Question 1

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

Accepted Answer

Samples where the selection of objects from one population is unrelated to the selection from the other population.

Question 2

A laboratory tests a new type of band for children and compares the mean wearing times to standard braces using two random groups of 20 children each. Which statistical test is most appropriate for analyzing the difference in mean wearing times?

Accepted Answer

Two-sample t-test for independent samples

Question 3

Which condition must be met to validly use the t-distribution for inference about $$\mu_1$ - \mu_2$ with independent samples?

Accepted Answer

Either both populations are normally distributed, or both sample sizes are at least 30.

Question 4

What is the formula for the standard error of the difference between two independent sample means?

Accepted Answer

$$\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 5

In a paired-sample (dependent) design, which formula is used to construct a confidence interval for the mean difference $$\mu_D$$?

Accepted Answer

$$\bar{y}_D \pm t \frac{s_D}{\sqrt{n}}$$

Question 6

A personnel manager wants to compare the mean number of sick days lost between day shift and night shift employees. Ten employees from each shift are randomly selected. What is the parameter of interest?

Accepted Answer

The difference in mean sick days lost between day and night shift employees ($$\mu_1$ - \mu_2$)

Question 7

Which of the following is a correct null hypothesis for a two-sample t-test comparing two population means?

Accepted Answer

$$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$

Question 8

In a study of blood pressure before and after biofeedback training, which statistical method should be used to test if the training reduces blood pressure?

Accepted Answer

Paired-sample t-test (dependent samples)

Question 9

If you reject the null hypothesis when it is actually true, what type of error have you made?

Accepted Answer

Type I error

Question 10

A confidence interval for $$\mu_1$ - \mu_2$ is calculated as $$\bar{y}_1 - \bar{y}_2 \pm t \sqrt{\frac{$$s_1^2$$}{$$n_1$$} + \frac{$$s_2^2$$}{$$n_2$$}}$$. What does it mean if this interval contains zero?

Accepted Answer

There is not enough evidence to conclude a significant difference between the two population means.

Question 11

Which of the following is NOT a required condition for using the t-distribution to compare two independent means?

Accepted Answer

The population standard deviations must be known.

Question 12

In a paired-sample experiment, what is the correct formula for the test statistic to test $$H_0$$: \mu_D = 0$$?

Accepted Answer

$$t = \frac{\bar{y}_D - 0}{s_D / \sqrt{n}}$$

Question 13

A researcher wants to estimate the mean reduction in blood pressure after training. Which confidence interval formula should be used?

Accepted Answer

$$\bar{y}_D \pm t \frac{s_D}{\sqrt{n}}$$

Question 14

Which scenario best describes when to use a paired-sample t-test?

Accepted Answer

When each subject is measured twice, such as before and after a treatment.

Question 15

Suppose you are testing $$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$ at $$\alpha = 0.05$$ and obtain a p-value of 0.03. What is the correct decision?

Accepted Answer

Reject $$H_0$$ because the p-value is less than $$\alpha.$$

Question 16

In the context of comparing two means, what is a Type II error?

Accepted Answer

Failing to reject the null hypothesis when it is actually false.

Question 17

A study compares the mean time to recovery for two treatments using independent samples. The sample sizes are both 25, and the populations are not known to be normal. Is it appropriate to use the t-distribution?

Accepted Answer

No, because the sample sizes are less than 30 and normality is not assumed.

Question 18

Which of the following is a correct alternative hypothesis for a two-sided test comparing two population means?

Accepted Answer

$$H_a: \mu_1$ - \mu_2$ 
eq 0$$

Question 19

In a medical study, patients' cholesterol levels are measured before and after a new drug. What is the population parameter of interest?

Accepted Answer

The mean difference in cholesterol levels before and after treatment ($$\mu_D)$$

Inferences Comparing Two Population Means: Independent and Dependent Samples

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