Finding Critical Values and Rejection Regions In Exercises 23–28, find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.


Right-tailed test, α = 0.08

Verified step by step guidance

1. Step 1: Understand the problem. This is a right-tailed z-test with a significance level (α) of 0.08. A right-tailed test means the rejection region is in the right tail of the standard normal distribution.
2. Step 2: Recall the relationship between the significance level (α) and the critical value. The critical value is the z-score that corresponds to the cumulative probability of 1 - α in the standard normal distribution.
3. Step 3: Use a z-table or statistical software to find the z-score that corresponds to a cumulative probability of 1 - α = 1 - 0.08 = 0.92. This z-score is the critical value for the test.
4. Step 4: Define the rejection region. For a right-tailed test, the rejection region consists of all z-scores greater than the critical value. This means if the test statistic falls in this region, you reject the null hypothesis.
5. Step 5: Visualize the rejection region on a standard normal distribution graph. Mark the critical value on the horizontal axis, shade the area to the right of this value to represent the rejection region, and label the area as α = 0.08.

Key Concepts

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. In hypothesis testing, it is determined based on the significance level (α) and the type of test (one-tailed or two-tailed). For a right-tailed test, the critical value corresponds to the z-score that marks the threshold for the upper tail of the distribution.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a right-tailed test, this region is located to the right of the critical value. It represents the area under the curve where the probability of observing a test statistic is less than the significance level (α), indicating that the observed result is statistically significant.

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). It is a threshold set by the researcher before conducting the test, commonly used values are 0.05, 0.01, and in this case, 0.08. The choice of α influences the critical value and the size of the rejection region, impacting the test's sensitivity.