Q: A sample of 36 policies has a mean processing time of 50 days and a standard deviation of 20 days. What is the margin of error for a 90% confidence interval if $$t_{\alpha/2} = 1.69$$?

$$E = 1.69 \times \frac{20}{\sqrt{36}} = 5.633$$

Q: You want to estimate the mean SAT Math score increase with a margin of error of 5 points and a population standard deviation of 50. What sample size is needed for a 95% confidence level? (Use $$z_{\alpha/2} = 1.96)$$

$$n = \left[ \frac{1.96 \times 50}{5} \right]^2 = 384.16$$, so 385 students

Q: Which of the following is the correct formula for determining the sample size $$n$$ needed to estimate a population proportion $$p$$ when no prior estimate is available?

$$n = \frac{(z_{\alpha/2})^2$ \times 0.25}{E^2$}$$

Question 1

Which of the following best describes a parameter in statistics?

Accepted Answer

A numerical summary of a population.

Question 2

A researcher wants to estimate the proportion of college students who own a car. She surveys 200 students and finds that 90 own a car. What is the point estimate for the population proportion?

Accepted Answer

$$\frac{90}{200} = 0.45$$

Question 3

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

Accepted Answer

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

Question 4

A 95% confidence interval for a population proportion is (0.30, 0.40). What is the margin of error?

Accepted Answer

$$E = \frac{0.40 - 0.30}{2} = 0.05$$

Question 5

If you increase the confidence level from 95% to 99% while keeping the sample size and sample proportion the same, what happens to the width of the confidence interval?

Accepted Answer

The width increases.

Question 6

Which of the following is a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

All of the above.

Question 7

A 90% confidence interval for the mean processing time of insurance policies is (44.4, 55.6) days. What is the point estimate of the mean?

Accepted Answer

$$\frac{44.4 + 55.6}{2} = 50$$

Question 8

Which distribution should be used to construct a confidence interval for a population mean when the population standard deviation $$\sigma is $$unknown and the sample size is small?

Accepted Answer

Student's t-distribution

Question 9

A sample of 36 policies has a mean processing time of 50 days and a standard deviation of 20 days. What is the margin of error for a 90% confidence interval if $$t_{\alpha/2} = 1.69$$?

Accepted Answer

$$E = 1.69 	imes \frac{20}{\sqrt{36}} = 5.633$$

Question 10

Which of the following statements about the Student t-distribution is TRUE?

Accepted Answer

It has heavier tails than the standard normal distribution.

Question 11

You want to estimate the mean SAT Math score increase with a margin of error of 5 points and a population standard deviation of 50. What sample size is needed for a 95% confidence level? (Use $$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$n = \left[ \frac{1.96 	imes 50}{5} ight]^2 = 384.16$$, so 385 students

Question 12

A 99% confidence interval for the mean box office receipts is (5.22, 31.62) million dollars. What is the margin of error?

Accepted Answer

$$E = \frac{31.62 - 5.22}{2} = 13.2$$

Question 13

Which of the following is the correct formula for determining the sample size $$n$$ needed to estimate a population proportion $$p$$ when no prior estimate is available?

Accepted Answer

$$n = \frac{(z_{\alpha/2})^2$ 	imes 0.25}{E^2$}$$

Question 14

A 95% confidence interval for the proportion of workers who aspire to have their boss’s job is (0.134, 0.266). What is the point estimate?

Accepted Answer

$$\frac{0.134 + 0.266}{2} = 0.20$$

Question 15

Which of the following best explains why a 95% confidence interval does not always contain the true population parameter?

Accepted Answer

Because 5% of such intervals, by chance, will not contain the true parameter.

Question 16

If a sample proportion $$\hat{p} = 0.46 is $$obtained from a sample of $$n = 500$$, and the critical value for a 90% confidence interval is $$z_{\alpha/2} = 1.645$$, what is the margin of error?

Accepted Answer

$$E = 1.645 	imes \sqrt{\frac{0.46 	imes 0.54}{500}} = 0.037$$

Question 17

Which of the following is NOT a requirement for constructing a confidence interval for a population mean when $$\sigma is $$unknown?

Accepted Answer

The population standard deviation must be known.

CH 7. Estimating Parameters and Determining Sample Sizes: Confidence Intervals for Proportions and Means

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