Question 1

Which of the following formulas is used to calculate the test statistic for the Student t-distribution when estimating a population mean with unknown population standard deviation?

Accepted Answer

$$ t = \frac{\overline{X} - \mu}{\frac{s}{\sqrt{n}}} $$

Question 2

A researcher collects a simple random sample of size $$n = 25$$ from a normally distributed population with unknown standard deviation. What is the degrees of freedom for the t-distribution in this scenario?

Accepted Answer

$$ 24 $$

Question 3

Which of the following conditions justifies the use of the Student t-distribution for constructing a confidence interval for the population mean?

Accepted Answer

The population is normally distributed and the population standard deviation is unknown.

Question 4

Given a sample mean $$\overline{X} = 8.953 g$$, sample standard deviation $$s = 0.25 g$$, and sample size $$n = 38$$, construct the 95% confidence interval for the mean weight of Reese’s Peanut Butter Cup Miniatures. Use $$t_{\alpha/2} = 2.028$$ for 37 degrees of freedom.

Accepted Answer

$$ (8.872, 9.034) $$

Question 5

Which of the following best describes the effect of increasing sample size on the shape of the Student t-distribution?

Accepted Answer

The t-distribution approaches the standard normal distribution as sample size increases.

Question 6

A 95% confidence interval for a population mean is calculated as $$(8.872, 9.034). $$What does this interval mean in the context of the Reese’s Peanut Butter Cup Miniatures example?

Accepted Answer

We are 95% confident that the true mean weight of the cups is between 8.872 g and 9.034 g.

Question 7

When should the standard normal (z) distribution be used instead of the Student t-distribution for constructing a confidence interval for the population mean?

Accepted Answer

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

Question 8

A sample of 15 patients has a mean cholesterol level of 210 mg/dL with a sample standard deviation of 20 mg/dL. Construct the 99% confidence interval for the population mean cholesterol level. Use $$t_{0.005, 14} = 2.977.$$

Accepted Answer

$$ (196.66, 223.34) $$

Question 9

Which of the following is the correct formula for the margin of error $$E$$ when using the Student t-distribution?

Accepted Answer

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

Question 10

A company claims that their packages contain 340.2 g of product. A sample of 38 packages has a mean of 8.953 g per item. Based on a 95% confidence interval of $$(8.872, 9.034)$$, what can you conclude about the company’s claim?

Accepted Answer

There is evidence that the packages may not contain the claimed amount, since 340.2/38 = 8.953 g is at the edge of the interval.

Question 11

If the population is not normally distributed and $$n \u003c 30$$, which method should be used to estimate the population mean?

Accepted Answer

A nonparametric method such as bootstrapping.

Question 12

Which of the following statements about degrees of freedom in the context of the t-distribution is correct?

Accepted Answer

Degrees of freedom equal the sample size minus one: $$df = n - 1.$$

Question 13

A sample of size $$n = 50 is $$drawn from a population with unknown standard deviation. Which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Student t-distribution.

Question 14

Which of the following is NOT a requirement for estimating the population mean using the Student t-distribution?

Accepted Answer

The population standard deviation is known.

Question 15

Why are the tails of the Student t-distribution thicker than those of the standard normal distribution?

Accepted Answer

Because the sample standard deviation is used as an estimate, introducing more variability.

Estimating Parameters: Confidence Intervals for the Population Mean

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