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.


Left-tailed test, α=0.02

Verified step by step guidance

1. Step 1: Understand the type of test and significance level. This is a left-tailed z-test with a significance level (α) of 0.02. A left-tailed test means the rejection region is located in the lower tail of the standard normal distribution.
2. Step 2: Recall the relationship between the significance level and the critical value. The critical value corresponds to the z-score that leaves an area of α (0.02) in the left tail of the standard normal distribution.
3. Step 3: Use a z-table or statistical software to find the z-score that corresponds to the cumulative probability of 0.02. This z-score is the critical value for the test. In MathML, the critical value can be expressed as: $z=-Z$_α, where Zα is the z-score for α = 0.02.
4. Step 4: Define the rejection region. For a left-tailed test, the rejection region includes all z-scores less than the critical value. In MathML, the rejection region can be expressed as: $z<-Z$_α.
5. Step 5: Visualize the graph. Draw a standard normal distribution curve, mark the critical value on the left tail, and shade the area to the left of the critical value to represent the rejection region. Label the critical value and the significance level (α = 0.02) on the graph.

Key Concepts

Critical Value

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For a left-tailed test, it is the z-score that corresponds to the specified level of significance (α). In this case, with α = 0.02, the critical value indicates the point below which we would reject the null hypothesis, helping to define the rejection region.

Rejection Region

The rejection region is the range of values for which the null hypothesis is rejected. In a left-tailed test, this region lies to the left of the critical value on the z-distribution. For α = 0.02, the rejection region includes all z-scores less than the critical value, indicating that if the test statistic falls within this region, we conclude that the sample provides sufficient evidence to reject 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 measures how many standard deviations an element is from the mean. In this context, the z-test is applied to assess whether the sample data significantly deviates from the hypothesized population mean under the specified significance level.