Question 1

A 2016 Pew Research Center survey of 4787 randomly selected U.S. adults showed that 79% of the respondents shop online. If you want to estimate the current proportion of adults who shop online with a 95% confidence level and a margin of error of no more than 3 percentage points, what is the minimum sample size required if you have no prior estimate of the population proportion?

Accepted Answer

$$n = \left( \frac{1.96}{0.03} \$$right)^2$$ (0.5)(0.5) = 1067.11$$, so 1068$

Question 2

Which of the following is the correct interpretation of 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 3

A sample of 950 undergraduate students found that 53% take online courses. Construct a 95% confidence interval for the population proportion $$p. $$Use $$z_{\alpha/2} = 1.96.$$

Accepted Answer

$$0.498 \u003c p \u003c 0.562$$

Question 4

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

Accepted Answer

The population must be normally distributed.

Question 5

Given a confidence interval for $$p of$$ $$(0.498, 0.562)$$, what is the point estimate $$\hat{p}$$ and the margin of error $$E$$?

Accepted Answer

$$\hat{p} = 0.53$$, $$E = 0.032$$

Question 6

If you want to estimate a population proportion $$p$$ with a margin of error $$E$$ and you have a prior estimate $$\hat{p}$$, which formula should you use to determine the required sample size $$n$$?

Accepted Answer

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

Question 7

A researcher wants to estimate the proportion of patients who respond to a new drug. She wants a 99% confidence interval with a margin of error of 0.04. She has no prior estimate for $$p. $$What critical value $$z_{\alpha/2}$$ should she use?

Accepted Answer

$$z_{\alpha/2} = 2.575$$

Question 8

Which of the following statements about the margin of error $$E is $$correct?

Accepted Answer

It is the maximum likely amount by which the sample statistic misses the population parameter.

Question 9

Suppose you increase the confidence level from 95% to 99% while keeping the sample size and sample proportion the same. What happens to the width of the confidence interval for $$p$$?

Accepted Answer

The interval becomes wider.

Question 10

A pollster wants to estimate the proportion of voters who support a new policy. She wants to be 90% confident that her estimate is within 2 percentage points of the true proportion. She has no prior estimate for $$p. $$What sample size should she use? (Use $$z_{\alpha/2} = 1.645)$$

Accepted Answer

$$n = \left( \frac{1.645}{0.02} \$$right)^2$$ (0.5)(0.5) = 1690.6$$, so 1691$

Question 11

Which of the following is a disadvantage of using a point estimate to estimate a population parameter?

Accepted Answer

Point estimates are almost never exactly equal to the true values.

Question 12

A confidence interval for a population proportion $$p is $$constructed as $$\hat{p} \pm E. $$Which of the following best describes $$E$$?

Accepted Answer

The margin of error, calculated as $$E = z_{\alpha/2} \sqrt{\frac{\hat{p}\hat{q}}{n}}$$

Question 13

If a sample of 1000 people yields a sample proportion $$\hat{p} = 0.4$$, and you want a 95% confidence interval, what is the margin of error $$E$$? (Use $$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$E = 1.96 \sqrt{\frac{0.4 	imes 0.6}{1000}} = 0.0304$$

Question 14

Which of the following best describes the relationship between confidence level and the value of $$\alpha$$?

Accepted Answer

Confidence level = $$1 - \alpha$$

Question 15

A confidence interval for a population proportion $$p is $$calculated as $$0.53 \pm 0.0317. $$Which of the following statements is correct?

Accepted Answer

We are confident that the true population proportion is between 0.498 and 0.562.

Question 16

If the requirements for constructing a confidence interval for $$p$$ are not met (for example, fewer than 5 successes or failures), what is the most likely consequence?

Accepted Answer

The confidence interval may not be valid because the sampling distribution is not approximately normal.

Question 17

Which of the following is the best point estimate of the population proportion $$p$$?

Accepted Answer

The sample proportion $$\hat{p}$$

Question 18

A company wants to estimate the proportion of customers who are satisfied with their service. They survey 400 customers and find that 320 are satisfied. What is the 95% confidence interval for the true proportion of satisfied customers? (Use $$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$0.775 \u003c p \u003c 0.845$$

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