Q: If the sample proportion $$\hat{p} is 0.6$$, the sample size $$n is 400$$, and $$z^* = 1.96$$, what is the margin of error $$E$$ for a 95% confidence interval? (Round to four decimal places.)

$$E = 1.96 \times \sqrt{\dfrac{0.6 \times 0.4}{400}} = 0.0480$$

Question 1

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

Accepted Answer

A range of values that is likely to contain the value of the population parameter $$p.$$

Question 2

Given a sample of $$n = 750$$ voters, with $$x = 305$$ voting for a candidate, what is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{305}{750} = 0.407$$

Question 3

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

Accepted Answer

$$\sqrt{\dfrac{p(1-p)}{n}}$$

Question 4

If a 95% confidence interval for a proportion is calculated as $$0.3715 \u003c p \u003c 0.4418$$, which of the following statements is correct?

Accepted Answer

We are 95% confident that the true population proportion $$p is $$between 0.3715 and 0.4418.

Question 5

What is the critical value $$z^*$$ for a 95% confidence interval?

Accepted Answer

$$z^* = 1.960$$

Question 6

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

Accepted Answer

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

Question 7

A newspaper reports: 'Based on a poll of 750 randomly selected people, we estimate that 40.7% of voters would vote for the Governor. The margin of error is 3.5%.' What is the correct interpretation of the margin of error?

Accepted Answer

The true proportion of voters who would vote for the Governor is likely within 3.5% of the sample estimate.

Question 8

If the confidence interval for the proportion of voters who would vote for a candidate is entirely below 45%, what can you conclude?

Accepted Answer

It is unlikely that the candidate will receive 45% or more of the votes.

Question 9

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

Accepted Answer

Calculate the population mean.

Question 10

Suppose you want to construct a 99% confidence interval for a population proportion. Which of the following changes would result in a wider confidence interval?

Accepted Answer

Increasing the confidence level from 95% to 99%.

Question 11

A poll of 1,000 people finds that 520 support a new policy. What is the 95% confidence interval for the proportion of supporters? (Use $$z^* = 1.96$$ and round to four decimal places.)

Accepted Answer

$$0.4892 \u003c p \u003c 0.5508$$

Question 12

Which of the following best describes the degree of confidence (confidence level) in a confidence interval?

Accepted Answer

The probability that the population parameter is contained in the confidence interval in repeated samples.

Question 13

If the sample proportion $$\hat{p} is 0.6$$, the sample size $$n is 400$$, and $$z^* = 1.96$$, what is the margin of error $$E$$ for a 95% confidence interval? (Round to four decimal places.)

Accepted Answer

$$E = 1.96 	imes \sqrt{\dfrac{0.6 	imes 0.4}{400}} = 0.0480$$

Question 14

Which of the following statements about the critical value $$z^* is $$correct?

Accepted Answer

$$z^* is $$the value that marks the vertical boundary for the area $$\dfrac{\alpha}{2} in $$the right tail of the standard normal distribution.

Question 15

A researcher wants to decrease the margin of error in a confidence interval for a population proportion. Which of the following actions will achieve this?

Accepted Answer

Increase the sample size $$n.$$

Question 16

If a 90% confidence interval for a proportion is $$0.35 \u003c p \u003c 0.45$$, what is the width of the confidence interval?

Accepted Answer

$$0.10$$

Question 17

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 calculated intervals would contain the true population proportion.

Question 18

A poll finds $$\hat{p} = 0.52$$ with $$n = 1000. $$What is the standard error of $$\hat{p}$$? (Round to four decimal places.)

Accepted Answer

$$SE = \sqrt{\dfrac{0.52 	imes 0.48}{1000}} = 0.0158$$

Question 19

Which of the following would NOT affect the width of a confidence interval for a population proportion?

Accepted Answer

The value of the population mean.

Confidence Intervals for Population Proportions

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