioned outcomes will occur without having to take any extra uncertainty into account or not.

## Formula

In the 2-proportion z-test, the following formula is given below-

$$\text{Z=}\frac{\left({p}_{1}-{p}_{2}\right)-0}{\sqrt{\widehat{p}\left(1-\widehat{p}\right)\left(\frac{1}{{n}_{1}}+\frac{1}{{n}_{2}}\right)}}$$Here, $\widehat{p}$ is pooled sample proportion, p1 is the first sample proportion, p2 is the second sample proportion, n1 is the first sample size, and n2 is the second sample size.

## What do We Mean by Sample Proportion?

If there is no way to predict a random variable that can vary from one sample to another, then it is called a sample proportion. The sample proportion is denoted by the term ‘p’ and it is a random variable. It comes with a mean and standard deviation as well.

## When can We Run a 2-Proportion Z-Test?

Usually, we can use the 2-proportion z-test only if the conditions given below are satisfied-

- The sample size of the required sample needs to be higher, and the minimum value should be 30. If not, the z-test is not the best option to use.
- For every set of points we use, we should make sure that the data points we take are not at all dependent on each other. Simply put, we only can use the data points that do not affect any other data point.
- Whenever we are going to perform the z-test, the data that we will take should already be in a normally distributed form. In a normal distribution, the data is already well arranged. However, if the sample size is extremely large, then this point can be ignored.
- Whatever data is selected in the test should be chosen completely randomly from the sample space. It should be so chosen that there is an equal chance for each of the items to get selected.
- If possible, the sample sizes chosen should have equal value. This will give the best z-test results.

## How can We Perform a 2-Proportion Z-Test?

To do a successful 2-proportion z-test, we need to follow the steps given below properly.

- At first, we have to identify the event and what are the different possibilities that can happen. Based on the two main possibilities, the null hypothesis H
_{0}and the alternate hypothesis H_{a}will be selected. - Next, we have to choose the alpha level in the 2-proportion z-test. The alpha level is also known as the significance level, and it states the possibility of the rejection of the null hypothesis H
_{0}, even when it is true. - Next, we have to find the value of z. We use a z-table for doing so. Usually, we need to find the critical value that z can hold.
- After that, we need to calculate the value of the z-test using the formula given above.
- The value of the test statistic z is then compared with the critical value of z. After that, we decide whether the null hypothesis H
_{0}can be rejected or supported. If supported, the null hypothesis holds. If not, the alternative hypothesis holds.

## How can we say that Two Proportions are Statistically Different or Not?

The most common method by which we can compare the two proportions is to compare their value. We have to keep in mind that both of the proportions are estimated values, and the difference between the two values can occur if the samples get chosen randomly from the same population or sample space.

With the help of a hypothesis test, we can find out whether or not the resulting difference in the evaluated proportions is enough to determine a change in the sample space proportion itself.

## Practice Problem

While testing corona vaccines, we have two vaccines, vaccine A and vaccine B. After finding out the results, we see that vaccine A has worked in 18 patients among a total of 2000, and vaccine B has worked in 34 patients among a total of 1000. Find out the individual sample proportions and the pooled sample proportion.

⇒ In vaccine A, there are 18 successes among 2000 cases.

Therefore, we have-

${p}_{1}=\frac{18}{2000}=\frac{9}{1000}$

Similarly,

${p}_{2}=\frac{34}{1000}=\frac{17}{500}$

Also, the pooled proportion is-

$\widehat{p}=\frac{18+34}{2000+1000}=\frac{52}{3000}=\frac{13}{750}$

This is the required answer.

## Context and Application

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelor of Science Mathematics
- Master of Science Mathematics

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