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 standard deviation is unknown and the sample size is greater than 30.

Question 4

A sample of 38 Reese’s Peanut Butter Cup Miniatures has a total weight of 340.2 g. What is the sample mean weight per cup?

Accepted Answer

$$ 8.953  g$$

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 constructed using a sample mean of $$\overline{X} = 8.953$$, sample standard deviation $$s = 0.5$$, and sample size $$n = 38. $$What is the correct formula for the margin of error $$E$$ using the t-distribution?

Accepted Answer

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

Question 7

If a population is not normally distributed and the sample size is less than 30, which method should be used to estimate the population mean?

Accepted Answer

Use the bootstrapping method or another nonparametric method.

Question 8

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

Accepted Answer

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

Question 9

A 95% confidence interval for a population mean is calculated as $$(8.548, 9.042). $$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 a Reese’s Peanut Butter Cup Miniature is between 8.548 g and 9.042 g.

Question 10

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

Accepted Answer

The population standard deviation must be known.

Question 11

A researcher wants to estimate the mean systolic blood pressure of adults in a city. The population standard deviation is unknown, and a sample of 50 adults is taken. Which distribution should be used to construct the confidence interval?

Accepted Answer

Student t-distribution

Question 12

Which of the following best explains why the t-distribution is used instead of the normal distribution when the population standard deviation is unknown?

Accepted Answer

The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.

Question 13

Suppose you want to construct a 99% confidence interval for a population mean using a sample of size $$n = 15. $$Which critical value should you use?

Accepted Answer

$$ t_{\alpha/2} $$ with $$df = 14$$

Question 14

A sample mean is calculated as $$\overline{X} = 8.953$$ with a 95% confidence interval of $$(8.548, 9.042). If $$the package label claims the mean is 9.0 g, what can you conclude?

Accepted Answer

The label claim is plausible because 9.0 g is within the confidence interval.

Question 15

Which of the following is a correct interpretation of the shaded regions in the t-distribution curve for a two-tailed test at the 5% significance level?

Accepted Answer

Each shaded region represents 2.5% of the total area, corresponding to the probability of observing a value as extreme or more extreme than the critical value under the null hypothesis.

Estimating Parameters and Determining Sample Sizes: Confidence Intervals for Population Mean and Proportion

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