Type 2 Error
Calculating the Probability of a Type II Error
To properly interpret the results of a test of hypothesis requires that you be able to judge the pvalue of the test. However, to do so also requires that you have an understanding of the relationship between Type I and Type II errors. Here, we describe how the probability of a Type II error is computed. A Type II error occurs when a false null hypothesis is not rejected. For example, if a rejection region is as follows:
xbar < 127.06 or xbar > 132.94
and the null hypothesis is false, then the probability of a Type II error is defined as
= P(127.06 < xbar < 132.94 (given that H0 is false)
The condition that the null hypothesis is false only tells us that the mean is not equal to 130. If we want to compute , we need to specify a value for . Suppose that we want to determine the probability of making a Type II error when, in actual fact, = 135, 131, 139, and/or any other value.
A Windmill Example:
The feasibility of constructing a profitable electricityproducing windmill depends on the average velocity of the wind. For a certain type of windmill, the average wind speed would have to exceed 20 mph in order for its construction to be feasible. To test whether or not a particular site is appropriate for this windmill, 50 readings of the wind velocity are taken, and the average is calculated. The test is designed to answer the question, is the site feasible? That is, is there sufficient evidence to conclude that the average wind velocity exceeds 20 mph? We want to test the following hypotheses. H0: A 20 HA: A > 20
If, when the test is conducted, a Type I error is committed (rejecting the null hypothesis when it is true), we would conclude mistakenly that the average wind velocity exceeds 20 mph. The consequence of this decision is that the windmill would be built on an inappropriate site. Because this error is quite
To properly interpret the results of a test of hypothesis requires that you be able to judge the pvalue of the test. However, to do so also requires that you have an understanding of the relationship between Type I and Type II errors. Here, we describe how the probability of a Type II error is computed. A Type II error occurs when a false null hypothesis is not rejected. For example, if a rejection region is as follows:
xbar < 127.06 or xbar > 132.94
and the null hypothesis is false, then the probability of a Type II error is defined as
= P(127.06 < xbar < 132.94 (given that H0 is false)
The condition that the null hypothesis is false only tells us that the mean is not equal to 130. If we want to compute , we need to specify a value for . Suppose that we want to determine the probability of making a Type II error when, in actual fact, = 135, 131, 139, and/or any other value.
A Windmill Example:
The feasibility of constructing a profitable electricityproducing windmill depends on the average velocity of the wind. For a certain type of windmill, the average wind speed would have to exceed 20 mph in order for its construction to be feasible. To test whether or not a particular site is appropriate for this windmill, 50 readings of the wind velocity are taken, and the average is calculated. The test is designed to answer the question, is the site feasible? That is, is there sufficient evidence to conclude that the average wind velocity exceeds 20 mph? We want to test the following hypotheses. H0: A 20 HA: A > 20
If, when the test is conducted, a Type I error is committed (rejecting the null hypothesis when it is true), we would conclude mistakenly that the average wind velocity exceeds 20 mph. The consequence of this decision is that the windmill would be built on an inappropriate site. Because this error is quite