Question 1

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. In a random sample of 200 patients, 30 reported side effects. Construct a 95% confidence interval for the true proportion of patients experiencing side effects. Which formula should be used for the margin of error?

Accepted Answer

$$z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 2

Which of the following is NOT an assumption required for constructing a confidence interval for a single population proportion?

Accepted Answer

The sample size is at least 30.

Question 3

A researcher is testing whether the mean cholesterol level in a population differs from 200 mg/dL. A random sample of 40 patients yields $$\bar{x} = 210$$, $$s = 25. $$Which test statistic should be used?

Accepted Answer

$$t_0$$ = \frac{\bar{x} - \$$mu_0$$}{s/\sqrt{n}}$$

Question 4

In hypothesis testing, what is a Type I error?

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 5

A hospital wants to compare the mean recovery times of two different physical therapy treatments. Patients are randomly assigned to each treatment. Which statistical test is most appropriate?

Accepted Answer

Two-sample t-test for independent means (assuming unequal variances)

Question 6

Which of the following best describes the margin of error in a confidence interval?

Accepted Answer

The maximum expected difference between the point estimate and the true parameter value.

Question 7

A study compares the effectiveness of two vaccines. Out of 1000 people, 600 received vaccine A and 400 received vaccine B. 30 people in group A and 10 in group B got sick. What is the pooled sample proportion $$\hat{p}$$ used in the two-sample z-test for proportions?

Accepted Answer

$$\hat{p} = \frac{30 + 10}{600 + 400}$$

Question 8

Which of the following scenarios is most appropriate for a paired t-test?

Accepted Answer

Measuring blood pressure before and after administering a drug to the same patients.

Question 9

A 99% confidence interval for a mean is wider than a 95% confidence interval for the same data because:

Accepted Answer

The critical value $$t_{\alpha/2} is $$larger for 99% confidence.

Question 10

Which of the following is a correct formula for the sample size needed to estimate a population proportion with margin of error $$E$$ and prior estimate $$\hat{p}$$?

Accepted Answer

$$n = \hat{p}(1-\hat{p}) \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$

Question 11

A researcher uses bootstrapping to estimate a confidence interval for the median income in a city. What is the main advantage of bootstrapping in this context?

Accepted Answer

It does not require the population to be normally distributed.

Question 12

In a two-sample t-test for independent means with unequal variances, which formula is used to estimate the degrees of freedom?

Accepted Answer

$$df = \frac{\left( \frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$} \$$right)^2$$}{\frac{1}{n_1$-1} \left( \frac{s_1$^2}{n_1$} \$$right)^2$$ + \frac{1}{n_2$-1} \left( \frac{s_2$^2}{n_2$} \$$right)^2$$}$$

Question 13

A clinical trial tests whether a new medication lowers blood pressure more than a placebo. The null hypothesis is $$H_0$$: \$$mu_1$$ = \$$mu_2. $$The alternative is $$H_1$$: \$$mu_1$$ \u003c \$$mu_2. $$What type of test is this?

Accepted Answer

Left-tailed test

Question 14

Which of the following statements about p-values is TRUE?

Accepted Answer

A small p-value indicates strong evidence against the null hypothesis.

Question 15

A school district wants to estimate the average number of hours students spend on homework per week. They want the margin of error to be no more than 1 hour, with 95% confidence. The population standard deviation is estimated to be 4 hours. What is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \cdot s}{E} \$$right)^2$$

Question 16

Which of the following is a robust test, meaning it is relatively unaffected by violations of the normality assumption when sample sizes are large?

Accepted Answer

t-test for mean

Question 17

A randomization test is most appropriate when:

Accepted Answer

The sampling distribution is unknown or the sample size is small.

Question 18

A two-sample confidence interval for the difference between two proportions is constructed. Which of the following is the correct formula for the margin of error?

Accepted Answer

$$z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 19

Which of the following best describes practical significance?

Accepted Answer

The observed effect is large enough to be meaningful in the real world.

Statistics Unit 3: Estimation and Hypothesis Testing (Chapters 9-11) Study Guide

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