Question 1

Which of the following statements correctly interprets a 95% confidence interval for a population proportion $$p$$ given by $$0.405 \u003c p \u003c 0.455$$?

Accepted Answer

We are 95% confident that the interval from 0.405 to 0.455 actually does contain the true value of the population proportion $$p.$$

Question 2

A sample of 950 undergraduate students found that 53% take online courses. What is the best point estimate for the population proportion $$p of $$all undergraduates who take online courses?

Accepted Answer

$$0.53$$

Question 3

Which formula gives the margin of error $$E$$ for estimating a population proportion $$p$$ using the Wald confidence interval?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$$

Question 4

If you want to be 99% confident that your sample proportion is within 0.04 of the true population proportion, and you have no prior estimate for $$\hat{p}$$, what formula should you use to determine the minimum sample size $$n$$?

Accepted Answer

$$n = \dfrac{z_{\alpha/2}^2 \cdot 0.25}{E^2$}$$

Question 5

A 95% confidence interval for a population proportion $$p is$$ $$(0.58, 0.81). $$What is the margin of error $$E$$?

Accepted Answer

$$E = \dfrac{0.81 - 0.58}{2} = 0.115$$

Question 6

Which of the following is NOT a requirement for constructing a confidence interval for a population proportion $$p$$?

Accepted Answer

The population size must be at least 10 times the sample size.

Question 7

Given a 95% confidence level, what is the corresponding critical value $$z_{\alpha/2}$$?

Accepted Answer

$$1.96$$

Question 8

A poll reports a 95% confidence interval for the proportion of adults who shop online as $$(0.76, 0.82). $$What is the point estimate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{0.76 + 0.82}{2} = 0.79$$

Question 9

Which method for constructing a confidence interval for a population proportion is considered to have coverage probability often less than the selected confidence level, especially for small samples or proportions near 0 or 1?

Accepted Answer

Wald confidence interval

Question 10

A researcher wants to estimate the proportion of patients who respond to a new drug. She wants to be 90% confident that her estimate is within 0.05 of the true proportion. She has no prior estimate for $$\hat{p}. $$What is the minimum sample size she should use? (Use $$z_{\alpha/2} = 1.645$$ for 90% confidence.)

Accepted Answer

$$n = \dfrac{1.645^2$ \cdot 0.25}{0.05^2$} = 271$$

Question 11

Which of the following is a correct statement about the effect of sample size on the reliability of poll results?

Accepted Answer

The reliability of poll results depends on the sample size and sampling method, not the population size.

Question 12

Which confidence interval method is based on the exact binomial distribution and is considered too conservative?

Accepted Answer

Clopper-Pearson method

Question 13

A survey of 4787 adults found that 79% shop online. If you want to be 95% confident that your sample proportion is within 0.03 of the true proportion, and you use $$\hat{p} = 0.79$$, what is the minimum sample size required? (Use $$z_{\alpha/2} = 1.96.)$$

Accepted Answer

$$n = \dfrac{1.96^2$ \cdot (0.79)(0.21)}{0.03^2$} = 709$$

Question 14

Which of the following is a benefit of the plus four confidence interval method compared to the Wald method?

Accepted Answer

It has a coverage probability closer to the selected confidence level.

Question 15

Which of the following is the correct formula for the lower limit of a Wald confidence interval for a population proportion $$p$$?

Accepted Answer

$$\hat{p} - z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$$

Question 16

A 90% confidence interval for a population proportion $$p is$$ $$(0.32, 0.48). $$What is the point estimate $$\hat{p}$$ and the margin of error $$E$$?

Accepted Answer

$$\hat{p} = 0.40,\ E = 0.08$$

7.1 Estimating Population Proportions and Determining Sample Sizes

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