Question 1

Which of the following statements best describes the main difference between the Student’s t-distribution and the standard normal (z) distribution?

Accepted Answer

The t-distribution is more spread out and has heavier tails than the z-distribution, especially for small sample sizes.

Question 2

Suppose you are testing whether the mean fill of apple juice bottles is less than the target value of 64.05 oz. You collect a random sample of 22 bottles and find $$\bar{y} = 64.0073$$, $$s = 0.0446. $$What is the correct formula for the test statistic?

Accepted Answer

$$t = \frac{\bar{y} - \mu_0$}{s / \sqrt{n}}$$

Question 3

Why do we use $$n-1 in $$the denominator when calculating the sample standard deviation $$s$$?

Accepted Answer

Because dividing by $$n-1$$ corrects for the bias introduced by estimating the population mean with the sample mean.

Question 4

A researcher wants to construct a 99% confidence interval for the mean length-to-width ratio of bones, using a sample of 41 bones with $$\bar{y} = 9.258$$ and $$s = 1.204. $$What is the correct formula for the confidence interval?

Accepted Answer

$$\bar{y} \pm t_{0.005,40} \cdot \frac{s}{\sqrt{n}}$$

Question 5

Which of the following is NOT an assumption or condition that must be checked before using the one-sample t-test?

Accepted Answer

The sample size must be greater than 100.

Question 6

If you have a sample size of $$n = 30$$ and want to find the critical value for a two-tailed t-test at the 0.05 significance level, which value from the t-table should you use?

Accepted Answer

$$t_{0.025,29}$$

Question 7

A quality control manager wants to estimate the mean fill amount of bottles to within 0.01 oz with 99% confidence. If the estimated standard deviation is $$s = 0.0446$$, what formula should be used to determine the required sample size?

Accepted Answer

$$n = \left( \frac{z_{0.005} \cdot s}{ME} \$$right)^2$$

Question 8

Which of the following best explains why the t-distribution approaches the normal distribution as sample size increases?

Accepted Answer

Because the estimate of the population standard deviation becomes more accurate as sample size increases.

Question 9

In the context of hypothesis testing, what does a p-value represent?

Accepted Answer

The probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Question 10

A 95% confidence interval for the mean is calculated as $$[8.75, 9.77]. $$Which of the following statements is correct?

Accepted Answer

We are 95% confident that the true population mean lies between 8.75 and 9.77.

Question 11

Which of the following scenarios would most likely violate the Nearly Normal Condition for using the t-distribution?

Accepted Answer

A histogram of the data shows two distinct peaks.

Question 12

Why is it important to use a smaller significance level (such as $$\alpha = 0.01) $$when the consequences of a Type I error are severe?

Accepted Answer

Because it reduces the probability of incorrectly rejecting the null hypothesis.

Question 13

Which of the following is a correct interpretation of a two-tailed hypothesis test at the 0.05 significance level?

Accepted Answer

We reject the null hypothesis if the p-value is less than or equal to 0.05.

Question 14

A sample of 41 bones has a mean length-to-width ratio of 9.258 and a standard deviation of 1.204. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{1.204}{\sqrt{41}}$$

Question 15

Which of the following is a practical implication of finding a statistically significant result in a t-test?

Accepted Answer

The result may not be practically important, and further subject-matter knowledge is needed to assess its real-world impact.

Question 16

Which of the following best describes the relationship between confidence intervals and hypothesis tests?

Accepted Answer

A (1-\alpha)100% confidence interval contains all null hypothesis values that would not be rejected by a two-sided test at significance level \alpha.

Chapter 18: Inferences About Means – The t-Distribution and Small Sample Inference

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