Q: Which formula gives the upper and lower confidence limits for the difference between two means with independent samples and known standard deviations?

$$\text{UCL}_{\bar{x}_1 - \bar{x}_2} = (\bar{x}_1 - \bar{x}_2) + z_{\alpha/2} \$$sigma_{\bar{x}$$_1 - \bar{x}_2}$$ and $$\text{LCL}_{\bar{x}_1 - \bar{x}_2} = (\bar{x}_1 - \bar{x}_2) - z_{\alpha/2} \$$sigma_{\bar{x}$$_1 - \bar{x}_2}$$

Question 1

A government agency wants to test if there is a difference in average salaries between federal and private-sector employees. Which of the following is the correct null and alternative hypothesis for this two-population mean test?

Accepted Answer

$$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$ (no difference); H_1$: \mu_1$ - \mu_2$ 
eq 0 ($$difference exists)

Question 2

Given independent samples with known population standard deviations, which formula correctly calculates the standard error of the difference between two means?

Accepted Answer

$$\$$sigma_{\bar{x}$$_1 - \bar{x}_2} = \sqrt{\frac{\$$sigma_1$$^2}{n_1$} + \frac{\$$sigma_2$$^2}{n_2$}}$$

Question 3

A hypothesis test comparing two means with independent samples and known standard deviations yields a z-test statistic of 2.24. If the significance level is $$\alpha = 0.05$$, what is the correct decision?

Accepted Answer

Reject $$H_0$$ because $$|z| = 2.24 \u003e 1.96$$

Question 4

Which formula is used to calculate the z-test statistic for the difference between two means with independent samples and known population standard deviations?

Accepted Answer

$$z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1$ - \mu_2$)_{H_0$}}{\$$sigma_{\bar{x}$$_1 - \bar{x}_2}}$$

Question 5

If the p-value for a two-tail hypothesis test is 0.025 and the significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$ because $$p-$$value $$\u003c \alpha$$

Question 6

Which formula gives the upper and lower confidence limits for the difference between two means with independent samples and known standard deviations?

Accepted Answer

$$	ext{UCL}_{\bar{x}_1 - \bar{x}_2} = (\bar{x}_1 - \bar{x}_2) + z_{\alpha/2} \$$sigma_{\bar{x}$$_1 - \bar{x}_2}$$ and $$	ext{LCL}_{\bar{x}_1 - \bar{x}_2} = (\bar{x}_1 - \bar{x}_2) - z_{\alpha/2} \$$sigma_{\bar{x}$$_1 - \bar{x}_2}$$

Question 7

When comparing two population means with independent samples and unknown but equal variances, which formula is used to calculate the pooled variance?

Accepted Answer

$$s_p^2$ = \frac{(n_1$ - 1)s_1$^2 + (n_2$ - 1)s_2$^2}{(n_1$ - 1) + (n_2$ - 1)}$$

Question 8

A clinical trial tests whether a new drug is better than a placebo. Which is the correct null hypothesis?

Accepted Answer

$$H_0$$: There is no difference between the new treatment and the placebo.

Question 9

In a matched-pair (dependent samples) hypothesis test, what is the formula for the matched-pair difference?

Accepted Answer

$$d = x_1$ - x_2$

Question 10

Which formula is used to calculate the mean of the matched-pair differences in a dependent samples test?

Accepted Answer

$$\bar{d} = \frac{\sum_{i=1}^n d_i}{n}$$

Question 11

A store manager wants to test if end-aisle displays result in higher sales than middle-aisle displays using a matched-pair t-test. Which hypothesis statement is correct?

Accepted Answer

$$H_0$$: \mu_d = 0$$ (no difference); H_1$: \mu_d \u003e 0 ($$end-aisle sales are higher)

Question 12

Which formula is used to calculate the t-test statistic for a matched-pair hypothesis test?

Accepted Answer

$$t = \frac{\bar{d} - (\mu_d)_{H_0$}}{s_d / \sqrt{n}}$$

Question 13

When comparing two population proportions with independent samples, which formula is used to estimate the standard error for the difference in proportions?

Accepted Answer

$$\sqrt{\frac{p_1$(1-p_1$)}{n_1$} + \frac{p_2$(1-p_2$)}{n_2$}}$$

Question 14

A bank wants to compare the proportion of young customers who pay bills online to older customers. Which hypothesis statement is correct for a two-tail test?

Accepted Answer

$$H_0$$: $$p_1$$ - $$p_2 = 0$; H_1$: p_1$ - p_2$ 
eq 0$$

Question 15

In a hypothesis test comparing two proportions, what is the formula for the pooled estimate of the overall proportion?

Accepted Answer

$$\hat{p} = \frac{x_1$ + x_2$}{n_1$ + n_2$}$$

Question 16

Which formula is used to calculate the z-test statistic for the difference between two proportions?

Accepted Answer

$$z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1$ - p_2$)_{H_0$}}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}}$$

Hypothesis Tests Comparing Two Populations: Means and Proportions

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