Statistics 2

In-Class Exercise 1

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

Stating the Hypotheses

For each claim, state H_0 and H_1, and identify the test as left-tailed, right-tailed, or two-tailed.

1. A manufacturer claims that the mean lifetime of its light bulbs is 1,200 hours.
2. A nutritionist claims that the mean daily sugar intake of teenagers is more than 65 grams.
3. A bank claims that the mean waiting time at its branches is less than 5 minutes.

Question 2

z Test for a Mean

A researcher claims that the mean monthly spending on food delivery exceeds NT$1,800. A sample of 50 users has a mean of NT$1,935. The population standard deviation is NT$420. Use \alpha=0.05.

1. State H_0 and H_1.
2. Compute the test statistic z=(\bar{X}-\mu)/(\sigma/\sqrt{n}) in R.
3. Find the critical value with qnorm() and state the decision.
4. Compute the P-value with pnorm() and confirm that it leads to the same decision.

Question 3

Errors and Critical Values

1. In the context of Question 2, describe what a Type I error and a Type II error would mean.
2. Find the two critical values for a two-tailed test at \alpha=0.01 with qnorm().
3. Explain in one sentence what happens to the chance of a Type I error when \alpha is lowered from 0.05 to 0.01.