Q: Given a sample proportion $$\hat{p} = 0.64$$, sample size $$n = 3634$$, and $$z^* = 1.96$$ for a 95% confidence interval, what is the margin of error $$E$$?

$$E = 1.96 \times \sqrt{\frac{0.64 \times 0.36}{3634}}$$

Q: In a real-world scenario, a researcher wants to estimate the proportion of voters who support a new policy. She wants the margin of error to be no more than $$0.04 at a 95$$% confidence level. She has no prior estimate for $$p. $$What is the minimum sample size she should use?

$$n = \frac{0.25 \times (1.96)^2$}{0.04^2$}$$

Q: A researcher wants to estimate the proportion of students who own a laptop. She wants to be 95% confident that her estimate is within $$0.05 of $$the true proportion. She has no prior estimate for $$p. $$What sample size should she use?

$$n = \frac{0.25 \times (1.96)^2$}{0.05^2$}$$

Question 1

Suppose a random sample of 350 people is drawn from a population in which the true proportion of women is $$p = 0.53. $$What is the expected value (mean) of the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.53$$

Question 2

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

Accepted Answer

$$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 3

A survey of 2019 adults found that 1252 frequently worry about their financial situation. What is the point estimate for the population proportion $$p$$?

Accepted Answer

$$\frac{1252}{2019}$$

Question 4

Given a sample proportion $$\hat{p} = 0.64$$ from a sample of $$n = 3634$$ and a margin of error $$E = 0.015 at 95$$% confidence, what is the 95% confidence interval for the population proportion $$p$$?

Accepted Answer

$$0.625 \leq p \leq 0.655$$

Question 5

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

Accepted Answer

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

Question 6

A poll reports a 95% confidence interval for a population proportion as $$0.602 to$$ $$0.638. $$What does this mean?

Accepted Answer

We are 95% confident that the true population proportion is between $$0.602$$ and $$0.638$$

Question 7

If the sample size increases while keeping the confidence level the same, what happens to the margin of error for a confidence interval for a population proportion?

Accepted Answer

The margin of error decreases

Question 8

Which formula would you use to determine the minimum sample size $$n$$ required to estimate a population proportion $$p$$ within a specified margin of error $$E at a $$given confidence level?

Accepted Answer

$$n = \frac{p(1-p)z^2$}{E^2$}$$

Question 9

A judge is accused of systematically excluding women from jury duty. In a random sample of 150 prospective jurors, only 35 are women. If the population proportion of women is $$0.53$$, what is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\frac{35}{150}$$

Question 10

For a sample of size $$n = 350$$ and $$p = 0.53$$, what is the value of $$np(1-p)$$? Is the normal approximation appropriate?

Accepted Answer

$$87.115$$, so the normal approximation is appropriate

Question 11

If you want to be 99% confident that your sample proportion is within $$0.03 of $$the true population proportion, and you have no prior estimate for $$p$$, what value should you use for $$p in $$the sample size formula?

Accepted Answer

$$0.5$$

Question 12

A 90% confidence interval for a population proportion is calculated as $$0.602 to$$ $$0.638. If $$the confidence level is increased to 99%, what will happen to the width of the confidence interval?

Accepted Answer

The interval will become wider

Question 13

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

Accepted Answer

It is the maximum likely difference between the sample proportion and the true population proportion

Question 14

Given a sample proportion $$\hat{p} = 0.64$$, sample size $$n = 3634$$, and $$z^* = 1.96$$ for a 95% confidence interval, what is the margin of error $$E$$?

Accepted Answer

$$E = 1.96 	imes \sqrt{\frac{0.64 	imes 0.36}{3634}}$$

Question 15

In a real-world scenario, a researcher wants to estimate the proportion of voters who support a new policy. She wants the margin of error to be no more than $$0.04 at a 95$$% confidence level. She has no prior estimate for $$p. $$What is the minimum sample size she should use?

Accepted Answer

$$n = \frac{0.25 	imes (1.96)^2$}{0.04^2$}$$

Question 16

A sample of 2019 adults found that 1252 frequently worry about their financial situation. Construct a 90% confidence interval for the population proportion. (Use $$z^* = 1.645$$ for 90% confidence.)

Accepted Answer

$$0.602 \leq p \leq 0.638$$

Question 17

If a sample proportion $$\hat{p} is $$much lower than the population proportion $$p$$, and the probability of observing such a low $$\hat{p} is $$very small, what can you conclude?

Accepted Answer

There is evidence that the sample may not be random or that the process is biased

Question 18

Which of the following statements about confidence intervals is correct?

Accepted Answer

A 95% confidence interval means that if we took many samples, about 95% of the intervals would contain the true population proportion

Question 19

A researcher wants to estimate the proportion of students who own a laptop. She wants to be 95% confident that her estimate is within $$0.05 of $$the true proportion. She has no prior estimate for $$p. $$What sample size should she use?

Accepted Answer

$$n = \frac{0.25 	imes (1.96)^2$}{0.05^2$}$$

Sampling Distributions and Confidence Intervals for Proportions

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