Q: A company wants to compare the mean sales of two unrelated store locations. The sample sizes are both 10, sample means are 246.4 and 202.3, and sample variances are 42.5420 and 32.5271. What is the pooled sample variance $$S_p^2$?

$$S_p^2$ = \frac{9 \times 42.5420 + 9 \times 32.5271}{18} = 1433.9167$$

Question 1

A business analyst wants to compare the average monthly sales between two independent store locations. The population variances are unknown but assumed equal. Which test should the analyst use?

Accepted Answer

A pooled-variance t test

Question 2

Which of the following is the correct formula for the pooled sample variance $$S_p^2$ when comparing two independent samples with unknown but equal variances?

Accepted Answer

$$S_p^2$ = \frac{(n_1$ - 1)S_1$^2 + (n_2$ - 1)S_2$^2}{n_1$ + n_2$ - 2}$$

Question 3

A researcher is testing $$H_0$$: \$$mu_1$$ = \$$mu_2$$ versus $$H_1$$: \$$mu_1$$ 
eq \$$mu_2$$ using a pooled-variance t test. The calculated $$t_{STAT} is 2.6041 $$and the critical value at $$\alpha = 0.05 is 2.1009. $$What is the correct conclusion?

Accepted Answer

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

Question 4

Which of the following is an assumption for the pooled-variance t test?

Accepted Answer

The population variances are unknown but assumed equal.

Question 5

Given two independent samples with $$n_1 = 12$, n_2$ = 15$$, what are the degrees of freedom for the pooled-variance t test?

Accepted Answer

$$df = n_1$ + n_2$ - 2 = 25$$

Question 6

A company wants to test if the mean price of a product differs between two stores using paired data (same products at both stores). Which test should be used?

Accepted Answer

A paired difference t test

Question 7

What is the formula for the test statistic in a paired difference t test?

Accepted Answer

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

Question 8

In a two-sample hypothesis test, when should you use the F distribution?

Accepted Answer

When testing for equality of variances between two populations

Question 9

Which of the following correctly describes the rejection region for a two-tailed pooled-variance t test at significance level $$\alpha$$?

Accepted Answer

Reject $$H_0 if$$ $$t_{STAT}  t_{\alpha/2}$$

Question 10

A paired difference test is performed on the prices of 7 items at two stores. The sample mean difference is $$\overline{D} = -3.2781$$, $$S_D = 6.0101$$, and $$n = 7. $$What is the value of the test statistic?

Accepted Answer

$$t_{STAT} = \frac{-3.2781 - 0}{6.0101 / \sqrt{7}} = -1.4471$$

Question 11

Which formula gives the confidence interval for the difference in means for two independent samples with unknown but equal variances?

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 12

When performing a paired difference test, which assumption is required for the validity of the t test?

Accepted Answer

The differences between paired observations are normally distributed

Question 13

A company wants to compare the mean sales of two unrelated store locations. The sample sizes are both 10, sample means are 246.4 and 202.3, and sample variances are 42.5420 and 32.5271. What is the pooled sample variance $$S_p^2$?

Accepted Answer

$$S_p^2$ = \frac{9 	imes 42.5420 + 9 	imes 32.5271}{18} = 1433.9167$$

Question 14

In the context of the F distribution, which sample variance should be placed in the numerator when calculating $$F_{STAT}$$?

Accepted Answer

The larger sample variance

Question 15

Which of the following is the correct formula for the sample standard deviation of paired differences $$S_D$$?

Accepted Answer

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

Question 16

A researcher performs a two-tailed paired difference test and obtains $$t_{STAT} = -1.4471$$ with critical values $$\pm 2.4469 at$$ $$\alpha = 0.05. $$What is the correct decision?

Accepted Answer

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

Question 17

Which of the following is NOT a benefit of using the paired difference test?

Accepted Answer

It requires independent samples.

Question 18

In an F-test for equality of variances, if $$F_{STAT}$$ exceeds the critical value $$F_{\alpha}$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$; the variances are significantly different.

Two-Sample Tests: Hypothesis Testing for Means and Variances

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