Q: Which formula is used to test the equality of two population variances?

$$F_{STAT} = \frac{S_1$^2}{S_2$^2}$$,$$\text{ where the larger variance is the numerator}$$

Question 1

Which of the following formulas correctly represents the pooled variance $$S_p^2$ for two independent samples with 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 2

A company wants to compare the mean productivity of two independent departments. Which test statistic should be used if the population variances are unknown but assumed equal?

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 3

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

Accepted Answer

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

Question 4

Which of the following is the correct formula for the confidence interval for the difference in means $$\mu_1$ - \mu_2$ for two independent samples with equal variances?

Accepted Answer

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

Question 5

A researcher wants to test if a training program reduced the number of customer complaints using before-and-after data for each employee. Which test is most appropriate?

Accepted Answer

Paired sample t-test

Question 6

Given the following paired differences: $$D_1 = -2$, D_2$ = -14$$, $$D_3 = -1$, D_4$ = 0$$, $$D_5 = -4$, what is the mean paired difference $$\overline{D}$$?

Accepted Answer

$$\overline{D} = \frac{-2 + (-14) + (-1) + 0 + (-4)}{5} = -4.2$$

Question 7

Which formula is used to test the equality of two population variances?

Accepted Answer

$$F_{STAT} = \frac{S_1$^2}{S_2$^2}$$,$$	ext{ where the larger variance is the numerator}$$

Question 8

In a test for the difference between two proportions, what is the formula for the pooled proportion $$\overline{p}$$?

Accepted Answer

$$\overline{p} = \frac{X_1$ + X_2$}{n_1$ + n_2$}$$

Question 9

Which of the following is the correct test statistic for comparing two population proportions?

Accepted Answer

$$Z_{STAT} = \frac{(p_1$ - p_2$) - (\pi_1$ - \pi_2$)}{\sqrt{\overline{p}(1-\overline{p})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}}$$

Question 10

A financial analyst wants to test if there is a difference in dividend yield between NYSE and NASDAQ stocks. What is the null hypothesis for this two-sample test?

Accepted Answer

$$H_0$$: \mu_{NYSE} = \mu_{NASDAQ}$$

Question 11

When conducting an F-test for equality of variances, what are the degrees of freedom for the numerator and denominator?

Accepted Answer

$$df_1$$ = $$n_1 - 1$, df_2$ = n_2$ - 1$$

Question 12

Which of the following scenarios is most appropriate for a paired sample t-test?

Accepted Answer

Measuring the effect of a diet by comparing weights before and after the diet for the same individuals.

Question 13

If the calculated $$t_{STAT}$$ for a two-sample test is greater than the critical value at $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

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

Question 14

Which formula gives 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 15

A business wants to compare the proportion of customers satisfied with service before and after a policy change using the same group of customers. Which test should be used?

Accepted Answer

Paired sample t-test

Question 16

Which formula is used to calculate the confidence interval for the mean paired difference $$\mu_D$$?

Accepted Answer

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

Two-Sample Tests in Business Statistics: Means, Proportions, and Variances

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