Q: A researcher wants to estimate the mean tensile strength of an alloy. A sample of $$n=15$$ yields $$\overline{x}=39.3$$ and $$s=2.6. $$Find the 90% confidence interval for the mean, given $$t_{0.05,14}=1.761.$$

$$(39.3 \pm 1.761 \times \frac{2.6}{\sqrt{15}})$$

Question 1

Which of the following statements best describes the main difference between the t-distribution and the standard normal (z) distribution?

Accepted Answer

The t-distribution has thicker tails than the z-distribution, especially for small sample sizes.

Question 2

Suppose you are testing the mean weight of canned fruit from a factory. The population standard deviation is unknown, and you collect a sample of $$n=12$$ cans. Which test statistic should you use to test the hypothesis $$H_0$$: \mu = 400$$?

Accepted Answer

$$t = \frac{\overline{x} - 400}{s/\sqrt{n}}$$

Question 3

What does the 'degrees of freedom' ($$df) $$represent in the context of the t-distribution for a one-sample mean test?

Accepted Answer

The number of independent values that can vary in the sample, calculated as $$n-1.$$

Question 4

A researcher wants to estimate the mean tensile strength of an alloy. A sample of $$n=15$$ yields $$\overline{x}=39.3$$ and $$s=2.6. $$Find the 90% confidence interval for the mean, given $$t_{0.05,14}=1.761.$$

Accepted Answer

$$(39.3 \pm 1.761 	imes \frac{2.6}{\sqrt{15}})$$

Question 5

Which of the following is TRUE about the relationship between the t-distribution and the standard normal distribution as the sample size increases?

Accepted Answer

The t-distribution approaches the standard normal distribution as the degrees of freedom increase.

Question 6

A brewery is testing a new process and collects a sample of $$n=5$$ times to travel a route: 29, 27, 30, 26, 28 minutes. The sample mean is 28, and the sample standard deviation is 1.58. Test $$H_0$$: \mu = 30$$ vs H_1$: \mu \u003c 30 at$$ $$\alpha=0.05$$ ($$t_{0.05,4} = -2.132). $$What is the correct conclusion?

Accepted Answer

Reject $$H_0$$; the new route is significantly faster at the 5% level.

Question 7

If the p-value for a t-test is 0.12 and the significance level $$\alpha is 0.05$$, what is the correct interpretation?

Accepted Answer

Fail to reject the null hypothesis because the p-value is greater than $$\alpha.$$

Question 8

Which of the following formulas gives the 95% confidence interval for the mean when the population standard deviation is known?

Accepted Answer

$$\overline{x} \pm z_{0.025} \frac{\sigma}{\sqrt{n}}$$

Question 9

A sample of $$n=10$$ yields $$\overline{x}=194.8$$, $$s=13.14. $$Test $$H_0$$: \mu=200$$ vs H_1$: \mu\u003c200 at$$ $$\alpha=0.05. $$The test statistic is $$t = -1.2514$$ and the critical value is $$-1.833. $$What is the correct decision?

Accepted Answer

Fail to reject $$H_0$$; there is not enough evidence that the mean is less than 200.

Question 10

Why is the t-distribution used instead of the z-distribution when the population variance is unknown and the sample size is small?

Accepted Answer

Because the t-distribution accounts for the additional uncertainty in estimating the population variance from the sample.

Question 11

Given a two-sided hypothesis test $$H_0$$: \mu = \$$mu_0 vs$$ $$H_1$$: \mu 
eq \$$mu_0$$ with sample size $$n=9$$, sample mean $$\overline{x}=8.3$$, and $$s=1.2$$, what is the 95% confidence interval for $$\mu if$$ $$t_{0.025,8}=2.306$$?

Accepted Answer

$$(8.3 \pm 2.306 	imes \frac{1.2}{\sqrt{9}})$$

Question 12

Which of the following best describes the meaning of a 90% confidence interval for a population mean?

Accepted Answer

If we repeatedly take samples and calculate intervals, about 90% of those intervals will contain the true population mean.

Question 13

A company wants to test if the mean weight of their canned fruit is less than the standard 400g. A sample of $$n=30$$ yields $$\overline{x}=390$$, $$\sigma=50. $$Calculate the test statistic for $$H_0$$: \mu=400$$ vs H_1$: \mu\u003c400.$$

Accepted Answer

$$z = \frac{390-400}{50/\sqrt{30}}$$

Question 14

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

Accepted Answer

Its variance is always less than 1.

Question 15

In a left-tailed t-test, the p-value is defined as:

Accepted Answer

$$P(t \leq t_0$)$$ where $$t_0 is $$the observed test statistic.

Question 16

Why did William Gosset publish his work on the t-distribution under the pseudonym 'Student'?

Accepted Answer

Because his employer, Guinness, did not allow employees to publish under their own names.

Inference for the Mean in Normal Populations with Small Samples: t-Distribution, Confidence Intervals, and Hypothesis Testing

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