Question 1

A researcher wants to estimate the proportion of adults in a city who support a new public transportation initiative. Which of the following conditions must be satisfied for the sampling distribution of the sample proportion $$\hat{p} to be $$approximately normal?

Accepted Answer

Both $$np \geq 10$$ and $$n(1-p) \geq 10$$ must be satisfied.

Question 2

Suppose you collect a random sample of 200 voters and find that 120 support a new policy. Construct a 95% confidence interval for the population proportion $$p of $$supporters. Which formula should you use?

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

A pharmaceutical company wants to estimate the mean recovery time for a new drug. Which of the following conditions is necessary for the sampling distribution of the sample mean $$\bar{x} to be $$approximately normal?

Accepted Answer

The sample size must be at least 30, or the population distribution must be normal.

Question 4

A 95% confidence interval for a population mean is calculated as $$(12.4, 15.6). $$What is the margin of error?

Accepted Answer

$$1.6$$

Question 5

A hospital administrator wants to determine the sample size needed to estimate the proportion of patients satisfied with their care within a margin of error of $$0.04 at a 95$$% confidence level. No preliminary estimate of $$p is $$available. Which formula should be used?

Accepted Answer

$$n = \dfrac{z$$^{*2}$$ \cdot 0.25}{E^2$}$$

Question 6

Which of the following best describes the null hypothesis $$H_0 in a $$one-sample $$t-$$test for a population mean?

Accepted Answer

$$H_0$$: \mu = \$$mu_0$$

Question 7

A p-value of $$0.03 is $$obtained in a hypothesis test with a significance level $$\alpha = 0.05. $$What is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 8

A nutritionist wants to compare the mean sodium content of two brands of soup. Which formula should be used to construct a 95% confidence interval for the difference in means, assuming independent samples and equal variances?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{s_p^2$ \left( \dfrac{1}{n_1$} + \dfrac{1}{n_2$} \right)}$$

Question 9

In a clinical trial, the same group of patients is measured before and after receiving a treatment. What type of data is this?

Accepted Answer

Paired data

Question 10

A paired sample $$t-$$test is used to analyze the effect of a new teaching method on student scores. Which of the following is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu_d = 0$$

Question 11

A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. A sample of 25 individuals yields a sample mean of 210 mg/dL and a sample standard deviation of 20 mg/dL. What is the value of the test statistic for a two-sided $$t-$$test?

Accepted Answer

$$t = \dfrac{210 - 200}{20/\sqrt{25}} = 2.5$$

Question 12

Which of the following statements about the margin of error in a confidence interval is correct?

Accepted Answer

Increasing the sample size decreases the margin of error.

Question 13

A study compares the proportion of smokers in two different cities. Which formula is appropriate for constructing a confidence interval for the difference in proportions?

Accepted Answer

$$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 14

A scientist wants to test whether a new fertilizer increases crop yield. The null hypothesis is $$H_0$$: \mu = 50$$ bushels/acre. The sample mean is 53, the sample standard deviation is 4, and the sample size is 16. What is the t$ statistic?

Accepted Answer

$$t = \dfrac{53 - 50}{4/\sqrt{16}} = 3$$

Question 15

Which of the following is NOT a requirement for constructing a valid confidence interval for a population mean?

Accepted Answer

The population standard deviation must be known.

Question 16

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

Accepted Answer

A higher confidence level requires a larger critical value $$z^*.$$

Question 17

A researcher collects paired data on blood pressure before and after a new medication in 10 patients. The mean difference is $$-5$$ mmHg with a standard deviation of $$4$$ mmHg. What is the 95% confidence interval for the mean difference?

Accepted Answer

$$-5 \pm t^* \dfrac{4}{\sqrt{10}}$$

Sampling Distributions, Confidence Intervals, and Hypothesis Testing: Study Guide

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