Question 1

Suppose a random sample of 200 people is taken from a city, and 60 of them support a new recycling program. What is the point estimate for the proportion of all city residents who support the program?

Accepted Answer

$\hat{p} = \frac{60}{200} = 0.3$

Question 2

Which of the following statements about the sampling distribution of the sample proportion $\hat{p}$ is TRUE?

Accepted Answer

The mean of the sampling distribution of $\hat{p}$ equals the population proportion $p$.

Question 3

A poll finds that 76% of Americans believe the state of moral values is getting worse. For a sample size of $n=60$, what is the standard deviation of the sampling distribution of $\hat{p}$?

Accepted Answer

$\sigma_{\hat{p}} = \sqrt{\frac{0.76(1-0.76)}{60}} = 0.055$

Question 4

Which condition must be satisfied to use the normal approximation for the sampling distribution of $\hat{p}$?

Accepted Answer

$np(1-p) \geq 10$

Question 5

In a survey of 800 teenagers, 272 report texting while driving. What is the 95% confidence interval for the proportion of teens who text while driving? (Use $z^* = 1.96$)

Accepted Answer

$0.307$ to $0.373$

Question 6

What happens to the margin of error when the confidence level increases from 95% to 99% for a fixed sample size?

Accepted Answer

The margin of error increases, resulting in a wider confidence interval.

Question 7

Which formula correctly represents the margin of error $E$ for a confidence interval for a population proportion?

Accepted Answer

$E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Question 8

A sample of 90 school-aged children finds 24 are overweight. If the population proportion is 0.188, is this sample proportion unusual?

Accepted Answer

Yes, because the sample proportion $\hat{p} = 0.2667$ is much higher than the population proportion and occurs with low probability.

Question 9

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

Accepted Answer

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

Question 10

Which of the following will result in a narrower confidence interval for a population proportion, assuming all other factors are constant?

Accepted Answer

Increasing the sample size $n$

Question 11

What is the critical value $z^*$ for a 99% confidence interval?

Accepted Answer

$z^* = 2.575$

Question 12

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

Accepted Answer

The sample must be larger than 5% of the population.

Question 13

A researcher wants to estimate the proportion of voters in a city who support a new law. Which sampling method is most appropriate for constructing a confidence interval?

Accepted Answer

Simple random sampling

Question 14

If the sample proportion $\hat{p}$ is in the tails of the sampling distribution, what is likely to happen to the confidence interval?

Accepted Answer

It may not contain the true population proportion.

Question 15

Which of the following best describes the effect of increasing the sample size $n$ on the standard deviation of the sampling distribution of $\hat{p}$?

Accepted Answer

The standard deviation decreases.

Question 16

A 95% confidence interval for a population proportion is calculated as $\hat{p} \pm 1.96 \sigma_{\hat{p}}$. What does the value 1.96 represent?

Accepted Answer

The critical value $z^*$ for a 95% confidence level

Question 17

Which of the following is a correct formula for the lower bound of a $(1-\alpha)100\%$ confidence interval for a population proportion?

Accepted Answer

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

Question 18

Why is it incorrect to say that there is a 95% probability that a specific confidence interval contains the population proportion?

Accepted Answer

Because the population proportion is a fixed value, not a random variable.

Question 19

A city council wants to estimate the proportion of residents who support a new park. They sample 500 residents and find 200 in favor. What is the 95% confidence interval for the proportion supporting the park? (Use $z^* = 1.96$)

Accepted Answer

$0.362$ to $0.438$

Confidence Intervals for Proportions: Concepts, Calculations, and Applications

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