Q: Suppose you want to estimate the proportion of California residents that are left-handed with a margin of error of 3% and a 99% confidence level. What is the minimum sample size required if you have no prior estimate for $$p$$?

$$n = \frac{(2.575)^2$}{4 \times (0.03)^2$} = 1842$$

Question 1

Which of the following best describes a point estimate in the context of estimating a population proportion?

Accepted Answer

A single value used to estimate a population parameter.

Question 2

Suppose a survey of 362 drive-thru orders at McDonald's found that 33 were inaccurate. What is the sample proportion $$\hat{p} of $$inaccurate orders?

Accepted Answer

$$\hat{p} = \frac{33}{362} = 0.0912$$

Question 3

What does a 95% confidence interval for a population proportion mean?

Accepted Answer

If we take many samples and construct intervals, about 95% of those intervals will contain the true population proportion.

Question 4

Which formula is used to calculate the margin of error $$E$$ for a confidence interval for a population proportion?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 5

Given a 95% confidence interval for a population proportion is $$0.0615 \u003c p \u003c 0.121$$, what is the margin of error $$E$$?

Accepted Answer

$$E = \frac{0.121 - 0.0615}{2} = 0.02975$$

Question 6

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

Accepted Answer

The sample must be a voluntary response sample.

Question 7

If you want a 99% confidence interval, what is the critical value $$z_{\alpha/2}$$ you should use?

Accepted Answer

$$z_{\alpha/2} = 2.575$$

Question 8

A poll of 1002 people finds that 70% say they voted. Voting records show that 61% of eligible people actually voted. The 95% confidence interval for the proportion who say they voted is $$0.666 \u003c p \u003c 0.734. $$What can you conclude?

Accepted Answer

The actual proportion (0.61) is not contained in the interval, so more people say they voted than actually did.

Question 9

Which formula should you use to determine the minimum sample size $$n$$ needed to estimate a population proportion $$p$$ with margin of error $$E$$ and confidence level $$z_{\alpha/2}$$, when $$\hat{p} is $$unknown?

Accepted Answer

$$n = \frac{(z_{\alpha/2})^2$}{4E^2$}$$

Question 10

Suppose you want to estimate the proportion of California residents that are left-handed with a margin of error of 3% and a 99% confidence level. What is the minimum sample size required if you have no prior estimate for $$p$$?

Accepted Answer

$$n = \frac{(2.575)^2$}{4 	imes (0.03)^2$} = 1842$$

Question 11

Which of the following statements about confidence intervals is correct?

Accepted Answer

A confidence interval provides a range of values that is likely to contain the true population proportion.

Question 12

What is the effect of increasing the confidence level from 95% to 99% on the width of the confidence interval for a population proportion?

Accepted Answer

The interval becomes wider.

Question 13

Which of the following is a common mistake when interpreting a confidence interval?

Accepted Answer

Saying there is a 95% chance that the true value is in the interval for this sample.

Question 14

A bag of M&Ms is found to have a 95% confidence interval for the proportion of green M&Ms as $$0.131 to$$ $$0.267. $$What is the point estimate for the proportion?

Accepted Answer

$$\hat{p} = \frac{0.267 + 0.131}{2} = 0.199$$

Question 15

Which of the following best describes a critical value in the context of confidence intervals?

Accepted Answer

A value that separates significant statistics from those that are not significant.

Question 16

If a margin of error is given as a percentage, what must you do before using it in the sample size formula?

Accepted Answer

Convert it to a decimal.

Question 17

Which calculator function is used to construct a confidence interval for a population proportion on a TI-83/84 calculator?

Accepted Answer

1-PropZInt

Question 18

Why is it important that the sample for a confidence interval is a simple random sample and not a voluntary response sample?

Accepted Answer

Voluntary response samples can introduce bias and do not represent the population accurately.

Question 19

If you want to estimate a population proportion with a margin of error of 0.05 and a 95% confidence level, and you have no prior estimate for $$p$$, what sample size should you use?

Accepted Answer

$$n = \frac{(1.96)^2$}{4 	imes (0.05)^2$} = 384$$

Estimating a Population Proportion and Confidence Intervals

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