Question 1

Which of the following best describes the difference between independent and dependent samples in hypothesis testing?

Accepted Answer

Independent samples are those where the data from one sample does not affect the data from the other sample, while dependent samples involve paired or related data.

Question 2

A researcher wants to compare the effectiveness of two teaching methods by analyzing the final exam scores of students from two different classes, each taught with a different method. Which type of samples is this scenario using?

Accepted Answer

Independent samples

Question 3

Which formula is used to calculate the test statistic for the difference between two population proportions ($$p_1$$ and $$p_2) $$when testing $$H_0$$: $$p_1$$ = $$p_2$$?

Accepted Answer

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

Question 4

In a study, a group of patients' blood pressure is measured before and after a new medication is administered. Which statistical test is most appropriate to determine if the medication had a significant effect?

Accepted Answer

Paired (dependent) samples t-test

Question 5

Which of the following is the correct formula for the test statistic in a paired (dependent) samples t-test?

Accepted Answer

$$t = \frac{\bar{d}}{s_d / \sqrt{n}}$$

Question 6

A confidence interval for the difference between two independent means ($$\mu_1$ - \mu_2$) is constructed as follows:

Accepted Answer

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

Question 7

When testing the equality of two population standard deviations, which distribution is used for the test statistic?

Accepted Answer

F-distribution

Question 8

Suppose you are testing $$H_0$$: \$$mu_1$$ = \$$mu_2$$ versus $$H_1$$: \$$mu_1$$ 
eq \$$mu_2$$ using two independent samples with unknown but equal variances. Which test should you use?

Accepted Answer

Pooled two-sample t-test

Question 9

A pharmaceutical company wants to estimate the difference in the proportion of patients experiencing side effects between two drugs. Which formula should they use to determine the required sample size for each group to estimate the difference within a margin of error $$E at $$confidence level $$1-\alpha$$?

Accepted Answer

$$n = \frac{(z_{\alpha/2})^2$ (p_1$(1-p_1$) + p_2$(1-p_2$))}{E^2$}$$

Question 10

In the context of hypothesis testing for two means, which of the following statements is TRUE regarding the use of the pooled two-sample t-test?

Accepted Answer

It is used when the population variances are assumed to be equal but unknown.

Question 11

A researcher is comparing the variances of two manufacturing processes. The sample variances are $$s_1^2 = 25$ and s_2$^2 = 16$$ with sample sizes $$n_1 = 10$ and n_2$ = 12. $$What is the value of the F test statistic?

Accepted Answer

$$F = \frac{25}{16} = 1.5625$$

Question 12

Which of the following is NOT a step in the traditional hypothesis testing procedure for two samples?

Accepted Answer

Calculating the population mean

Question 13

A group of students takes a practice test, then receives tutoring, and takes a second test. The mean difference in scores is $$\bar{d} = 5$$, the standard deviation of differences is $$s_d = 8$$, and $$n = 16. $$What is the value of the test statistic for testing if the tutoring had an effect?

Accepted Answer

$$t = \frac{5}{8/\sqrt{16}} = \frac{5}{2} = 2.5$$

Question 14

If the p-value for a two-sample hypothesis test is 0.03 and the significance level $$\alpha is 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 15

Which of the following scenarios would require the use of the F-distribution?

Accepted Answer

Testing the equality of two population variances

Question 16

A 95% confidence interval for the difference in means between two independent samples is calculated to be (2.1, 5.3). What does this interval suggest?

Accepted Answer

The first mean is likely between 2.1 and 5.3 units higher than the second mean.

Question 17

Which of the following is a correct interpretation of a left-tailed test in the context of comparing two means?

Accepted Answer

The alternative hypothesis is $$H_1$$: \$$mu_1$$ - \$$mu_2$$ \u003c 0$$

Question 18

A study compares the effectiveness of two diets by measuring weight loss in two independent groups. The sample means are $$\bar{x}_1 = 8 kg$$, $$\bar{x}_2 = 5 kg$$, $$s_1 = 2$ kg, s_2$ = 3 kg$$, $$n_1 = 20$, n_2$ = 25. $$What is the standard error for the difference in means?

Accepted Answer

$$\sqrt{\frac{2^2$}{20} + \frac{3^2$}{25}} = \sqrt{0.2 + 0.36} = \sqrt{0.56} \approx 0.748$$

Question 19

Which of the following statements about confidence intervals for the difference between two proportions is correct?

Accepted Answer

The confidence interval is constructed as $$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}.$$

Inference for Two Samples: Proportions, Means, and Variances

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