Question

Medical personnel are required to report suspected cases of child abuse. Because some diseases have symptoms that mimic those of child abuse, doctors who see a child with these symptoms must decide between two competing hypotheses: \(H_{0}\) " symptoms are due to child abuse \(H_{a}:\) symptoms are due to disease (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) The article "Blurred Line Between Illness, Abuse Creates Problem for Authorities" (Macon Telegraph, February 28,2000 ) included the following quote from a doctor in Atlanta regarding the consequences of making an incorrect decision: "If it's disease, the worst you have is an angry family. If it is abuse, the other kids (in the family) are in deadly danger." a. For the given hypotheses, describe Type I and Type II errors. b. Based on the quote regarding consequences of the two kinds of error, which type of error does the doctor quoted consider more serious? Explain.

Short answer

a. Type I error: diagnosing a disease as child abuse (wrongly rejecting \(H_{0}\)), Type II error: missing a child abuse case and attributing symptoms to a disease (falsely accepting \(H_{0}\)). b. The doctor seems to consider Type II errors as the more serious error due to the potential danger to other children in the family.

Step-by-step solution

Understanding Type I Errors

A Type I error occurs when one unnecessarily rejects a true null hypothesis. In this scenario, a Type I error would occur if the medical personnel wrongly diagnose a child's disease symptoms as child abuse. That is, they incorrectly reject \(H_{0}\) when it is true.

Understanding Type II Errors

A Type II error occurs if the null hypothesis is falsely accepted when it is indeed false. In this scenario, a Type II error would occur if the medical personnel miss a case of child abuse, interpreting the symptoms as disease-based when in fact the child is a victim of abuse. This effectively means accepting \(H_{0}\) when it is false.

Interpreting the quote

The quote from the doctor implies that incorrectly diagnosing a disease as child abuse (Type I error) primarily results in an angry family. However, misdiagnosing abuse as a disease (Type II error) puts other children in potential harm's way. This suggests the doctor views Type II errors as more serious due to the potentially dangerous consequences.