Question 1

Which of the following best describes a confidence interval for a population proportion?

Accepted Answer

A range of values used to estimate the true value of a population proportion.

Question 2

Given a sample proportion $$\hat{p}$$, sample size $$n$$, and critical value $$z_{\alpha/2}$$, which formula correctly calculates the margin of error $$E$$ for a confidence interval?

Accepted Answer

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

Question 3

A poll surveyed 1013 adults and found 669 worried about identity theft. What is the best point estimate for the population proportion $$p$$?

Accepted Answer

$$\frac{669}{1013}$$

Question 4

If a 95% confidence interval for a population proportion is (0.69, 0.77), what is the margin of error?

Accepted Answer

0.04 (since $$\frac{0.77 - 0.69}{2} = 0.04)$$

Question 5

Which condition must be met to use the normal distribution for the sampling distribution of proportions?

Accepted Answer

$$np \u003e 5$$ and $$nq \u003e 5$$

Question 6

What does the critical value $$z_{\alpha/2}$$ represent in constructing a confidence interval?

Accepted Answer

The z-score with an area of $$\alpha/2 to $$its right.

Question 7

If you increase the confidence level from 95% to 99%, what happens to the margin of error?

Accepted Answer

It increases.

Question 8

Suppose previous research suggests the population proportion is 0.27. What formula should you use to determine the minimum sample size $$n$$ needed for a margin of error $$E at a $$given confidence level?

Accepted Answer

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

Question 9

Which of the following is an incorrect interpretation of a 95% confidence interval (0.63, 0.69)?

Accepted Answer

There is a 95% chance that the true population proportion will fall between 0.63 and 0.69.

Question 10

A researcher wants to be 90% confident that the sample percentage is in error by no more than three percentage points. What is the value of $$E in $$the sample size formula?

Accepted Answer

$$E = 0.03$$

Question 11

Which statement about the relationship between margin of error and sample size is correct?

Accepted Answer

Margin of error decreases as sample size increases.

Question 12

If a confidence interval for a population proportion is (0.63, 0.69), what is the point estimate $$\hat{p}$$?

Accepted Answer

$$\frac{0.63 + 0.69}{2}$$

Question 13

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

Accepted Answer

The population must be normally distributed.

Question 14

A confidence level of 95% means:

Accepted Answer

The process used will produce intervals containing the true proportion 95% of the time.

Question 15

Which formula gives the confidence interval for a population proportion $$p$$?

Accepted Answer

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

Question 16

In the context of estimating a population proportion, what does the Central Limit Theorem imply?

Accepted Answer

The sampling distribution of the sample proportion is approximately normal if $$np \u003e 5$$ and $$nq \u003e 5.$$

Question 17

If a poll reports a margin of error of 3% at a 95% confidence level, what does this mean?

Accepted Answer

We are 95% confident that the population proportion is within 3% of the sample proportion.

Question 18

Which of the following is a plausible reason for increasing the sample size when estimating a population proportion?

Accepted Answer

To decrease the margin of error.

Question 19

A confidence interval for a population proportion is calculated as (0.55, 0.65). What is the width of the interval?

Accepted Answer

0.10

Estimating Population Proportions and Determining Sample Sizes

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