Question 1

Which of the following best describes what a 95% confidence interval for a population proportion means?

Accepted Answer

If we took many random samples and built a confidence interval from each, about 95% of those intervals would contain the true population proportion.

Question 2

Given a sample proportion $$\hat{p}$$ and sample size $$n$$, which formula correctly calculates the standard error of $$\hat{p}$$?

Accepted Answer

$$SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

A survey of 1520 U.S. adults found that 68% use Facebook. The 95% confidence interval for the proportion is 0.656 to 0.704. What does this interval mean?

Accepted Answer

We are 95% confident that the true proportion of U.S. adults who use Facebook is between 65.6% and 70.4%.

Question 4

Which of the following statements is INCORRECT regarding confidence intervals?

Accepted Answer

A 95% confidence interval means that there is a 95% chance the true parameter is in our specific interval.

Question 5

If you increase the sample size $$n$$ while keeping the sample proportion $$\hat{p}$$ the same, what happens to the width of the confidence interval for the population proportion?

Accepted Answer

The width decreases because the standard error decreases.

Question 6

Which formula is used to construct a confidence interval for a population proportion?

Accepted Answer

$$\hat{p} \pm z^* SE(\hat{p})$$

Question 7

Suppose you repeat a survey 100 times, each time constructing a 95% confidence interval for the same population proportion. About how many of these intervals would you expect to contain the true proportion?

Accepted Answer

About 95 intervals

Question 8

Which of the following is a common misconception about confidence intervals?

Accepted Answer

The confidence interval tells us the probability that the parameter is in the interval.

Question 9

A researcher reports: 'I am 95% confident that between 23.4% and 38.2% of adults use Facebook.' What is wrong with this statement?

Accepted Answer

The interval does not match the sample proportion of 68%.

Question 10

Why can't we be 100% sure that our confidence interval contains the true population proportion?

Accepted Answer

Because the interval is based on a random sample and sampling variability means some intervals will miss the true value.

Question 11

Which of the following is a correct interpretation of a 95% confidence interval for a proportion?

Accepted Answer

If we repeated the sampling process many times, about 95% of the intervals would contain the true proportion.

Question 12

If the standard error of $$\hat{p} is 0.012 $$and the critical value $$z^*$$ for 95% confidence is 1.96, what is the margin of error?

Accepted Answer

$$1.96 	imes 0.012 = 0.02352$$

Question 13

Which of the following is NOT a valid way to interpret a confidence interval?

Accepted Answer

95% of individuals in the population are within the interval.

Question 14

A poll reports a 95% confidence interval for the proportion of voters supporting a candidate as (0.48, 0.54). Which of the following is a correct conclusion?

Accepted Answer

We are 95% confident that between 48% and 54% of all voters support the candidate.

Sampling Distribution Models and Confidence Intervals for Proportions

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