Question

Ann Landers, in her advice column of October 24 , 1994 (San Luis Obispo Telegram-Tribune), described the reliability of DNA paternity testing as follows: "To get a completely accurate result, you would have to be tested, and so would (the man) and your mother. The test is 100 percent accurate if the man is not the father and \(99.9\) percent accurate if he is." a. Consider using the results of DNA paternity testing to decide between the following two hypotheses: \(H_{0}\) " a particular man is the father \(H_{a}:\) a particular man is not the father In the context of this problem, describe Type I and Type II errors. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) b. Based on the information given, what are the values of \(\alpha\), the probability of Type I error, and \(\beta\), the probability of Type II error? c. Ann Landers also stated, "If the mother is not tested, there is a \(0.8\) percent chance of a false positive." For the hypotheses given in Part (a), what are the values of \(\alpha\) and \(\beta\) if the decision is based on DNA testing in which the mother is not tested?

Short answer

The probability of a Type I error (\(\alpha\)) is 0 and the probability of a Type II error (\(\beta\)) is 0.001 based on the given information. If the mother is not tested, the probability of a Type I error changes to 0.008 while the probability of a Type II error cannot be determined.

Step-by-step solution

Define Type I and Type II errors

In the context of this problem, a Type I error would occur if we conclude that a particular man is not the father (reject \(H_{0}\)) when he actually is (meaning \(H_{0}\) is true). A Type II error would occur if we conclude that a particular man is the father (fail to reject \(H_{0}\)) when he actually is not (meaning \(H_{0}\) is false).

Calculate the probability of Type I and Type II error

Based on the information given, the test is 100 percent accurate if the man is not the father, meaning the probability of Type I error \(\alpha\) is \(0.0\) (zero percent). The test is 99.9 percent accurate if the man is the father, thus the probability of Type II error \(\beta\) is \(1 - 0.999 = 0.001\) (0.1 percent).

Calculate the probability of Type I and Type II error without the mother's test

According to Ann Landers, if the mother is not tested, there is a 0.8 percent chance of a false positive. This means the probability of Type I error \(\alpha\) is \(0.008\) (0.8 percent). Since no information is given about the probability of deciding a man is the father when he is not, in this case, the probability of Type II error cannot be determined.