91Ó°ÊÓ

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 \(100 \%\) accurate if the man is not the father and \(99.9 \%\) 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 a Type I error, and \(\beta,\) the probability of a Type II error? c. Ann Landers also stated, "If the mother is not tested, there is a \(0.8 \%\) chance of a false positive." For the hypotheses given in Part (a), what is the value of \(\beta\) if the decision is based on DNA testing in which the mother is not tested?

Step by step solution

01

Description of Type I and Type II Errors

From a statistical testing perspective: \n • Type I error, denoted by \(\alpha\), is the probability of rejecting the null hypothesis \(H_{0}\) when it is actually true. In this context, it means claiming the man is not the father when he actually is. \n • Type II error, denoted by \(\beta\), is the probability of accepting the null hypothesis \(H_{0}\) when it is false. In this context, that means claiming the man is the father when he is not.

02

Finding \(\alpha\) and \(\beta\)

From the given text, • The probability of stating a man is not the father (rejecting \(H_{0}\)) when indeed he is, is the Type I error, \(\alpha\), which is \(0.1 \% = 0.001\). • The probability of stating a man is the father (accepting \(H_{0}\)) when, in reality, he is not, is the Type II error, \(\beta\), which is \(0.0 \% = 0.\)

03

Finding \(\beta\) without Mother's Result

If the mother is not tested, the probability of stating the man is the father when he is not (Type II error, \(\beta\)), is given as a \(0.8 \% = 0.008\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error

A Type I error occurs when we incorrectly reject the null hypothesis. In the context of DNA paternity testing, this means concluding that a particular man is not the father, when, in fact, he is. This is essentially a 'false positive' decision.

Here are some important points to remember about Type I errors:

• The probability of making a Type I error is denoted by \( \alpha \).
• In the given DNA testing scenario, \( \alpha \) is 0.1%, indicating a very low chance of mistakenly identifying a man as not being the father when he actually is.
• Minimizing \( \alpha \) is crucial in fields where false rejections have serious consequences, like criminal justice or medical testing.

Managing Type I errors often involves setting a very low \( \alpha \), thereby reducing the chance of making this mistake, although balancing it with the risks of Type II errors is always necessary.

Type II Error

A Type II error occurs when we fail to reject the null hypothesis when it is false. In paternity testing, this means incorrectly stating that a particular man is the father, even though he is not. This results in a 'false negative.'

Key aspects of Type II errors include:

• The probability of a Type II error is denoted by \( \beta \).
• In the original scenario where the mother is tested, \( \beta \) is 0%, meaning there's no risk of a false negative under full testing conditions.
• If the mother isn’t tested, \( \beta \) rises to 0.8%, indicating a small but increased risk of incorrectly confirming paternity.

Reducing \( \beta \) often requires increasing the sample size or using more accurate tests. Balancing Type I and Type II errors is essential for optimal testing efficiency.

DNA Paternity Testing

DNA paternity testing is a scientific method used to determine a biological relationship between a child and a potential father. It is based on comparing a child’s DNA profile with that of the alleged father to identify matches. This testing is highly reliable and widely used.

Important points about DNA paternity testing include:

• The test is 100% accurate in determining non-paternity (if the man is not the father).
• It is 99.9% accurate if the man is the father, meaning there's a small chance of a Type I error.
• Testing both the man and the mother reduces the risk of a Type II error to 0%, making results more definitive.

DNA paternity tests are powerful due to their ability to provide clear answers, but understanding potential errors is crucial for interpreting results.