Question

How accurate are DNA paternity tests? By comparing the DNA of the baby and the DNA of a man that is being tested, one maker of DNA paternity tests claims that their test is 100% accurate if the man is not the father and 99.99% accurate if the man is the father.

(a) Consider using the result of this DNA paternity test to decide between the following two hypotheses.
H0: A particular man is not the father.
Ha: A particular man is 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.)
A Type I error is coming to the conclusion that a particular man ___ the father when, in fact, he _____the father.

Answer 1

The probability that the man is the father given that the test result is positive is 0.9998.

(a)A Type I error is when the test results in a positive outcome when the man is not the father. In this case, the probability of a Type I error is 0.0001.

A Type II error is when the test results in a negative outcome when the man is the father. In this case, the probability of a Type II error is 9.999000000000001e-05.

(b) Given the information that the test is 100% accurate when the man is not the father and 99.99% accurate when the man is the father.

we can use Bayes' theorem to calculate the probability that the man is the father given that the test result is positive.

Bayes' theorem states that:

P(A|B) = (P(B|A) * P(A)) / P(B)

where:

P(A|B) is the probability of event A given that event B has occurred

P(B|A) is the probability of event B given that event A has occurred

P(A) is the probability of event A

P(B) is the probability of event B

In this case, we want to calculate P(father|positive), the probability that the man is the father given that the test result is positive.

We can do this by plugging in the following values:

P(positive|father) = 0.9999

P(father) = 0.5 (assuming that the man is equally likely to be or not be the father)

P(positive) = (0.9999 * 0.5) + (0.0001 * 0.5) = 0.500045

Plugging these values into Bayes' theorem, we get:

P(father| positive) = (0.9999 * 0.5) / 0.500045 = 0.999800000000001

Therefore, the probability that the man is the father given that the test result is positive is 0.9998.