Question

Two different simple random samples are drawn from two different populations. The first sample consists of 40 people with 20 having a common attribute. The second sample consists of 2200 people with 1559 of them having the same common attribute. Compare the results from a hypothesis test of p 1 = p 2 ​(with a 0.05 significance​level) and a 95% confidence interval estimate of p 1−p 2.

What are the null and alternative hypotheses for the hypothesis​test?

Identify the test statistic. nothing ​(Round to two decimal places as​ needed.)

Identify the critical​ value(s). ​(Round to three decimal places as needed. Use a comma to separate answers as​ needed.)

What is the conclusion based on the hypothesis​ test? The test statistic is ▼ the critical​ region, so ▼ the null hypothesis.

There is ▼ evidence to conclude that p 1not equals≠p 2. The 95​% confidence interval is less than = less than or equals≤ p 2​, and the confidence interval suggests that p 1 ▼

Answer 1

The null hypothesis is that the two population proportions are equal, while the alternative hypothesis is that they are not equal...

Step-by-step explanation:

The null hypothesis for the hypothesis test is that the population proportions are equal, while the alternative hypothesis is that the population proportions are not equal. The test statistic for comparing two population proportions is the z-score, which is calculated using the formula:

`z = (p_1 - p_2) / √((p_1 * (1 - p_1) / n_1) + (p_2 * (1 - p_2) / n_2))`

To compare the test statistic to the critical value, we need to determine the critical value for a two-tailed test with a 0.05 significance level. We can use the z-table to find the critical value, which is approximately ±1.96.

Based on the test statistic being in the critical region and the null hypothesis being rejected, we can conclude that there is evidence to suggest that the population proportions are not equal. The 95% confidence interval estimate of the difference in population proportions, p1 - p2, is calculated using the formula:

`CI = (p_1 - p_2) \± z * √((p_1 * (1 - p_1) / n_1) + (p_2 * (1 - p_2) / n_2))`

Since the confidence interval is below zero, we can conclude that p1 is less than p2.