Question 1

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

Accepted Answer

A sampling method in which the selection of an individual for one sample does not affect the selection of individuals for the other sample.

Question 2

Suppose a researcher wants to test whether the proportion of people who support a new policy differs between two cities. Which hypothesis test should be used?

Accepted Answer

A two-sample z-test for proportions

Question 3

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

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

A psychologist wants to compare the mean reaction times of two groups: those given a stimulus and those not given a stimulus. Which test is most appropriate if the samples are independent?

Accepted Answer

A two-sample t-test for means

Question 5

When constructing a $$(1-\alpha) 	imes 100\%$$ confidence interval for the difference between two population proportions, which formula is used?

Accepted Answer

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

Question 6

In a matched-pairs design, which formula is used to construct a confidence interval for the population mean difference?

Accepted Answer

$$\bar{d} \pm t_{\alpha/2} \frac{s_d}{\sqrt{n}}$$

Question 7

A researcher wants to estimate the difference in proportions of males and females who support a new law. She has no prior estimates. What value should she use for $$p_1$$ and $$p_2$$ when calculating sample size?

Accepted Answer

$$0.5$$ for both $$p_1$$ and $$p_2$$

Question 8

Which of the following is a requirement for using the normal approximation for the sampling distribution of the difference between two sample proportions?

Accepted Answer

Both $$n_1$$ \hat{p}_1 \geq 5$$ and n_1$ (1-\hat{p}_1) \geq 5$$, and both $$n_2$$ \hat{p}_2 \geq 5$$ and n_2$ (1-\hat{p}_2) \geq 5$$

Question 9

In the context of hypothesis testing for the difference between two independent means, which test statistic is used when population variances are assumed equal?

Accepted Answer

$$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1$ - \mu_2$)}{s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}}}$$

Question 10

A quality control manager wants to compare the mean time spent in line at two different amusement parks. Which assumption must be met to use the pooled variance t-test?

Accepted Answer

The population variances are equal.

Question 11

Which scenario best describes a matched-pairs design?

Accepted Answer

Measuring blood pressure before and after medication in the same group of patients.

Question 12

A researcher is testing $$H_0$$: $$p_1$$ = $$p_2$$ versus $$H_a: p_1$ 
eq p_2$ at $$\alpha = 0.05$$. If the calculated z-statistic is 2.10$, what is the correct conclusion?

Accepted Answer

Reject $$H_0 if$$ $$|z| \u003e 1.96$$; thus, reject $$H_0.$$

Question 13

When constructing a confidence interval for the difference between two independent means, which formula is used if population variances are not assumed equal?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 14

A medical researcher wants to know if a new drug reduces recovery time compared to a placebo. She randomly assigns patients to two groups. What type of samples are these?

Accepted Answer

Independent samples

Question 15

Which of the following is NOT a requirement for using a two-sample t-test for the difference between two means?

Accepted Answer

The samples must be dependent.

Question 16

A poll asks whether people favor capital punishment. To test if there is a significant difference in proportions between two age groups, which test should be used?

Accepted Answer

A two-sample z-test for proportions

Question 17

In a matched-pairs t-test, what is the null hypothesis?

Accepted Answer

$$H_0$$: \mu_d = 0$$

Question 18

A researcher wants to determine the sample size needed to estimate the difference in proportions with a specified margin of error $$E. $$Which formula should be used if no prior estimates are available?

Accepted Answer

$$n = \frac{1}{4} \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$

Question 19

Which of the following statements about the robustness of t-procedures is correct?

Accepted Answer

T-procedures are robust to moderate departures from normality, especially when sample sizes are large.

Inference on Two Population Parameters: Proportions and Means

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