Q: A researcher wants to estimate the average cholesterol level in a population. The population standard deviation is known to be $$\sigma = 15 mg/dL. A $$random sample of $$n=36$$ yields a sample mean of $$\bar{x} = 200 mg/dL. $$What is the 95% confidence interval for the population mean?

$$200 \pm 1.96 \times \frac{15}{\sqrt{36}}$$

Q: A linear regression analysis yields the equation $$\hat{y} = 2 + 3x. $$What is the predicted value of $$y$$ when $$x = 4$$?

$$\hat{y} = 2 + 3 \times 4 = 14$$

Question 1

A pharmaceutical company claims that their new drug increases the average recovery rate from a certain illness. In a clinical trial, the recovery times (in days) for 16 patients are recorded. The population standard deviation is unknown. Which statistical test should be used to determine if the drug significantly reduces recovery time compared to the known average of 10 days?

Accepted Answer

A one-sample t-test for the mean

Question 2

Suppose the probability that a randomly selected student passes a statistics exam is $$0.7. If 8 $$students are selected independently, what is the probability that exactly 6 of them pass? (Use the binomial formula.)

Accepted Answer

$$P(X=6) = \binom{8}{6} (0.7)^6$ (0.3)^2$

Question 3

A researcher wants to estimate the average cholesterol level in a population. The population standard deviation is known to be $$\sigma = 15 mg/dL. A $$random sample of $$n=36$$ yields a sample mean of $$\bar{x} = 200 mg/dL. $$What is the 95% confidence interval for the population mean?

Accepted Answer

$$200 \pm 1.96 	imes \frac{15}{\sqrt{36}}$$

Question 4

Which of the following statements about the normal distribution is TRUE?

Accepted Answer

Approximately 68% of the data falls within one standard deviation of the mean.

Question 5

A box plot is constructed for the weights of 100 newborn babies. Which of the following can be identified directly from the box plot?

Accepted Answer

The interquartile range (IQR)

Question 6

In a study, the probability that a randomly selected person is left-handed is $$0.12. If $$two people are selected independently, what is the probability that both are left-handed?

Accepted Answer

$$0.12 	imes 0.12$$

Question 7

A hospital wants to compare the mean recovery times for two different treatments. The population variances are unknown and the sample sizes are small. Which test should be used?

Accepted Answer

A two-sample t-test for the means

Question 8

Which of the following is the correct formula for the test statistic in a hypothesis test for a single proportion?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$ (1 - p_0$)}{n}}}$$

Question 9

A researcher performs a chi-square test for independence on a two-way table and obtains a p-value of 0.03. What is the correct interpretation at the 0.05 significance level?

Accepted Answer

There is sufficient evidence to conclude that the two variables are associated.

Question 10

A 95% confidence interval for the difference in means between two groups is $$(-2.1, 0.5). $$What can you conclude?

Accepted Answer

There is no significant difference between the two group means at the 0.05 level.

Question 11

Which of the following is NOT an assumption of the one-way ANOVA test?

Accepted Answer

The samples are paired.

Question 12

A linear regression analysis yields the equation $$\hat{y} = 2 + 3x. $$What is the predicted value of $$y$$ when $$x = 4$$?

Accepted Answer

$$\hat{y} = 2 + 3 	imes 4 = 14$$

Question 13

Which of the following scenarios best illustrates the use of conditional probability?

Accepted Answer

Calculating the probability that a patient has a disease given a positive test result.

Question 14

A sample of 25 light bulbs has a mean lifetime of 1200 hours with a sample standard deviation of 100 hours. Construct a 95% confidence interval for the mean lifetime. Which distribution should be used?

Accepted Answer

Student t-distribution

Question 15

Which of the following is the correct formula for the standard error of the difference between two independent sample means?

Accepted Answer

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

Question 16

A company claims that at least 80% of its products pass quality control. In a random sample of 100 products, 74 pass. What is the value of the test statistic for testing $$H_0$$: p = 0.8$$?

Accepted Answer

$$z = \frac{0.74 - 0.8}{\sqrt{\frac{0.8 	imes 0.2}{100}}}$$

Question 17

Which of the following best describes the p-value in hypothesis testing?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.

Question 18

A researcher wants to test if the mean systolic blood pressure differs between men and women. Which hypothesis test is appropriate?

Accepted Answer

A two-sample t-test for the means

Question 19

Which of the following is a key assumption for performing a hypothesis test for two proportions?

Accepted Answer

The samples are independent.

Question 20

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.5} = 0.4$$

ST 231 Winter 2026 Exam Guide: Key Topics in College Statistics

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