Question 1

Which of the following is TRUE about the t-distribution when estimating a population mean with unknown standard deviation?

Accepted Answer

It is symmetric about the mean and becomes closer to the z-distribution as sample size increases.

Question 2

A researcher wants to estimate the mean systolic blood pressure of a population. The population standard deviation is known to be 10 mmHg. What is the formula for the margin of error at a 95% confidence level?

Accepted Answer

$$E = z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}}$$

Question 3

Which of the following is a requirement for using the t-distribution to construct a confidence interval for a mean?

Accepted Answer

The population must be normally distributed or the sample size must be large (n \u003e 30).

Question 4

Given a sample mean of 133.93, a sample standard deviation of 9.04, and a sample size of 14, what is the 95% confidence interval for the population mean using the t-distribution? (Use $$t_{0.025, 13} \approx 2.160)$$

Accepted Answer

$$133.93 \pm 2.160 \cdot \dfrac{9.04}{\sqrt{14}}$$

Question 5

Why did Sir William Gossett recommend using a larger critical value (the t-statistic) instead of the z-statistic when estimating means with unknown population standard deviation?

Accepted Answer

Because the t-distribution accounts for the extra uncertainty from estimating the standard deviation with the sample.

Question 6

A scientist wants to estimate the average time for a three-minute hourglass timer. She collects 24 measurements with a sample mean of 179.42 seconds and a sample standard deviation of 3.45 seconds. What is the 95% confidence interval for the mean time? (Use $$t_{0.025, 23} = 2.069)$$

Accepted Answer

$$179.42 \pm 2.069 \cdot \dfrac{3.45}{\sqrt{24}}$$

Question 7

Which of the following scenarios requires the use of a nonparametric method or bootstrapping instead of the z or t distribution?

Accepted Answer

Population is not normally distributed and $$n \u003c 30.$$

Question 8

If you want to achieve a margin of error of 1 mmHg at a 99% confidence level, and the population standard deviation is 10 mmHg, what is the minimum sample size required? (Use $$z_{0.005} = 2.576)$$

Accepted Answer

$$n = \left[ \dfrac{2.576 \cdot 10}{1} \right]^2$$

Question 9

Which of the following is NOT a property of the t-distribution?

Accepted Answer

It is always narrower than the z-distribution.

Question 10

A sample of 14 blood pressure readings has a mean of 133.93 and a population standard deviation of 10 mmHg. What is the 95% confidence interval for the population mean? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$133.93 \pm 1.96 \cdot \dfrac{10}{\sqrt{14}}$$

Question 11

Which of the following best describes the effect of increasing the sample size on the width of a confidence interval for the mean?

Accepted Answer

The confidence interval becomes narrower.

Question 12

When should you use the z-distribution instead of the t-distribution for constructing a confidence interval for a mean?

Accepted Answer

When the population standard deviation is known and the population is normally distributed or $$n \u003e 30.$$

Question 13

A histogram of sample data is approximately symmetric and bell-shaped. Which assumption for using the t-interval is most likely satisfied?

Accepted Answer

The sample is random and the population is normal.

Question 14

Which of the following is the correct formula for the sample size needed to estimate a population mean with a specified margin of error $$E$$, confidence level, and known population standard deviation $$\sigma$$?

Accepted Answer

$$n = \left[ \dfrac{z_{\alpha/2} \cdot \sigma}{E} \right]^2$$

Question 15

A sample of 24 hourglass times has a mean of 179.42 seconds and a standard deviation of 3.45 seconds. What is the standard error of the mean?

Accepted Answer

$$\dfrac{3.45}{\sqrt{24}}$$

Estimating a Population Mean: Confidence Intervals and the t-Distribution

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