Q: A researcher wants to estimate the average systolic blood pressure of adults in a city. She collects a random sample of 40 adults and finds a sample mean of 122 mmHg with a sample standard deviation of 15 mmHg. Construct a 95% confidence interval for the population mean. Which of the following is the correct interval?

$$122 \pm 2.021 \times \dfrac{15}{\sqrt{40}}$$

Question 1

A researcher wants to estimate the average systolic blood pressure of adults in a city. She collects a random sample of 40 adults and finds a sample mean of 122 mmHg with a sample standard deviation of 15 mmHg. Construct a 95% confidence interval for the population mean. Which of the following is the correct interval?

Accepted Answer

$$122 \pm 2.021 	imes \dfrac{15}{\sqrt{40}}$$

Question 2

Which of the following best describes the meaning of a 95% confidence interval for a mean?

Accepted Answer

If we repeated the sampling process many times, about 95% of the calculated intervals would contain the true population mean.

Question 3

Which assumption must be checked before performing a one-sample t-test for a mean?

Accepted Answer

The data should be approximately normally distributed or the sample size should be large (Central Limit Theorem applies).

Question 4

A hospital administrator wants to know if the average length of stay for patients has changed from the historical mean of 5 days. A random sample of 25 patients has a mean stay of 5.6 days and a standard deviation of 1.2 days. What is the value of the test statistic for a one-sample t-test?

Accepted Answer

$$t = \dfrac{5.6 - 5}{1.2/\sqrt{25}}$$

Question 5

A one-sample t-test yields a p-value of 0.03. Which of the following is the correct interpretation if the significance level is 0.05?

Accepted Answer

There is sufficient evidence to reject the null hypothesis at the 0.05 significance level.

Question 6

Which of the following is NOT an assumption for constructing a confidence interval for the difference between two means using a two-sample t-test?

Accepted Answer

The population variances are equal.

Question 7

A clinical trial compares the mean recovery times of two drugs. Group A (n=30) has a mean of 8 days (SD=2), and Group B (n=35) has a mean of 10 days (SD=2.5). What is the correct formula for the standard error of the difference in means?

Accepted Answer

$$SE = \sqrt{\dfrac{2^2$}{30} + \dfrac{2.5^2$}{35}}$$

Question 8

Suppose a 95% confidence interval for the difference in mean test scores between two teaching methods is $$(-3.2, 1.5). $$What can you conclude about the hypothesis test for no difference in means at the 0.05 significance level?

Accepted Answer

Fail to reject the null hypothesis; there is not a significant difference.

Question 9

A two-sample t-test is performed to compare the mean cholesterol levels of two diets. The test yields a p-value of 0.12. Which of the following is the best interpretation?

Accepted Answer

There is not enough evidence to conclude a difference in mean cholesterol levels between the two diets.

Question 10

Which of the following scenarios would violate the independence assumption required for a two-sample t-test?

Accepted Answer

Comparing the before and after weights of the same group of patients.

Question 11

A researcher reports a 99% confidence interval for the mean number of hours college students sleep per night as $$(6.1, 7.3). $$Which of the following statements is correct?

Accepted Answer

We are 99% confident that the mean number of hours all college students sleep per night is between 6.1 and 7.3 hours.

Question 12

If the sample size increases while keeping the sample standard deviation constant, what happens to the width of the confidence interval for the mean?

Accepted Answer

The width decreases.

Question 13

A study compares the mean test scores of two independent groups. The 95% confidence interval for the difference in means is $$(2.0, 5.0). $$Which of the following is a correct interpretation?

Accepted Answer

There is evidence that the first group has a higher mean score than the second group.

Question 14

Which of the following is a correct step in performing a two-sample t-test for the difference between means?

Accepted Answer

Check that the samples are independent and the data are approximately normal.

Question 15

A researcher wants to use a confidence interval to approximate a hypothesis test for the difference between two means. Which of the following is true?

Accepted Answer

If the confidence interval for the difference includes zero, we fail to reject the null hypothesis.

Question 16

A sample of 16 patients has a mean cholesterol level of 210 mg/dL and a standard deviation of 20 mg/dL. What is the margin of error for a 90% confidence interval for the mean cholesterol level?

Accepted Answer

$$t^* 	imes \dfrac{20}{\sqrt{16}}$$ where $$t^* is $$the critical value for 15 degrees of freedom at 90% confidence.

Question 17

Which of the following is a reason to use a t-distribution instead of a z-distribution when constructing a confidence interval for a mean?

Accepted Answer

The population standard deviation is unknown and the sample size is small.

Question 18

A researcher performs a two-sample t-test and obtains a 95% confidence interval for the difference in means as $$(0.5, 4.2). $$What does this interval suggest about the p-value for the test?

Accepted Answer

The p-value is less than 0.05.

Question 19

Which of the following is a correct interpretation of a p-value in the context of a hypothesis test for a mean?

Accepted Answer

The probability of obtaining a result as extreme or more extreme than the observed result, assuming the null hypothesis is true.

Confidence Intervals and t-Tests for Means: Study Guide

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