Q: A poll finds that 340 out of 800 teenagers text while driving. Construct a 95% confidence interval for the proportion of teenagers who text while driving. What is the correct lower bound?

$$0.34 - 1.96 \sqrt{\frac{0.34 \times 0.66}{800}} = 0.307$$

Q: A medical researcher wants to estimate the proportion of patients who respond to a new drug within a margin of error of 0.03 and 99% confidence. No prior estimate is available. What sample size should be used?

$$n = \frac{(2.575)^2$ \times 0.25}{0.03^2$} = 1848$$

Q: A sample of 16 Toyota Camry cars has a mean mpg of 28.1 and a sample standard deviation of 2.38. What is the margin of error for a 95% confidence interval for the mean mpg?

$$E = 2.131 \times \frac{2.38}{\sqrt{16}} = 1.27$$

Q: A scientist wants to estimate the mean weight of a species of fish within 0.5 kg with 95% confidence. The sample standard deviation is 2.0 kg. What sample size is required?

$$n = \left( \frac{1.96 \times 2.0}{0.5} \$$right)^2$$ = 61.47$$, so 62 fish

Question 1

A researcher wants to estimate the proportion of college students who own a car. In a random sample of 500 students, 220 report owning a car. What is the point estimate for the population proportion?

Accepted Answer

$$\frac{220}{500} = 0.44$$

Question 2

Which formula is used to calculate the margin of error for a confidence interval for a population proportion at a 95% confidence level?

Accepted Answer

$$E = 1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

A poll finds that 340 out of 800 teenagers text while driving. Construct a 95% confidence interval for the proportion of teenagers who text while driving. What is the correct lower bound?

Accepted Answer

$$0.34 - 1.96 \sqrt{\frac{0.34 	imes 0.66}{800}} = 0.307$$

Question 4

If you increase the level of confidence from 95% to 99% for a confidence interval, what happens to the margin of error?

Accepted Answer

The margin of error increases

Question 5

Which of the following statements about confidence intervals is TRUE?

Accepted Answer

A 95% confidence interval means that 95% of all possible samples will produce intervals containing the true parameter.

Question 6

A medical researcher wants to estimate the proportion of patients who respond to a new drug within a margin of error of 0.03 and 99% confidence. No prior estimate is available. What sample size should be used?

Accepted Answer

$$n = \frac{(2.575)^2$ 	imes 0.25}{0.03^2$} = 1848$$

Question 7

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

Accepted Answer

The t-distribution has heavier tails and varies with degrees of freedom

Question 8

A sample of 16 Toyota Camry cars has a mean mpg of 28.1 and a sample standard deviation of 2.38. What is the margin of error for a 95% confidence interval for the mean mpg?

Accepted Answer

$$E = 2.131 	imes \frac{2.38}{\sqrt{16}} = 1.27$$

Question 9

When constructing a confidence interval for a population mean using the t-distribution, which condition must be checked if the sample size is less than 30?

Accepted Answer

The sample data should be approximately normal with no outliers

Question 10

A scientist wants to estimate the mean weight of a species of fish within 0.5 kg with 95% confidence. The sample standard deviation is 2.0 kg. What sample size is required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 2.0}{0.5} \$$right)^2$$ = 61.47$$, so 62 fish

Question 11

Which of the following best describes the interpretation of a 99% confidence interval for a population mean?

Accepted Answer

If we take many samples, about 99% of the intervals constructed will contain the true mean.

Question 12

A critical value $$t_{0.10}$$ with 15 degrees of freedom is needed. Which statement is correct?

Accepted Answer

$$t_{0.10} is $$greater than the corresponding $$z_{0.10}$$ value because the t-distribution is more spread out.

Question 13

Which formula is used to calculate the sample size needed to estimate a population proportion with a specified margin of error $$E$$ and confidence level $$(1-\alpha)$$, given a prior estimate $$\hat{p}$$?

Accepted Answer

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

Question 14

Why does the t-distribution have heavier tails than the standard normal distribution?

Accepted Answer

Because the sample standard deviation $$s is $$used instead of the population standard deviation $$\sigma$$, introducing more variability.

Question 15

A survey reports a sample mean of 28.1 mpg and a sample standard deviation of 2.38 mpg for 16 cars. What is the 95% confidence interval for the population mean mpg?

Accepted Answer

$$28.1 \pm 2.131 	imes \frac{2.38}{4}$$, or (26.83, 29.37)

Question 16

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 17

If a prior estimate of $$p is $$unavailable, what value should be used for $$\hat{p}$$ when calculating the required sample size for a population proportion?

Accepted Answer

$$\hat{p} = 0.5$$

Ch.9-Estimating Population Proportions: Point Estimates and Confidence Intervals

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