Q: A researcher collects two independent samples and finds $$\hat{p}_1 = 0.60$$, $$n_1 = 100$, $$\hat{p}_2 = 0.50$$, n_2$ = 120. $$Calculate the pooled proportion $$\hat{p}.$$

$$\hat{p} = \frac{0.60 \times 100 + 0.50 \times 120}{100 + 120} = 0.545$$

Question 1

Which of the following is NOT a required condition for conducting a two-sample z-test for the difference between two population proportions?

Accepted Answer

The population standard deviations must be known.

Question 2

Suppose you want to test whether the proportion of people who wear seat belts is different in two states. Which of the following is the correct null hypothesis for a two-tailed test?

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$

Question 3

Given two independent random samples, with sample proportions $$\hat{p}_1$$ and $$\hat{p}_2$$, and sample sizes $$n_1$$ and $$n_2$$, what is the formula for the standard error used in the two-sample z-test for proportions?

Accepted Answer

$$\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}$$

Question 4

A medical study tests whether a new medication increases the recovery rate from PTSD compared to a placebo. Which of the following is the correct alternative hypothesis for a right-tailed test?

Accepted Answer

$$H_a: p_1$ \u003e p_2$

Question 5

In a two-sample z-test for proportions, what does the test statistic $$z$$ represent?

Accepted Answer

The number of standard errors the observed difference in sample proportions is from the hypothesized difference.

Question 6

If the calculated z-value falls in the rejection region for a right-tailed test at the 1% significance level, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is enough evidence to support the alternative hypothesis.

Question 7

Which of the following real-world scenarios is most appropriate for a two-sample z-test for proportions?

Accepted Answer

Comparing the proportion of smokers in two different cities.

Question 8

A researcher collects two independent samples and finds $$\hat{p}_1 = 0.60$$, $$n_1 = 100$, $$\hat{p}_2 = 0.50$$, n_2$ = 120. $$Calculate the pooled proportion $$\hat{p}.$$

Accepted Answer

$$\hat{p} = \frac{0.60 	imes 100 + 0.50 	imes 120}{100 + 120} = 0.545$$

Question 9

Which of the following statements best describes the meaning of a p-value in the context of a two-sample z-test for proportions?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

Question 10

A study finds $$z = 1.25$$ for a two-tailed test at $$\alpha = 0.10. $$The critical values are $$\pm 1.645. $$What is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis; there is not enough evidence to support a difference.

Question 11

Which of the following is a common misconception when interpreting the results of a two-sample z-test for proportions?

Accepted Answer

A non-significant result proves the two population proportions are equal.

Question 12

In the context of the seat belt study, which of the following would increase the power of the two-sample z-test for proportions?

Accepted Answer

Increasing the sample sizes in both states.

Question 13

Which formula is used to calculate the test statistic $$z$$ for the difference between two proportions?

Accepted Answer

$$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}}$$

Question 14

A researcher wants to compare the effectiveness of two vaccines. Which of the following best describes the type of data and test required?

Accepted Answer

Categorical data; two-sample z-test for proportions.

Question 15

If $$n_1 = 50$, $$\hat{p}_1 = 0.4$$, n_2$ = 60$$, $$\hat{p}_2 = 0.6$$, and $$\alpha = 0.05$$ for a two-tailed test, which of the following is the correct rejection region?

Accepted Answer

$$z  1.96$$

Hypothesis Testing with Two Samples: Two-Sample z-Test for Proportions

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