Q: A researcher wants to test if the variation in weights of pennies before 1983 is greater than after 1983. The sample statistics are: Before 1983: $$n = 35$$, $$s = 0.03910 g $$After 1983: $$n = 37$$, $$s = 0.01648 g $$What is the correct null hypothesis for this F-test?

$$H_0$$: \$$sigma^2$$_{\text{before}} = \$$sigma^2$$_{\text{after}}$$

Question 1

Which of the following scenarios best describes a situation where a paired (dependent) samples t-test should be used?

Accepted Answer

Comparing the cholesterol levels of patients before and after taking a new medication.

Question 2

Given the following matched pairs data for measured and reported weights (in pounds):
Measured: 152.6, 149.3, 174.8, 119.5, 194.9, 180.3, 215.4, 239.6
Reported: 150, 148, 170, 119, 185, 180, 224, 239
What is the sample mean of the differences $$\bar{d} ($$measured - reported)?

Accepted Answer

$$\bar{d} = 1.425$$

Question 3

What is the correct formula for the test statistic in a paired samples t-test?

Accepted Answer

$$t = \frac{\bar{d} - \mu_d}{s_d/\sqrt{n}}$$

Question 4

Suppose you conduct a paired t-test and obtain a confidence interval for $$\mu_d$$ that contains 0. What does this imply?

Accepted Answer

There is not a significant difference between the two population means.

Question 5

Which of the following is a requirement for using the paired samples t-test?

Accepted Answer

The differences are from a population that is approximately normal, or $$n \u003e 30.$$

Question 6

In the context of testing two variances using the F-test, why must the sample with the larger variance always be placed in the numerator?

Accepted Answer

To ensure the test statistic $$F is $$always greater than or equal to 1, allowing use of the right tail of the F distribution.

Question 7

Given the following sample statistics for two populations:
Population 1: $$n_1 = 12$, s_1$ = 12.4194$$
Population 2: $$n_2 = 10$, s_2$ = 15.5306$$
What is the value of the F test statistic for comparing variances?

Accepted Answer

$$F = \frac{15.5306^2$}{12.4194^2$} = 1.563$$

Question 8

Which Excel function would you use to find the P-value for a right-tailed F-test with test statistic $$F$$, numerator degrees of freedom $$df_1$$, and denominator degrees of freedom $$df_2$$?

Accepted Answer

F.DIST.RT(F, df1, df2)

Question 9

A researcher wants to test if the variation in weights of pennies before 1983 is greater than after 1983. The sample statistics are:
Before 1983: $$n = 35$$, $$s = 0.03910 g
$$After 1983: $$n = 37$$, $$s = 0.01648 g
$$What is the correct null hypothesis for this F-test?

Accepted Answer

$$H_0$$: \$$sigma^2$$_{	ext{before}} = \$$sigma^2$$_{	ext{after}}$$

Question 10

If the calculated F statistic is close to 1 in a test comparing two variances, what does this suggest?

Accepted Answer

There is evidence in favor of the null hypothesis that the variances are equal.

Question 11

A study compares the heights of fathers and their first sons using paired data. Which is the correct null hypothesis for this paired t-test?

Accepted Answer

$$H_0$$: \mu_d = 0$$

Question 12

Which of the following statements about the F distribution is correct?

Accepted Answer

It is a continuous, asymmetric distribution that is always nonnegative and skewed to the right.

Question 13

Suppose you are testing $$H_0$$: \$$sigma_1^2$$ = \$$sigma_2^2$ using the F-test. If your calculated F statistic is much larger than the critical value, what is the appropriate conclusion?

Accepted Answer

Reject $$H_0$$; there is evidence that the variances are different.

Question 14

Which of the following is NOT a requirement for using the F-test to compare two variances?

Accepted Answer

The two samples must be dependent.

Question 15

A confidence interval for the mean difference $$\mu_d is $$calculated as $$\bar{d} \pm E. If$$ $$E = 2.5$$ and $$\bar{d} = 1.4$$, what is the confidence interval?

Accepted Answer

$$-1.1 \u003c \mu_d \u003c 3.9$$

Question 16

Which of the following best describes the relationship between the P-value method, critical value method, and confidence interval method in hypothesis testing for two means from dependent samples?

Accepted Answer

All three methods yield the same conclusion.

Hypothesis Testing for Two Means (Dependent Samples) and Two Variances

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