Statistics 2

In-Class Exercise 3

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

z Test for a Proportion

A university claims that 62% of its graduates find a job within three months. In a survey of 420 graduates, 275 had found a job within three months. Use \alpha=0.05 and a two-tailed test.

  1. State H_0 and H_1.
  2. Verify that np\ge5 and nq\ge5.
  3. Compute z=(\hat{p}-p)/\sqrt{pq/n} in R.
  4. Find the critical values with qnorm() and state the decision.

Question 2

Chi-Square Test for a Variance

A machine is supposed to fill bottles with a standard deviation of 4.3 millilitres. A sample of 25 bottles has a standard deviation of 5.6 millilitres. The quality manager suspects that the variation has increased. Use \alpha=0.05 and a right-tailed test.

  1. State H_0 and H_1 in terms of \sigma^2.
  2. Compute \chi^2=(n-1)s^2/\sigma^2 in R.
  3. Find the critical value with qchisq() and d.f.=n-1, then state the decision.

Question 3

The P-Value Approach

  1. For the test in Question 1, compute the P-value with pnorm(). Remember to double the tail area for a two-tailed test.
  2. For the test in Question 2, compute the P-value with pchisq().
  3. State the decision rule of the P-value approach, and confirm that both tests give the same decision as the critical value approach.