Question 1

A pharmaceutical company wants to test the effectiveness of a new drug. They measure blood pressure in the same group of patients before and after taking the drug. Which type of confidence interval should be used to estimate the mean difference in blood pressure?

Accepted Answer

Confidence interval for the mean difference in dependent (paired) samples

Question 2

Which formula gives the confidence interval for the mean difference in paired (dependent) samples?

Accepted Answer

$$ \bar{d} \pm t_{n-1,\alpha/2} \frac{s_d}{\sqrt{n}} $$

Question 3

A company wants to compare the average salaries of employees in two unrelated departments. Which assumptions must be met to use the confidence interval formula for independent samples with known variances?

Accepted Answer

Samples are randomly and independently drawn, both populations are normal, and population variances are known

Question 4

Given two independent samples with unknown but assumed equal variances, what is the formula for the pooled variance $$s_p^2$?

Accepted Answer

$$ s_p^2$ = \frac{(n_x-1)s_x^2$ + (n_y-1)s_y^2$}{n_x + n_y - 2} $$

Question 5

A tech company tests two types of processors for speed. CPU1: $$n_1=17$, $$\bar{x}_1=3004$$, s_1$=74. $$CPU2: $$n_2=14$, $$\bar{x}_2=2538$$, s_2$=56. $$Assuming equal variances, what is the degrees of freedom for the t-distribution used in the confidence interval?

Accepted Answer

$$ n_1$ + n_2$ - 2 = 29 $$

Question 6

Which formula is used to calculate the confidence interval for the difference between two independent means when population variances are known?

Accepted Answer

$$ (\bar{x} - \bar{y}) \pm z_{\alpha/2} \sqrt{\frac{\sigma_x^2$}{n_x} + \frac{\sigma_y^2$}{n_y}} $$

Question 7

A researcher wants to compare the proportion of men and women who have college degrees. In a random sample, 26 of 50 men and 28 of 40 women have degrees. What is the point estimate for the difference in proportions?

Accepted Answer

$$ 0.52 - 0.70 = -0.18 $$

Question 8

Which formula gives the standard error for the difference between two population proportions?

Accepted Answer

$$ \sqrt{\frac{\hat{p}_x(1-\hat{p}_x)}{n_x} + \frac{\hat{p}_y(1-\hat{p}_y)}{n_y}} $$

Question 9

In which scenario is it appropriate to use the confidence interval formula for the difference between two independent means with unknown and unequal variances?

Accepted Answer

Comparing the average sales of two unrelated stores with unknown and unequal variances

Question 10

A business analyst wants to estimate the difference in mean customer satisfaction scores between two independent branches, assuming population variances are unknown and unequal. Which formula should be used for the confidence interval?

Accepted Answer

$$ (\bar{x} - \bar{y}) \pm t_{v,\alpha/2} \sqrt{\frac{s_x^2$}{n_x} + \frac{s_y^2$}{n_y}} $$

Question 11

Which of the following is NOT an assumption for constructing a confidence interval for the difference between two population proportions?

Accepted Answer

Population variances are known

Question 12

A company wants to know if a new training program increases employee productivity. They measure productivity before and after training for each employee. What is the point estimate for the mean difference?

Accepted Answer

$$ \sum_{i=1}^n d_i / n $$ where $$d_i = x_i - y_i$$

Question 13

If the confidence interval for the difference between two means includes zero, what can be concluded?

Accepted Answer

There is not enough evidence to conclude the means are different

Question 14

A market researcher wants to compare the proportion of customers who prefer Brand A versus Brand B. In a sample of 100 customers, 60 prefer Brand A and 40 prefer Brand B. What is the point estimate for the difference in proportions?

Accepted Answer

$$ 0.60 - 0.40 = 0.20 $$

Question 15

Which formula is used to calculate the margin of error for the confidence interval of the mean difference in paired samples?

Accepted Answer

$$ t_{n-1,\alpha/2} \frac{s_d}{\sqrt{n}} $$

Question 16

A hospital compares recovery times for two unrelated treatments. The sample sizes are small, and population variances are unknown and unequal. Which t-distribution degrees of freedom formula should be used?

Accepted Answer

$$ v = \frac{\left(\frac{s_x^2$}{n_x} + \frac{s_y^2$}{n_y}\$$right)^2$$}{\frac{\left(\frac{s_x^2$}{n_x}\$$right)^2$$}{n_x-1} + \frac{\left(\frac{s_y^2$}{n_y}\$$right)^2$$}{n_y-1}} $$

Question 17

Which of the following best describes the purpose of forming a confidence interval for the difference between two population means?

Accepted Answer

To estimate, with a specified level of confidence, the range in which the true difference between two population means lies

Estimation: Additional Topics – Confidence Intervals for Differences in Means and Proportions

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