Question 1

Which of the following statements best describes the t-distribution?

Accepted Answer

It is bell-shaped and symmetric about the mean, with thicker tails than the normal distribution.

Question 2

A researcher collects a random sample of 20 blood pressure readings from a population with an unknown standard deviation. Which distribution should be used to construct a 95% confidence interval for the mean?

Accepted Answer

The t-distribution with 19 degrees of freedom.

Question 3

What is the formula for the margin of error $$E$$ when constructing a confidence interval for the mean with unknown population standard deviation?

Accepted Answer

$$E = t_c \frac{s}{\sqrt{n}}$$

Question 4

A sample of 36 cars has a mean time on the lot of 9.75 days and a sample standard deviation of 2.39 days. What are the degrees of freedom for constructing a confidence interval for the mean?

Accepted Answer

$$35$$

Question 5

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

Accepted Answer

The standard deviation is always less than 1.

Question 6

As the degrees of freedom increase, what happens to the t-distribution?

Accepted Answer

It approaches the standard normal distribution.

Question 7

A random sample of 16 coffee shops yields a sample mean temperature of $$160^\circF $$and a sample standard deviation of $$10^\circF. $$What is the correct formula for the 95% confidence interval for the population mean temperature?

Accepted Answer

$$160 \pm t_{0.025,15} \frac{10}{\sqrt{16}}$$

Question 8

In which scenario would you use the standard normal (z) distribution instead of the t-distribution to construct a confidence interval for the mean?

Accepted Answer

When the population standard deviation is known, regardless of sample size.

Question 9

A sample of 25 houses has a mean construction cost of $$329,000 and a known population standard deviation of 50$$,000$. Which distribution should be used to construct a 95% confidence interval for the mean construction cost?

Accepted Answer

Standard normal (z) distribution.

Question 10

Which of the following best explains why the t-distribution has thicker tails than the normal distribution?

Accepted Answer

Because the sample mean is more variable when the population standard deviation is unknown.

Question 11

A 99% confidence interval for a population mean is calculated as (8.66, 10.84). What does this mean in context?

Accepted Answer

We are 99% confident that the population mean lies between 8.66 and 10.84.

Question 12

Which of the following is the correct formula for the confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_c \frac{s}{\sqrt{n}}$$

Question 13

If a sample size increases, what happens to the width of the confidence interval for the mean (assuming all else is constant)?

Accepted Answer

It decreases.

Question 14

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

Accepted Answer

The sample must be random and the population must be approximately normally distributed or the sample size must be at least 30.

Question 15

A researcher wants to estimate the average time students spend studying per week. She collects a random sample of 40 students and finds a sample mean of 15 hours with a sample standard deviation of 3 hours. What is the correct formula for the 95% confidence interval for the population mean?

Accepted Answer

$$15 \pm t_{0.025,39} \frac{3}{\sqrt{40}}$$

Confidence Intervals for the Mean (Sigma Unknown): Study Notes

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