Question 1

Which of the following is the correct formula for the test statistic when comparing two population proportions using hypothesis testing?

Accepted Answer

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

Question 2

In a randomized controlled trial, 33 infants using insecticide-treated bednets had 15 develop malaria, while 39 infants not using bednets had 27 develop malaria. What is the pooled sample proportion $$\hat{p}$$ used in the hypothesis test?

Accepted Answer

$$\hat{p} = \frac{15 + 27}{33 + 39}$$

Question 3

Which of the following statements about confidence intervals for the difference between two proportions is correct?

Accepted Answer

If the confidence interval for $$(p_1$ - p_2$)$$ does not include 0, we have evidence that $$p_1$$ and $$p_2$$ are different.

Question 4

When conducting a hypothesis test for the difference between two independent means, which formula is used for the test statistic when population variances are unknown and not assumed equal?

Accepted Answer

$$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1$ - \mu_2$)}{\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}}$$

Question 5

In the context of two means hypothesis testing, which scenario describes dependent samples?

Accepted Answer

Sample values from one population are paired with sample values from the other population, such as matched pairs.

Question 6

A researcher wants to test if the mean FICO score for high-interest mortgages is different from that for low-interest mortgages. Which type of test should be used?

Accepted Answer

Two means hypothesis test for independent samples

Question 7

Which formula is used to construct a confidence interval for the difference between two independent means?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm E$$ where $$E = t \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 8

When testing the mean difference between paired values (dependent samples), what is the correct formula for the test statistic?

Accepted Answer

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

Question 9

Which of the following is true about the F-test for comparing two variances?

Accepted Answer

The sampling distribution of $$F = \frac{s_1$^2}{s_2$^2} is $$not symmetric and depends on two different degrees of freedom.

Question 10

A large value of the F statistic in a two variances hypothesis test suggests which of the following?

Accepted Answer

There is evidence that $$\$$sigma_1$$^2$$ and $$\$$sigma_2$$^2$$ are different.

Question 11

In a study comparing the standard deviation of braking distances for 6-cylinder and 8-cylinder cars, which test should be used to determine if the variances are equal?

Accepted Answer

Two variances hypothesis test (F-test)

Question 12

Which of the following is a key assumption for the validity of the F-test for comparing two variances?

Accepted Answer

Both populations must be normally distributed.

Question 13

In the context of two means hypothesis testing, what is the degrees of freedom used when sample sizes are $$n_1$$ and $$n_2$$?

Accepted Answer

The smaller of $$n_1 - 1$ and n_2$ - 1$$

Question 14

A hospital administrator wants to compare the number of admissions from motor vehicle crashes on two Fridays. Which statistical test is most appropriate?

Accepted Answer

Two means hypothesis test for dependent samples (paired t-test)

Question 15

Which formula is used to construct a confidence interval for the mean difference in paired (dependent) samples?

Accepted Answer

$$\bar{d} \pm t \frac{s_d}{\sqrt{n}}$$

Question 16

Which of the following real-world scenarios would require a two proportions hypothesis test?

Accepted Answer

Comparing the proportion of infants developing malaria between those using bednets and those not using bednets.

Question 17

If the confidence interval for the difference between two means includes zero, what does this suggest?

Accepted Answer

There is no evidence of a significant difference between the two means.

Question 18

Which of the following is NOT a scenario where a two means hypothesis test for independent samples is appropriate?

Accepted Answer

Comparing the heights of the same individuals before and after a nutrition program.

Chapter 9: Inferences from Two Samples – Proportions, Means, and Variances

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