Question

(S 11.1) Random samples of sizes n₁ = 404 and n₂ = 310 were taken from two independent populations. In the first sample, 107 of the individuals met a certain criteria whereas in the second sample, 135 of the individuals met the same criteria. Run a 2PropZtest to test whether the proportions are different, and answer the following questions.

What is the value of p?, the pooled sample proportion?Round your response to at least 3 decimal places.

Calculate the z test statistic, for testing the null hypothesis that the population proportions are equal. Round your response to at least 2 decimal places.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P₁ = P₂
Alternative hypothesis: P₁ P₂

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)
p = 0.33894
SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.03574
z = (p₁ - p₂) / SE

z = - 4.77

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -4.77 or greater than 4.77

Thus, the P-value = less than 0.001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that the population proportions are equal.