 # Two Sample Hypothesis Testing for Proportions

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Two Sample Hypothesis Testing for Proportions

Sampling Distribution of the Difference between Two Proportions
Suppose that a simple random sample of size n1 is taken from a population where x1 of the individuals have a specified characteristic, and a simple random sample of size n2 is independently taken from a different population where x2 of the individuals have a specified characteristic. The sampling distribution of , where and , is approximately normal, with mean and standard deviation provided that and

Sampling Distribution of the Difference between Two Proportions
The standardized version of is then written as which has an approximate standard normal distribution.

The best point estimate of p is called the pooled estimate of p, denoted , where
Test statistic for Comparing Two Population Proportions

Hypothesis Test Regarding the Difference
between Two Population Proportions To test hypotheses regarding two population proportions, p1 and p2, we can use the steps that follow, provided that: the samples are independently obtained using simple random sampling, and and 3. n1 ≤ 0.05N1 and n2 ≤ 0.05N2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experiment.

Step 1: Determine the null and alternative hypotheses
Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

Step 2: Select a level of significance, , based
on the seriousness of making a Type I error.

Step 3: Compute the test statistic
where

P-Value Approach Step 4: Use Table V to estimate the P-value..

P-Value Approach Two-Tailed

P-Value Approach Left-Tailed

P-Value Approach Right-Tailed

P-Value Approach Step 5: If the P-value < , reject the null hypothesis. If the P-value ≥ α, fail to reject the null hypothesis

Step 6: State the conclusion in the context of the problem.

Parallel Example 1: Testing Hypotheses Regarding Two
Parallel Example 1: Testing Hypotheses Regarding Two Population Proportions An economist believes that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. He obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Test the economist’s claim at the =0.05 level of significance.

Solution We must first verify that the requirements are satisfied:
The samples are simple random samples that were obtained independently. x1=338, n1=800, x2=292 and n2=750, so 3. The sample sizes are less than 5% of the population size.

H0: p1 - p2=0 versus H1: p1 - p2 > 0
Solution Step 1: We want to determine whether the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. So, H0: p1 = p2 versus H1: p1 > p2 or, equivalently, H0: p1 - p2=0 versus H1: p1 - p2 > 0 Step 2: The level of significance is  = 0.05.

Solution Step 3: The pooled estimate of is: The test statistic is:

Solution: P-Value Approach
Step 4: Because this is a right-tailed test, the P-value is the area under the normal to the right of the test statistic z0= That is, P-value =P(Z > 1.33) ≈ 0.09.

Solution: P-Value Approach
Step 5: Since the P-value is greater than the level of significance =0.05, we fail to reject the null hypothesis.

Solution Step 6: There is insufficient evidence at the =0.05 level to conclude that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access.

Constructing a (1-) 100% Confidence Interval for the Difference between Two Population Proportions
To construct a (1-)100% confidence interval for the difference between two population proportions, the following requirements must be satisfied: the samples are obtained independently using simple random sampling, , and 3. n1 ≤ 0.05N1 and n2 ≤ 0.05N2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experiment.

Provided that these requirements are met, a
Constructing a (1-) 100% Confidence Interval for the Difference between Two Population Proportions Provided that these requirements are met, a (1-)100% confidence interval for p1-p2 is given by Lower bound: Upper bound:

Parallel Example 3: Constructing a Confidence Interval for
Parallel Example 3: Constructing a Confidence Interval for the Difference between Two Population Proportions An economist obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Find a 99% confidence interval for the difference between the proportion of urban households that have Internet access and the proportion of rural households that have Internet access.

Solution We have already verified the requirements for constructing a confidence interval for the difference between two population proportions in the previous example. Recall

Solution Thus, Lower bound = Upper bound =

Solution We are 99% confident that the difference between the proportion of urban households that have Internet access and the proportion of rural households that have Internet access is between and Since the confidence interval contains 0, we are unable to conclude that the proportion of urban households with Internet access is greater than the proportion of rural households with Internet access.