In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.


Right-tailed test, α=0.025

Verified step by step guidance

1. Step 1: Understand the problem. This is a right-tailed z-test with a significance level (α) of 0.025. The goal is to find the critical value(s) and the rejection region(s). A right-tailed test means we are looking for the critical value where the area to the right under the standard normal curve equals 0.025.
2. Step 2: Recall the relationship between the significance level (α) and the z-score. For a right-tailed test, the critical value corresponds to the z-score such that the cumulative probability to the left of the z-score is 1 - α. In this case, 1 - α = 1 - 0.025 = 0.975.
3. Step 3: Use a z-table or statistical software to find the z-score that corresponds to a cumulative probability of 0.975. 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 found in Step 3. This means if the test statistic falls in this region, we reject the null hypothesis.
5. Step 5: Visualize the result. On a standard normal distribution graph, shade the area to the right of the critical value (representing the rejection region). Label the critical value on the x-axis and indicate that the area under the curve in the rejection region is equal to α = 0.025.

Key Concepts

Critical Value

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For a right-tailed z-test, it is the z-score that corresponds to the specified level of significance (α). In this case, with α = 0.025, the critical value indicates the point beyond which the null hypothesis will be rejected.

Rejection Region

The rejection region is the area in the tail of the distribution where, if the test statistic falls, the null hypothesis is rejected. For a right-tailed test with α = 0.025, the rejection region is located to the right of the critical value. This region represents the outcomes that are statistically significant, indicating strong evidence against the null hypothesis.

Z-Test

A z-test is a statistical test used to determine if there is a significant difference between sample and population means when the population variance is known. It utilizes the standard normal distribution to calculate the z-score, which helps in comparing the sample mean to the population mean. In this context, the z-test is applied to assess the hypothesis under the specified significance level.