Question 1

Which of the following best describes independent sampling in the context of comparing two population proportions?

Accepted Answer

Individuals selected for one sample do not influence which individuals are selected for the second sample.

Question 2

A researcher wants to test if the proportion of students passing a lab-based math course is higher than those passing a lecture-based course. Which hypothesis structure should be used?

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$ versus $$H_A: p_1$ \u003e p_2$

Question 3

Given two independent samples with sample proportions $$\hat{p}_1$$ and $$\hat{p}_2$$, what is the formula for the standard error of the difference between the proportions?

Accepted Answer

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

Question 4

In the Nasonex clinical trial example, what is the pooled estimate $$\hat{p}$$ used for the hypothesis test?

Accepted Answer

$$\hat{p} = \frac{547 + 368}{2103 + 1671}$$

Question 5

Which of the following is NOT a requirement for constructing a confidence interval for the difference between two population proportions?

Accepted Answer

The population proportions must be equal.

Question 6

A political scientist wants to estimate the difference in proportions between Democrats and Republicans who trust state government, with a margin of error $$E = 0.03$$ and 95% confidence. If prior estimates are $$\hat{p}_1 = 0.62$$ and $$\hat{p}_2 = 0.55$$, what is the sample size formula?

Accepted Answer

$$n = \left[ \hat{p}_1(1-\hat{p}_1) + \hat{p}_2(1-\hat{p}_2) \right] \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$

Question 7

Suppose you want to construct a 90% confidence interval for $$p_1$$ - $$p_2. $$What is the value of $$z_{\alpha/2}$$ you should use?

Accepted Answer

$$z_{\alpha/2} = 1.645$$

Question 8

In the context of hypothesis testing for two proportions, what does a P-value of 0.002 indicate if the significance level is $$\alpha = 0.05$$?

Accepted Answer

There is strong evidence to reject the null hypothesis.

Question 9

Which formula gives the lower bound for a $$(1-\alpha)	imes 100\%$$ confidence interval for $$p_1$$ - $$p_2$$?

Accepted Answer

$$(\hat{p}_1 - \hat{p}_2) - z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 10

A study compares prices of 20 products sold at both Amazon and Walmart. Which sampling method is used?

Accepted Answer

Dependent (matched-pairs) sampling

Question 11

If the sample sizes for two groups are both less than 5% of their respective population sizes, why is this important for inference?

Accepted Answer

It ensures the independence necessary for a binomial experiment.

Question 12

In a two-proportion z-test, which of the following is the correct formula for the test statistic using the pooled estimate?

Accepted Answer

$$z_0$$ = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{$$n_1$$} + \frac{1}{$$n_2$$}\right)}}$$

Question 13

A confidence interval for $$p_1$$ - $$p_2 is $$calculated as (0.077, 0.141). What does it mean if this interval does not contain 0?

Accepted Answer

There is evidence of a significant difference between the two population proportions.

Question 14

Which of the following is a correct interpretation of a 90% confidence interval for the difference between two proportions?

Accepted Answer

We are 90% confident that the true difference between the population proportions lies within the interval.

Question 15

If prior estimates of $$p_1$$ and $$p_2$$ are unavailable, what formula should be used to determine the sample size for a desired margin of error $$E$$?

Accepted Answer

$$n = n_1$ = n_2$ = 0.5 \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$

Question 16

Which of the following scenarios would require the use of a matched-pairs (dependent) sampling method?

Accepted Answer

Comparing blood pressure before and after treatment in the same group of patients.

Question 17

In the classical approach to hypothesis testing for two proportions, what is the decision rule for a right-tailed test?

Accepted Answer

Reject $$H_0 if$$ $$z_0$$ \u003e z_{\alpha}$$

Question 18

A researcher finds $$z_0 = 2.85$ for a right-tailed test with $$\alpha = 0.05$$ and z$$_{0.05} = 1.645$$. What is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is sufficient evidence for a difference.

Inferences from Two Populations: Hypothesis Testing and Confidence Intervals for Two Proportions

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