Question 1

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

Accepted Answer

Samples drawn from two different populations, with no relationship between the individuals in each sample.

Question 2

A researcher wants to compare the proportion of students who pass a lab-based course versus a lecture-based course. She randomly selects students from each course. Which type of sampling is this?

Accepted Answer

Independent sampling, because the students are selected randomly from each course without pairing.

Question 3

Which of the following is a necessary condition for the sampling distribution of $\hat{p}_1 - \hat{p}_2$ to be approximately normal?

Accepted Answer

$n_1 p_1 > 5$, $n_1 (1-p_1) > 5$, $n_2 p_2 > 5$, $n_2 (1-p_2) > 5$

Question 4

Suppose you want to test the hypothesis $H_0: p_1 = p_2$ versus $H_a: p_1 
eq p_2$. Which test statistic should you use?

Accepted Answer

$z = \dfrac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\dfrac{1}{n_1} + \dfrac{1}{n_2})}}$

Question 5

Which of the following is NOT a step in testing a hypothesis about the difference between two population proportions using independent samples?

Accepted Answer

Calculate the pooled sample variance.

Question 6

A clinical trial compares the proportion of patients experiencing side effects in a treatment group and a control group. Which formula gives the standard error for the difference in sample proportions?

Accepted Answer

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

Question 7

Which of the following is the correct formula for a $(1-\alpha) 	imes 100\%$ confidence interval for $p_1 - p_2$ using independent samples?

Accepted Answer

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

Question 8

Which of the following is a condition that must be met to construct a valid confidence interval for the difference between two population proportions?

Accepted Answer

Each sample must be a simple random sample from its population.

Question 9

A political scientist wants to estimate the difference in the proportion of Democrats and Republicans who trust the state government. If no prior estimates are available, which value should be used for $p_1$ and $p_2$ when determining the required sample size?

Accepted Answer

$0.5$

Question 10

Which formula gives the minimum sample size required for each group to estimate $p_1 - p_2$ with margin of error $E$ and confidence level $(1-\alpha) 	imes 100\%$, assuming no prior estimates?

Accepted Answer

$n = \dfrac{0.25 \left(z_{\alpha/2}\right)^2}{E^2}$

Question 11

In a clinical trial, 200 out of 1000 patients in the treatment group and 150 out of 1000 in the control group experienced side effects. What is the point estimate for the difference in proportions?

Accepted Answer

$0.20 - 0.15 = 0.05$

Question 12

Which of the following best explains why we use a pooled proportion in hypothesis testing for two population proportions?

Accepted Answer

Because under the null hypothesis, we assume the two population proportions are equal.

Question 13

A Gallup poll surveys 1000 adults in May and 1007 adults in May of the following year about their views on moral values. Which statistical method is appropriate to compare the proportions who say values are 'getting worse'?

Accepted Answer

Construct a confidence interval for the difference between two independent proportions.

Question 14

Which of the following is the correct formula for the margin of error when constructing a confidence interval for $p_1 - p_2$?

Accepted Answer

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

Question 15

Which of the following scenarios would require the use of dependent samples rather than independent samples?

Accepted Answer

Comparing the proportion of customers who prefer a product before and after a marketing campaign.

Section 11.1 Inference about Two Population Proportions: Independent Samples

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