Question 1

A researcher wants to estimate the proportion of college students who own a laptop. In a random sample of 200 students, 150 report owning a laptop. What is the point estimate for the population proportion?

Accepted Answer

$$\hat{p} = \frac{150}{200} = 0.75$$

Question 2

Which of the following best describes the meaning of a 95% confidence interval for a population proportion?

Accepted Answer

We are 95% confident that the interval contains the true population proportion.

Question 3

Given a sample proportion $$\hat{p} = 0.6$$, sample size $$n = 100$$, and a 95% confidence level, what is the margin of error? (Use $$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$E = 1.96 \cdot \sqrt{\frac{0.6(1-0.6)}{100}} \approx 0.096$$

Question 4

Which of the following factors will decrease the margin of error in a confidence interval for a population proportion, all else held constant?

Accepted Answer

Increasing the sample size

Question 5

A sociologist wants to estimate the proportion of residents who speak only English at home. She wants a margin of error of 0.03 with 90% confidence and no prior estimate. What sample size should she use? (Use $$z_{\alpha/2} = 1.645)$$

Accepted Answer

$$n = 0.25 \left( \frac{1.645}{0.03} \$$right)^2$$ \approx 752$$

Question 6

Which of the following is a correct interpretation of a 99% confidence interval for the mean weight of pennies minted after 1982 being (2.452, 2.477) grams?

Accepted Answer

We are 99% confident that the true mean weight of pennies minted after 1982 is between 2.452 and 2.477 grams.

Question 7

A sample of 25 patients has a mean cholesterol level of 190 mg/dL and a sample standard deviation of 20 mg/dL. Construct a 95% confidence interval for the population mean cholesterol level. (Use $$t_{0.025, 24} = 2.064)$$

Accepted Answer

$$190 \pm 2.064 \cdot \frac{20}{\sqrt{25}} = 190 \pm 8.256$$

Question 8

Which property distinguishes the Student’s t-distribution from the standard normal distribution?

Accepted Answer

The t-distribution has heavier tails, especially for small sample sizes.

Question 9

A 90% confidence interval for a population proportion is calculated as $$(0.61, 0.65). $$Which of the following statements is correct?

Accepted Answer

We are 90% confident that the true population proportion is between 0.61 and 0.65.

Question 10

Which formula is used to determine the sample size needed to estimate a population mean $$\mu$$ within a specified margin of error $$E at a $$given confidence level, assuming the population standard deviation $$\sigma is $$known?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \$$right)^2$$

Question 11

A poll finds that 63% of 1783 registered voters favor a policy. What is the 90% confidence interval for the population proportion? (Use $$z_{\alpha/2} = 1.645)$$

Accepted Answer

$$0.63 \pm 1.645 \cdot \sqrt{\frac{0.63(1-0.63)}{1783}} \approx (0.61, 0.65)$$

Question 12

Which of the following is NOT a requirement for constructing a confidence interval for a population mean using the t-distribution?

Accepted Answer

The population standard deviation $$\sigma$$ must be known.

Question 13

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

Accepted Answer

The width decreases.

Question 14

A sample of 17 pennies has a sample mean weight of 2.464 grams and a sample standard deviation of 0.02 grams. What is the 99% confidence interval for the population mean? (Use $$t_{0.005, 16} = 2.921)$$

Accepted Answer

$$2.464 \pm 2.921 \cdot \frac{0.02}{\sqrt{17}} \approx (2.452, 2.477)$$

Question 15

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

Accepted Answer

As the sample size increases, the t-distribution approaches the standard normal distribution.

Question 16

A 95% confidence interval for a population mean is calculated as $$(45, 55). $$What does this mean?

Accepted Answer

We are 95% confident that the true population mean is between 45 and 55.

Estimating the Value of a Parameter: Population Proportion and Mean

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