Q: A sample of 100 voters shows 60% support for a candidate. What is the 95% confidence interval for the proportion of all voters who support the candidate? (Use $$z_{0.025} = 1.96)$$

$$0.6 \pm 1.96 \sqrt{\frac{0.6 \times 0.4}{100}}$$

Q: A researcher is analyzing the effect of hours studied (x) on exam score (y) using regression. The estimated regression equation is $$\hat{y} = 50 + 5x. $$What is the predicted exam score for a student who studied 6 hours?

$$\hat{y} = 50 + 5 \times 6 = 80$$

Question 1

A pharmaceutical company claims that their new drug is effective in 80% of patients. In a clinical trial, 10 patients are treated. What is the probability that exactly 8 patients will respond to the drug? Use the binomial probability formula.

Accepted Answer

$$P(X = 8) = \binom{10}{8} (0.8)^8$ (0.2)^2$

Question 2

A researcher wants to test if a new teaching method improves test scores. She collects scores from 30 students before and after using the method. Which statistical test is most appropriate?

Accepted Answer

Paired t-test

Question 3

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

Accepted Answer

$$\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$$

Question 4

A hospital records whether patients recover (Yes/No) and whether they received a new treatment (Yes/No). Which statistical tool is best for analyzing the association between treatment and recovery?

Accepted Answer

Contingency table

Question 5

In a simple linear regression, the equation is $$y = \$$beta_0$$ + \$$beta_1$$ x + \varepsilon. $$What does $$\$$beta_1$$ represent?

Accepted Answer

The expected change in $$y$$ for a one-unit increase in $$x$$

Question 6

A study tests whether the mean cholesterol level in a population is different from 200 mg/dL. Which is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu = 200$$

Question 7

A sample of 100 voters shows 60% support for a candidate. What is the 95% confidence interval for the proportion of all voters who support the candidate? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$0.6 \pm 1.96 \sqrt{\frac{0.6 	imes 0.4}{100}}$$

Question 8

Which of the following statements about the $$p-$$value in hypothesis testing is correct?

Accepted Answer

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

Question 9

A regression analysis yields $$r = 0.85. $$What does this value indicate?

Accepted Answer

A strong positive linear relationship between the variables

Question 10

A two-sample $$t-$$test is used to compare the means of two independent groups. Which of the following is an assumption of this test?

Accepted Answer

The two samples are independent and drawn from normally distributed populations with equal variances.

Question 11

A researcher is analyzing the effect of hours studied (x) on exam score (y) using regression. The estimated regression equation is $$\hat{y} = 50 + 5x. $$What is the predicted exam score for a student who studied 6 hours?

Accepted Answer

$$\hat{y} = 50 + 5 	imes 6 = 80$$

Question 12

Which of the following is the correct formula for the test statistic in a one-sample $$z-$$test for the mean when the population standard deviation is known?

Accepted Answer

$$z = \frac{\bar{x} - \mu_0$}{\sigma / \sqrt{n}}$$

Question 13

A contingency table is used to:

Accepted Answer

Summarize the relationship between two categorical variables

Question 14

In regression analysis, the $$R^2$$ value represents:

Accepted Answer

The proportion of variance in the dependent variable explained by the independent variable(s)

Question 15

A study compares the mean blood pressure of patients before and after a new medication. The data are paired. Which test should be used?

Accepted Answer

Paired t-test

Question 16

Which of the following is the correct formula for the expected frequency in a cell of a $$2 	imes 2$$ contingency table?

Accepted Answer

$$E = \frac{(	ext{row total}) 	imes (	ext{column total})}{	ext{grand total}}$$

Question 17

A researcher wants to estimate the mean height of adult males in a city. She collects a random sample of 36 men, with a sample mean of 175 cm and a sample standard deviation of 12 cm. What is the 95% confidence interval for the mean height? (Use $$t_{0.025, 35} \approx 2.03)$$

Accepted Answer

$$175 \pm 2.03 	imes \frac{12}{\sqrt{36}}$$

Question 18

In hypothesis testing, if the $$p-$$value is less than the significance level $$\alpha$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 19

A binomial experiment consists of 5 trials with probability of success $$p = 0.4. $$What is the expected number of successes?

Accepted Answer

$$E[X] = np = 5 	imes 0.4 = 2$$

Question 20

Which of the following is NOT an assumption of simple linear regression?

Accepted Answer

The dependent variable must be categorical

Final Exam Study Guide: Key Topics in College Statistics

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