Question

Critical Values. In Exercises 21–24, refer to the information in the given exercise and do the following.

a. Find the critical value(s).

b. Using a significance level of \(\alpha \)= 0.05, should we reject \({H_0}\)or should we fail to reject \({H_0}\)?

Short answer

a.The critical value is equal to -1.645.

b.The null hypothesis is rejected.

Step-by-step solution

Given information

A test statistic value of \(z = - 2.50\) is obtained, and the claim to be tested is \(p < 0.75\).

Hypotheses and tail of the test

In correspondence with the given claim, the following hypotheses are set up:

Null Hypothesis:\(p = 0.75\)

Alternative Hypothesis:\(p < 0.75\)

Since there is a lesser than sign in the alternative hypothesis, the test is left-tailed.

Critical value

a.

Referring to the standard normal table, the critical value of z corresponding to the left-tailed test at \(\alpha = 0.05\) is equal to -1.645.

Decision about the test

b.

If the absolute value of the test statistic is greater than the critical value, then the null hypothesis is rejected.

Here, the absolute value of the test statistic (2.50) is greater than the critical value (-1.645). Thus, the null hypothesis is rejected.