Question 1

A researcher wants to test whether the mean cholesterol level in a population is different from 200 mg/dL. She collects a random sample of 40 adults and finds a sample mean of 210 mg/dL with a sample standard deviation of 30 mg/dL. Which test statistic should she use to test her hypothesis?

Accepted Answer

Use the $$t-$$test statistic: $$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$$

Question 2

Which of the following is the correct formula for a 95% confidence interval for the population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_{0.025, n-1} \cdot \frac{s}{\sqrt{n}}$$

Question 3

A pharmaceutical company claims that their drug reduces blood pressure by at least 10 mmHg. In a clinical trial, the mean reduction is 8 mmHg with a standard deviation of 4 mmHg in 25 patients. What is the value of the test statistic to test the company's claim?

Accepted Answer

$$t = \frac{8 - 10}{4 / \sqrt{25}} = -2.5$$

Question 4

Which of the following statements about the $$t-$$distribution is TRUE?

Accepted Answer

The $$t-$$distribution is always symmetric and has heavier tails than the normal distribution.

Question 5

A hospital administrator wants to compare the mean length of stay for two different wards. Ward A has 30 patients with a mean stay of 5 days and standard deviation of 2 days. Ward B has 35 patients with a mean stay of 6 days and standard deviation of 2.5 days. What is the formula for the test statistic to compare the means?

Accepted Answer

$$t = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2$}{n_A} + \frac{s_B^2$}{n_B}}}$$

Question 6

A sample of 20 students has a mean test score of 75 and a standard deviation of 10. What is the 95% confidence interval for the mean score? (Use $$t_{0.025, 19} = 2.093)$$

Accepted Answer

$$75 \pm 2.093 \cdot \frac{10}{\sqrt{20}} = 75 \pm 4.68$$

Question 7

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

We are 95% confident that the interval contains the true population mean.

Question 8

A scientist wants to test if the mean weight of a species of fish is less than 500g. She collects a sample of 16 fish, finds a mean of 480g and a standard deviation of 40g. What is the $$t-$$statistic for her test?

Accepted Answer

$$t = \frac{480 - 500}{40 / \sqrt{16}} = -2.0$$

Question 9

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

Accepted Answer

The samples are dependent on each other.

Question 10

A researcher is testing the effect of a new teaching method. The mean score for the experimental group is 82 (SD = 6, n = 25) and for the control group is 78 (SD = 8, n = 25). What is the pooled standard deviation?

Accepted Answer

$$s_p = \sqrt{\frac{(24 \cdot 6^2$) + (24 \cdot 8^2$)}{48}} = 7.07$$

Question 11

A psychologist wants to know if therapy reduces anxiety scores. She measures anxiety before and after therapy for 12 patients. Which statistical test should she use?

Accepted Answer

Paired $$t-$$test

Question 12

A sample of 15 batteries has a mean lifetime of 120 hours and a standard deviation of 15 hours. What is the margin of error for a 99% confidence interval? ($$t_{0.005, 14} = 2.977)$$

Accepted Answer

$$2.977 \cdot \frac{15}{\sqrt{15}} = 11.54$$

Question 13

Which of the following scenarios requires the use of a one-tailed $$t-$$test?

Accepted Answer

Testing if a new fertilizer increases crop yield compared to the current average.

Question 14

A biologist wants to estimate the mean height of a plant species. She collects a sample of 36 plants, finds a mean of 22 cm and a standard deviation of 3 cm. What is the 90% confidence interval for the mean? ($$t_{0.05, 35} = 1.690)$$

Accepted Answer

$$22 \pm 1.690 \cdot \frac{3}{\sqrt{36}} = 22 \pm 0.845$$

Question 15

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

Accepted Answer

$$H_0$$: \$$mu_1$$ = \$$mu_2$$

Question 16

A nutritionist wants to test if the mean calorie intake of athletes is greater than 2500 kcal. She collects a sample of 20 athletes, finds a mean of 2600 kcal and a standard deviation of 200 kcal. What is the $$t-$$statistic?

Accepted Answer

$$t = \frac{2600 - 2500}{200 / \sqrt{20}} = 2.236$$

Question 17

Which of the following is a reason to use the $$t-$$distribution instead of the normal distribution?

Accepted Answer

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

Question 18

A company wants to know if the average time to assemble a product is less than 30 minutes. A sample of 10 workers has a mean time of 28 minutes and a standard deviation of 3 minutes. What is the $$t-$$statistic?

Accepted Answer

$$t = \frac{28 - 30}{3 / \sqrt{10}} = -2.11$$

Question 19

Which of the following is the correct formula for degrees of freedom in a two-sample $$t-$$test with equal variances?

Accepted Answer

$$df = n_1$ + n_2$ - 2$$

Review of Hypothesis Testing for Two Samples (Chapter 10)

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