Question 1

Which of the following conditions must be met to use the Central Limit Theorem (CLT) for the sampling distribution of the difference in means between two independent samples?

Accepted Answer

Both sample sizes must be at least 30: $$n_1$$ \geq 30$$ and n_2$ \geq 30.$$

Question 2

Suppose you are comparing the mean percent change in polyphenol levels for coffee drinkers and non-coffee drinkers. Which of the following is the correct formula for the standard error of the difference in sample means?

Accepted Answer

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

Question 3

A researcher wants to construct a 95% confidence interval for the difference in means between two groups. The calculator output shows the interval as $$(-3.958, -2.042). $$What does this interval suggest?

Accepted Answer

There is evidence that the mean of group 2 is greater than the mean of group 1.

Question 4

In a two-sample t-test, which of the following is the correct null hypothesis?

Accepted Answer

$$H_0$$: \$$mu_1$$ = \$$mu_2$$

Question 5

Given the histogram for the 'Coffee' group, which of the following best describes the shape of the distribution?

Accepted Answer

Roughly symmetric with no strong skew or outliers.

Question 6

If the sample sizes for both groups are less than 30, which condition must be checked before using a two-sample t-procedure?

Accepted Answer

Both populations should be approximately normal.

Question 7

A two-sample t-test for the difference in means yields a p-value of 0.0716. At a significance level of $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis; there is not convincing evidence of a difference.

Question 8

Which calculator function would you use to construct a confidence interval for the difference in means between two independent samples?

Accepted Answer

2-SampTInt

Question 9

In the context of the calcium and placebo experiment, what does a negative value in the response variable (decrease in systolic blood pressure) indicate?

Accepted Answer

An increase in systolic blood pressure.

Question 10

Which of the following is the correct formula for the degrees of freedom (df) using the conservative method for a two-sample t-test?

Accepted Answer

$$df = \min(n_1$ - 1, n_2$ - 1)$$

Question 11

A 90% confidence interval for the difference in mean percent change in polyphenol levels between coffee and no coffee groups is $$(2.59, 6.89). $$What does this interval suggest?

Accepted Answer

Coffee is more effective at increasing polyphenol levels.

Question 12

Which of the following best describes the effect of using the conservative method for degrees of freedom in a two-sample t-interval?

Accepted Answer

It increases the margin of error, making the interval wider.

Question 13

Which of the following is NOT an assumption required for a two-sample t-test for the difference in means?

Accepted Answer

The samples must be dependent.

Question 14

Given the calculator output for a one-sample t-interval: $$\bar{x} = 15$$, $$S_x = 2.566$$, $$n = 35$$, and a 95% confidence interval of $$(14.119, 15.881)$$, what is the correct interpretation?

Accepted Answer

We are 95% confident that the true mean lies between 14.119 and 15.881.

Question 15

If a two-sample t-test yields a p-value much less than 0.05, what is the most appropriate conclusion?

Accepted Answer

Reject the null hypothesis; there is convincing evidence of a difference in means.

Question 16

Which of the following best describes the histogram for the 'Calcium Group'?

Accepted Answer

The distribution is left-skewed.

Question 17

When using a two-sample t-test, what is the effect of having unequal sample sizes and variances?

Accepted Answer

You should not pool the variances and use the formula for unequal variances.

Question 18

Which calculator function would you use to test if a single sample mean is different from a hypothesized value?

Accepted Answer

T-Test

Question 19

If the histogram for a sample shows no strong skew or outliers, what does this suggest about the use of t-procedures?

Accepted Answer

It is appropriate to use t-procedures even if the sample size is less than 30.

Question 20

A two-sample t-test is performed and the output shows $$t = -6.30$$ and $$p = 8.99 	imes 10$$^{-8}$$. What does this result indicate?

Accepted Answer

There is extremely strong evidence against the null hypothesis.

Inference for a Difference in Means: Two-Sample T Procedures

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