Q: Which formula is used to calculate the margin of error for a confidence interval for a population mean when the population standard deviation is unknown?

$$E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}$$

Q: A university wants to estimate the mean one-way commute time for its students with a margin of error of ±3 minutes at 95% confidence. If the estimated standard deviation is 15 minutes, what is the minimum required sample size?

$$n = (1.96 \times 15 / 3)^2$ = 96.04$$, so $$n = 97$$

Question 1

The image above shows a normal distribution with two shaded regions in the tails. In the context of constructing a 95% confidence interval for a population proportion, what do the shaded regions represent?

Accepted Answer

The total area in both tails, representing $$\alpha$$, the probability that the true parameter is not captured by the interval.

Question 2

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

Accepted Answer

We are 95% confident that the interval contains the true population mean, based on the long-run success rate of the method.

Question 3

A researcher wants to estimate the proportion of students who use reusable water bottles. In a sample of 400 students, 248 report regular use. What is the point estimate for the population proportion?

Accepted Answer

$$p̂ = \frac{248}{400} = 0.620$$

Question 4

Which of the following is NOT a requirement for using the Wald confidence interval for a population proportion?

Accepted Answer

The population must be normally distributed.

Question 5

For a 95% confidence interval for a population proportion, what is the critical value $$z_{\alpha/2}$$?

Accepted Answer

$$z_{\alpha/2} = 1.960$$

Question 6

A 95% confidence interval for a population proportion is reported as (0.58, 0.81). What is the margin of error?

Accepted Answer

$$E = \frac{0.81 - 0.58}{2} = 0.115$$

Question 7

If a researcher wants to decrease the margin of error for a confidence interval while keeping the confidence level the same, what should they do?

Accepted Answer

Increase the sample size.

Question 8

Which formula is used to calculate the margin of error for a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$E = t_{\alpha/2} 	imes \frac{s}{\sqrt{n}}$$

Question 9

A sample of 25 students has a mean commute time of 34.0 minutes and a sample standard deviation of 12.0 minutes. For a 95% confidence interval, what are the degrees of freedom?

Accepted Answer

df = 24

Question 10

Which of the following changes will make a confidence interval for a mean wider, assuming all other factors are held constant?

Accepted Answer

Decreasing the sample size.

Question 11

A university wants to estimate the mean one-way commute time for its students with a margin of error of ±3 minutes at 95% confidence. If the estimated standard deviation is 15 minutes, what is the minimum required sample size?

Accepted Answer

$$n = (1.96 	imes 15 / 3)^2$ = 96.04$$, so $$n = 97$$

Question 12

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

Accepted Answer

The t distribution has heavier tails than the normal distribution, especially for small sample sizes.

Question 13

If a 99% confidence interval is desired instead of a 95% confidence interval, what will happen to the width of the interval (all else equal)?

Accepted Answer

The interval will become wider.

Question 14

A 95% confidence interval for a mean is reported as (21.4, 27.8). What is the sample mean?

Accepted Answer

$$\bar{x} = \frac{21.4 + 27.8}{2} = 24.6$$

Question 15

Which of the following is a correct interpretation of a 90% confidence interval for a population proportion (0.41, 0.49)?

Accepted Answer

We are 90% confident that the interval 0.41 to 0.49 contains the true population proportion.

Question 16

When planning a study to estimate a population proportion with no prior estimate, which value should be used for $$p̂ to $$ensure a conservative sample size calculation?

Accepted Answer

$$p̂ = 0.50$$

Question 17

Which of the following is a limitation of the Wald confidence interval for a population proportion?

Accepted Answer

It can have poor coverage with small samples or proportions near 0 or 1.

Question 18

A sample of 18 observations is used to construct a t confidence interval for a mean. What is the correct degrees of freedom?

Accepted Answer

df = 17

Question 19

Which of the following is NOT a correct reason to use the t distribution instead of the z distribution when constructing a confidence interval for a mean?

Accepted Answer

The sample is from a categorical variable.

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