Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 56 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 56 cases: n = 56; ; S = 20; ; Test Statistic = 1.5; Critical Value = 1.282. Decision = Reject . What critical value should the manager use to determine the rejection region?

Answer 1

To determine the critical value for the rejection region in this hypothesis test, we need to consider the significance level (\(\alpha\)) and the type of test being performed. The manager is conducting a one-tailed test (specifically testing if the mean number of defective bulbs is greater than 20).

Given:

• Significance level (\(\alpha\)) = 0.10
• Test Type = One-tailed (right tail)

For a one-tailed test at \(\alpha = 0.10\), we need to find the critical value from the standard normal distribution (Z distribution). The critical value marks the point beyond which we will reject the null hypothesis.

Using Z-tables or standard normal distribution calculators, we find that:

The critical value \(z\) for \(\alpha = 0.10\) in a one-tailed test is approximately 1.282.

This means that the manager's critical value of 1.282 is correct for determining the rejection region for her hypothesis test. If the calculated test statistic exceeds this critical value, she will reject the null hypothesis. Since the test statistic provided is 1.5, which exceeds 1.282, the decision to reject the null hypothesis is appropriately justified.