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

Understanding Type I and Type II errors

A Type I error would be concluding that the man is not a father (rejecting \(H_0\)) when he actually is. This is a false positive and the probability of this happening is given as \(0.1 \%\). A Type II error would be determining that the man is the father (not rejecting \(H_0\)) when he actually isn't. The false negative accuracy isn't given and will be assumed to be negligible.

02

Calculating the probability of Type I and II errors (\(\alpha\) and \(\beta\))

The probability of a Type I error (\(\alpha\)) is the likelihood of incorrectly rejecting \(H_0\), which is given here as \(0.1 \%\). So, \(\alpha = 0.001\). The probability for a Type II error (\(\beta\)) involves the false negative rate, however, this value is not directly given in the exercise material. However, as stated in the Step 1, one might assume it is negligible or 0 since there is no hint towards a possible false negative.

03

Calculating new \(\beta\) when mother is not tested

We are informed that if the mother is not tested, there is a \(0.8 \%\) chance of a false positive. This changes the value for \(\alpha\) to 0.008. Since Type I and Type II errors are mutually exclusive, and there's no mention of any false negatives when the mother isn't tested, this doesn't directly affect the \(\beta\), hence it would stay the same.

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Key Concepts

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

Type I Error

In the context of hypothesis testing, a Type I error occurs when we incorrectly reject the null hypothesis (\(H_0\)), which is the statement we initially assume to be true. Here's how it works in the case of paternity testing:

• The null hypothesis (\(H_0\)) is that the particular man is the father.
• A Type I error would mean concluding the man is not the father, when in fact, he is, leading to a false positive result.

In the example provided, this false positive rate is 0.1%, meaning the test inaccurately indicates that the man is not the father 0.1% of the time when he actually is. This probability is denoted by \(\alpha\), which is \(0.001\) in this case.

Type II Error

On the flip side, a Type II error happens when we fail to reject the null hypothesis (\(H_0\)) even though the alternative hypothesis (\(H_a\)) is true. In paternity testing, here's what that entails:

• The null hypothesis (\(H_0\)) states that the man is the father.
• A Type II error occurs when we conclude that the man is the father, when in reality, he is not, which leads to a false negative.

Although the false negative rate for the paternity test isn't directly provided, it is suggested to be negligible, implying there is a very low chance of mistakenly identifying a non-father as the father. This probability is denoted by \(\beta\). In different scenarios, understanding the likelihood of a Type II error is crucial for assessing the test's reliability.

Paternity Testing

Paternity testing is a form of genetic testing employed to determine whether a man is the biological father of a child. This procedure has vital implications, often used in legal, personal, and medical contexts. Here’s a breakdown of the process involved:

• Sample Collection: DNA samples are typically collected from the child, the alleged father, and often the mother, using cheek swabs or blood samples.
• DNA Analysis: The collected samples undergo analysis to compare DNA profiles. Scientists look for shared genetic markers indicative of parentage.
• Result Interpretation: Based on the comparison, the test determines with high probability if the man is the biological father.

In the given scenario, involving a DNA test, the result is virtually certain when showing the man is not the father (100% accuracy) and highly probable (99.9% accuracy) when indicating he is the father. The implications of these results rest on understanding the probabilities of Type I and Type II errors, reinforcing the importance of accuracy in biological testing.

DNA Testing

DNA testing is a powerful scientific tool used in various fields, including paternity testing, criminal investigations, and ancestry research. It works by examining an individual's unique genetic code to extract important information. Here’s an overview of how DNA testing is relevant in paternity testing:

• Genetic Markers: DNA testing focuses on comparing specific genetic markers shared between individuals, crucial in confirming biological relationships.
• Accurate Results: With advancements in technology, DNA tests can offer near-certain results, as seen with paternity tests yielding 99.9% accuracy under certain conditions.
• Influence of Additional Samples: The inclusion of the mother’s DNA can further refine the test's accuracy, reducing errors like false positives.

As demonstrated in the paternity test scenario, when the mother’s sample is omitted, the chance of a false positive increases to 0.8%. Therefore, carefully considering the samples and interpreting DNA testing results is fundamental in making precise biological determinations.