Statistics 2

In-Class Exercise 4

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

z Test for Two Means

Two campuses report their students’ scores on the same English placement test. The population standard deviations are known.

	n	\bar{X}	\sigma
Campus A	45	78.2	9.4
Campus B	50	74.6	10.1

Test the claim that the two campuses have different mean scores at \alpha=0.05.

1. State H_0 and H_1.
2. Compute z=(\bar{X}_1-\bar{X}_2)/\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2} in R.
3. Find the critical values with qnorm() and state the decision.

Question 2

t Test for Two Independent Means

Two training programmes are compared. The population variances are unknown and are not assumed equal.

	n	\bar{X}	s
Programme 1	14	32.5	4.2
Programme 2	12	28.9	5.6

Test the claim that Programme 1 has a higher mean score at \alpha=0.05.

1. State H_0 and H_1.
2. Compute t=(\bar{X}_1-\bar{X}_2)/\sqrt{s_1^2/n_1+s_2^2/n_2} in R.
3. Use d.f.=\min(n_1-1,\ n_2-1), find the critical value with qt(), and state the decision.

Question 3

Confidence Interval for the Difference

Using the data in Question 1:

1. Construct the 95% confidence interval for \mu_1-\mu_2.
2. Does the interval contain 0?
3. Explain in one or two sentences how the interval leads to the same conclusion as the hypothesis test.