Introduction




You might have learned about testing the population mean/s. In this article, you will learn how to perform the hypothesis test for the population proportion. Consider the following scenarios.

To find the proportion of iPad users in the school, study the proportion of defective items in the lot, compare proportions of blue marbles in two bags, etc. we need to test the claim. In all these situations, the population proportion/s (p) is/are the parameter/s of interest.

To test the claim about population proportion or need to compare population proportions, we collect the sampled data from population/s. Then we make the inferences and conclusions based on these samples. There are some assumptions that should be satisfied to carry out hypothesis testing.


Assumptions for proportion test

  1. The data should be a simple random sample.
  2. The data should have binary outcomes. In other words, the samples should come from the binomial distribution.
  3. The observations in the data should be independent.
  4. If there are two samples then these samples should be independent of each other.
  5. Both the values n*p ≥ 10, n*(1-p) ≥ 10. Also, n*p*(1-p) ≥ 5.

In addition, if the sample size is n < 0.05*N then this assures the observations are independent.

If the data follows all these conditions, then one can use proportion testing. There are two tests in proportion testing. For the test for a single proportion, there is a one-sample proportion test while for the comparison of two proportions there is a two-sample proportion test.


Steps

Let us see what the steps are involved in the proportion test one by one.

Step 1: Fix the null and alternative hypotheses

The null and alternative hypotheses for proportion testing are as follows.



Step 2: Chose the correct formula to calculate the test statistic

Use the following formula to find the test statistic for the one-sample Z proportion test.



Here p-hat is the sample proportion; p0 is the proportion, at which you need to test the claim, n is the sample size.

Use the following formula to find the test statistic for the two-sample Z proportion test.



Where p1-hat and p2-hat are sample proportion, n1, n2 are sample sizes, p is the pooled proportion. Use the following formula to find pooled proportion.


The test statistic in both the proportion tests follows the standard normal distribution or Z distribution. Therefore, these tests are also known as Z proportion tests.


Step 3: Fix the significance level (α)

Most of the time the value of α is given in the question. If you do not know the value of α then just take α=0.05.


Step 4: Find the critical value/s or p-value

The critical values or p-value are essential in making the decision, whether to reject or fail to reject the null hypothesis. You can use either the statistical tables or technology like excel or TI84 calculator or R-Studio, etc. to find critical values as well as p-value.


Step 5: Make the decisions based on critical value/ p-value

Based on critical values

Using critical values, you can decide the rejection region. If the value of test statistic found in step 2 lies in the rejection region, then reject the null hypothesis at α level of significance, otherwise fail to reject the null hypothesis.

 

Based on p-value

If the p-value ≤ α then reject the null hypothesis otherwise fail to reject the null hypothesis.


Step 6: Conclusion based on the decision

If you reject the null hypothesis then write that, “There is sufficient evidence to support the claim in alternative hypothesis”. This is the same as writing, “There is sufficient evidence to reject the claim in null hypothesis”.

If you fail to reject the null hypothesis then write that, “There is insufficient evidence to support the claim in alternative hypothesis”. This is the same as writing, “There is insufficient evidence to reject the claim in null hypothesis”.

Write the conclusion based on the test that you are using.