Q: If a sample of 100 births includes 14 premature babies, what is the standard error for the sample proportion of premature births?

$$SE(\hat{p}) = \sqrt{\frac{0.14 \times 0.86}{100}} \approx 0.035$$

Q: A sample of 150 people finds that 90 support a new law. Construct a 90% confidence interval for the proportion of supporters. (Use $$z^* = 1.645.)$$

$$0.60 \pm 1.645 \times \sqrt{\frac{0.60 \times 0.40}{150}} \approx 0.60 \pm 0.066 = (0.534, 0.666)$$

Question 1

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

Accepted Answer

To give a range of plausible values for the population proportion, based on sample data.

Question 2

A researcher wants to estimate the proportion of voters in a city who support a new policy. She surveys 200 randomly selected voters and finds that 120 support the policy. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{120}{200} = 0.60$$

Question 3

Which of the following is NOT a required condition for constructing a one-proportion z-interval?

Accepted Answer

The population must be normally distributed.

Question 4

What is the formula for the standard error of the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 5

A 95% confidence interval for a population proportion is calculated as $$0.71 \pm 0.078. $$Which of the following is the correct interpretation?

Accepted Answer

We are 95% confident that the true proportion is between 0.632 and 0.788.

Question 6

If a sample of 100 births includes 14 premature babies, what is the standard error for the sample proportion of premature births?

Accepted Answer

$$SE(\hat{p}) = \sqrt{\frac{0.14 	imes 0.86}{100}} \approx 0.035$$

Question 7

Which of the following statements about confidence intervals is INCORRECT?

Accepted Answer

A confidence interval is about the sample statistic, not the population parameter.

Question 8

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

Accepted Answer

$$z^* = 2.576$$

Question 9

A sample of 150 people finds that 90 support a new law. Construct a 90% confidence interval for the proportion of supporters. (Use $$z^* = 1.645.)$$

Accepted Answer

$$0.60 \pm 1.645 	imes \sqrt{\frac{0.60 	imes 0.40}{150}} \approx 0.60 \pm 0.066 = (0.534, 0.666)$$

Question 10

Which of the following is a correct statement about the margin of error in a confidence interval for a proportion?

Accepted Answer

It reflects the sampling variability and the confidence that the interval captures the parameter.

Question 11

Suppose you want to increase the precision of your confidence interval for a population proportion. Which of the following actions would achieve this?

Accepted Answer

Increase the sample size.

Question 12

A pollster reports a 95% confidence interval for the proportion of voters who favor a candidate as (0.48, 0.56). Which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the true proportion of voters who favor the candidate is between 0.48 and 0.56.

Question 13

Which of the following is the correct formula for a one-proportion z-interval?

Accepted Answer

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

Question 14

A sample of 200 students finds that 50 prefer online classes. Which of the following conditions for a one-proportion z-interval is NOT satisfied?

Accepted Answer

None; all conditions are satisfied.

Question 15

If the confidence level is increased from 90% to 99%, what happens to the width of the confidence interval for a proportion?

Accepted Answer

The interval becomes wider.

Question 16

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

Accepted Answer

There is a 95% probability that the true parameter is in the calculated interval.

Question 17

A city health department wants to estimate the proportion of residents who received a flu shot last year. They randomly sample 400 residents and find 120 received a flu shot. What is the 95% confidence interval for the true proportion? (Use $$z^* = 1.96.)$$

Accepted Answer

$$0.30 \pm 1.96 	imes \sqrt{\frac{0.30 	imes 0.70}{400}} \approx 0.30 \pm 0.045 = (0.255, 0.345)$$

Sampling Distribution Models and Confidence Intervals for Proportions

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