Question 1

Which of the following is an assumption required for the pooled-variance t test when comparing two independent sample means?

Accepted Answer

The populations must be normally distributed or both sample sizes must be at least 30.

Question 2

A business analyst wants to compare the average monthly sales between two store locations using independent samples. The sample variances are unknown but assumed equal. Which test should be used?

Accepted Answer

Pooled-variance t test

Question 3

Given two independent samples with $n_1 = 12$, $n_2 = 15$, $S_1^2 = 25$, and $S_2^2 = 36$, what is the pooled variance $S_p^2$?

Accepted Answer

$S_p^2 = \frac{(12-1)25 + (15-1)36}{12+15-2} = 31.47$

Question 4

Which formula correctly calculates the test statistic for the pooled-variance t test for two independent samples?

Accepted Answer

$t_{STAT} = \frac{(\overline{X}_1 - \overline{X}_2) - (\mu_1 - \mu_2)}{\sqrt{S_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$

Question 5

A researcher performs a two-tailed pooled-variance t test and obtains $t_{STAT} = 2.6041$ with a critical value $t_{0.025,18} = 2.1009$. What is the correct conclusion at $\alpha = 0.05$?

Accepted Answer

Reject $H_0$; there is evidence of a difference in means.

Question 6

In a paired difference test, what is the formula for the test statistic $t_{STAT}$?

Accepted Answer

$t_{STAT} = \frac{\overline{D} - \mu_D}{S_D / \sqrt{n}}$

Question 7

Which scenario is most appropriate for using a paired difference test?

Accepted Answer

Comparing the mean blood pressure of patients before and after treatment.

Question 8

A paired difference test yields $t_{STAT} = -1.4471$ and the critical values are $\pm 2.4469$ at $\alpha = 0.05$. What is the correct decision?

Accepted Answer

Do not reject $H_0$; there is insufficient evidence of a difference.

Question 9

What is the formula for the confidence interval for the difference between two means ($\mu_1 - \mu_2$) when variances are unknown but assumed equal?

Accepted Answer

$(\overline{X}_1 - \overline{X}_2) \pm t_{\alpha/2} \sqrt{S_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}$

Question 10

When using the F test to compare two variances, which sample variance should be placed in the numerator?

Accepted Answer

The larger sample variance.

Question 11

A company wants to test if the variance in sales between two departments is different. If $S_1^2 = 50$ and $S_2^2 = 30$, what is the value of $F_{STAT}$?

Accepted Answer

$F_{STAT} = \frac{50}{30} = 1.67$

Question 12

Which of the following best describes the rejection region for a two-tailed F test for equality of variances?

Accepted Answer

Reject $H_0$ if $F_{STAT} > F_{\alpha/2}$ or $F_{STAT} < F_{1-\alpha/2}$.

Question 13

Which of the following is NOT a valid assumption for the paired difference test?

Accepted Answer

The sample sizes must be equal.

Question 14

In the context of two-sample tests, what is the main advantage of using paired samples over independent samples?

Accepted Answer

Paired samples eliminate variation among subjects.

Question 15

A business analyst uses Excel to construct a box plot to check the normality assumption for a pooled-variance t test. What is the purpose of this step?

Accepted Answer

To visually assess whether the data are approximately normally distributed.

Question 16

If the p-value for a two-sample t test is 0.0179 and $\alpha = 0.05$, what is the correct decision?

Accepted Answer

Reject $H_0$; there is evidence of a difference.

Question 17

Which formula gives the sample standard deviation of the paired differences in a paired difference test?

Accepted Answer

$S_D = \sqrt{\frac{\sum_{i=1}^n (D_i - \overline{D})^2}{n-1}}$

Question 18

A company wants to estimate the confidence interval for the mean difference in sales between two locations using paired samples. Which formula should be used?

Accepted Answer

$\overline{D} \pm t_{\alpha/2} \frac{S_D}{\sqrt{n}}$

Two-Sample Tests: Comparing Means and Variances in Business Statistics

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