Question 1

Which of the following statements best describes when to use the Student's t-distribution instead of the standard normal (z) distribution in hypothesis testing?

Accepted Answer

When the sample size is small (n \u003c 30) and the population standard deviation is unknown.

Question 2

A researcher collects a sample of n = 15 and wants to test a hypothesis about the population mean. The population standard deviation is unknown. Which formula should be used to calculate the test statistic?

Accepted Answer

$$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$$

Question 3

Which property distinguishes the t-distribution from the standard normal distribution?

Accepted Answer

The t-distribution has thicker tails than the standard normal distribution.

Question 4

A local telephone company claims the average call length is 8 minutes. A sample of 18 calls has a mean of 7.8 minutes and a standard deviation of 0.5 minutes. At $$\alpha = 0.05$$, what are the degrees of freedom for the appropriate t-test?

Accepted Answer

$$df = 17$$

Question 5

Given a two-tailed test with $$\alpha = 0.10$$ and $$n = 12$$, what are the critical values for the t-distribution?

Accepted Answer

$$-1.796$$ and $$1.796$$

Question 6

Which of the following best explains the concept of degrees of freedom in the context of estimating a sample mean?

Accepted Answer

It is the number of values in the sample that are free to vary after one parameter (the mean) has been estimated.

Question 7

A used car dealer claims the mean price of a car is at least $$20,500. A sample of 14 cars has a mean of 19$$,850 and a standard deviation of $1,084. What is the correct null hypothesis for testing the dealer's claim?

Accepted Answer

$$H_0$$: \mu \geq 20,500$$

Question 8

In a left-tailed t-test with $$\alpha = 0.05$$ and $$n = 21$$, what is the critical value for the rejection region?

Accepted Answer

$$-1.725$$

Question 9

If the calculated t-statistic falls within the non-rejection region in a two-tailed test, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis.

Question 10

A sample of 19 water samples from a river has a mean pH of 6.7 and a standard deviation of 0.24. The company claims the mean pH is 6.8. At $$\alpha = 0.05$$, what is the value of the test statistic?

Accepted Answer

$$t = \frac{6.7 - 6.8}{0.24 / \sqrt{19}} \approx -1.816$$

Question 11

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

Accepted Answer

The t-distribution is always skewed to the left.

Question 12

Why are the tails of the t-distribution 'thicker' than those of the standard normal distribution?

Accepted Answer

Because the t-distribution accounts for additional uncertainty due to estimating the population standard deviation from the sample.

Question 13

A researcher is conducting a right-tailed t-test with $$\alpha = 0.01$$ and $$n = 17. $$What is the critical value for the rejection region?

Accepted Answer

$$2.583$$

Question 14

Which of the following steps is NOT part of the hypothesis testing procedure for a small sample mean using the t-distribution?

Accepted Answer

Estimate the population standard deviation using the sample mean.

Question 15

If a sample size is less than 30 and the population is not normally distributed, which of the following is the best course of action?

Accepted Answer

Transform the data to make it approximately normal or use a nonparametric test.

Question 16

A two-tailed t-test is performed with $$\alpha = 0.05$$ and $$n = 18. $$What are the critical values for the rejection regions?

Accepted Answer

$$-2.110$$ and $$2.110$$

Question 17

Which of the following real-world scenarios is most appropriate for a one-tailed t-test?

Accepted Answer

Testing if a new medication decreases blood pressure compared to the current standard.

Hypothesis Testing for the Mean with the Student's t-Distribution (Small Samples)

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