Question 1

Which of the following best describes the difference between independent and dependent sampling when comparing two population means?

Accepted Answer

Independent sampling means the selection of individuals in one group does not influence the selection in the other group, while dependent sampling (matched pairs) means the selection in one group is related to the other.

Question 2

A researcher wants to compare the mean blood pressure of patients before and after a new treatment. Which statistical test is most appropriate?

Accepted Answer

Paired t-test for dependent samples

Question 3

Suppose you have two independent samples with sample means $$\bar{y}_1$$ and $$\bar{y}_2$$, and sample standard deviations $$s_1$$ and $$s_2$$, with sample sizes $$n_1$$ and $$n_2. $$What is the formula for the standard error of the difference in means?

Accepted Answer

$$SE = \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 4

Which of the following is NOT an assumption required for the two-sample t-test for independent samples?

Accepted Answer

The variances of the two populations must be equal.

Question 5

A shoe manufacturer wants to test if the average left foot is longer than the average right foot. Which sampling method is more appropriate and why?

Accepted Answer

Measure left and right feet of the same individuals; this is dependent sampling and controls for individual variation.

Question 6

Given the following summary statistics for two independent samples: $$n_1 = 14$, $$\bar{y}_1 = 7.88$$, s_1$ = 1.02$$; $$n_2 = 14$, $$\bar{y}_2 = 8.43$$, s_2$ = 1.01. $$What is the value of the test statistic for testing $$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$?

Accepted Answer

$$t = \frac{7.88 - 8.43}{\sqrt{\frac{1.02^2$}{14} + \frac{1.01^2$}{14}}} = -1.44$$

Question 7

If the 95% confidence interval for the difference in means (Group 1 - Group 2) is $$(-1.34, 0.24)$$, what can you conclude?

Accepted Answer

Zero is contained in the interval, so there is no significant difference between the means.

Question 8

When is it appropriate to use the pooled t-test instead of the standard two-sample t-test?

Accepted Answer

When the variances of the two populations are known to be equal and both groups are normally distributed.

Question 9

What is the formula for the pooled variance $$s_{pooled}^2$$ when assuming equal variances for two samples?

Accepted Answer

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

Question 10

A clinical trial randomly assigns 430 subjects to receive either a new medication or a placebo. Which type of sampling is this, and which test should be used to compare mean outcomes?

Accepted Answer

Independent sampling; use a two-sample t-test.

Question 11

Which of the following is a correct null hypothesis for a two-sample t-test comparing means?

Accepted Answer

$$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$

Question 12

In the context of the concrete strength example, if the pooled standard deviation is $$s_{pooled} = 467.0$$, $$n_1 = 10$, n_2$ = 9$$, $$\bar{y}_1 = 4483$$, $$\bar{y}_2 = 3669$$, what is the value of the test statistic for $$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$?

Accepted Answer

$$t = \frac{4483 - 3669}{467.0 \sqrt{\frac{1}{10} + \frac{1}{9}}} = 3.80$$

Question 13

Which of the following is a reason to avoid using the pooled t-test unless the equal variance assumption is justified?

Accepted Answer

If the variances are not equal, the pooled t-test can give misleading results.

Question 14

A 95% confidence interval for the difference in means is calculated as $$\bar{y}_1 - \bar{y}_2 \pm t_{\alpha/2, df} 	imes SE. $$What does it mean if this interval does NOT contain zero?

Accepted Answer

There is evidence of a significant difference between the two population means at the 5% significance level.

Question 15

Which of the following is the correct formula for determining the sample size needed to estimate the difference in two means with margin of error $$ME$$ and confidence level $$1-\alpha$$?

Accepted Answer

$$n = \frac{(z_{\alpha/2})^2$ (s_1$^2 + s_2$^2)}{ME^2$}$$

Question 16

Which of the following is a potential pitfall when using two-sample t methods?

Accepted Answer

Using two-sample methods when the samples are not independent.

Question 17

In a randomized experiment, why might it be reasonable to assume equal variances between groups?

Accepted Answer

Randomization tends to produce groups with similar population variances.

Comparing Means for Independent Samples: Two-Sample t-Tests and Pooled t-Tests

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