Question

Medical personnel are required to report suspected cases of child abuse. Because some diseases have symptoms that are similar to 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 not due to child abuse (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 IIIness, 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 is considered more serious by the doctor quoted? Explain.

Short answer

a. Type I error: misdiagnosing disease as child abuse; results in an angry but innocent family. Type II error: misdiagnosing child abuse as disease; results in potentially endangering other kids in the family. b. Based on the quote, the doctor considers a Type II error more serious, as it leaves other children potentially at risk.

Step-by-step solution

Identify Type I and Type II Errors in Context

The null hypothesis \(H_{0}\) is that the symptoms are due to child abuse and the alternative hypothesis \(H_{a}\) is that the symptoms are not due to child abuse. - A Type I error would occur if a doctor concludes that the symptoms are due to child abuse (\(H_{a}\)) when in reality, they are due to some other disease (\(H_{0}\)). In this situation, the family would be falsely accused of child abuse. - A Type II error would occur if a doctor concludes that the symptoms are due to some other disease (\(H_{0}\)) when in reality, they are due to child abuse (\(H_{a}\)). In this case, the child may remain in a situation of abuse.

Analyze the Quote for Perception of Error Severity

As per the quote, if the disease is misdiagnosed as child abuse (Type I error), the worst outcome is an angry family. However, if child abuse is misdiagnosed as disease (Type II error), other children in the family remain in danger. From the doctor's perspective, it seems that a Type II error, where child abuse goes unreported and other children are potentially at risk, is considered more serious.

Key concepts

Type I and Type II Errors

Statistical hypothesis testing is a critical tool used to make decisions based on data. Two potential errors, known as Type I and Type II errors, can occur in this process, and understanding them is essential for interpreting test results.

A Type I error happens when we reject the null hypothesis when it is actually true. Imagine sounding an unnecessary alarm — this is akin to a false positive. Within the context of the medical scenario, a Type I error would lead to a family being wrongfully accused of child abuse, causing distress and possible social and legal ramifications for the family.

Consequences of Type I Error

In terms of impact, this error inflicts emotional stress on the family and could lead to a breakdown in the relationship between the medical professional and the patient. Despite these consequences, families can recover over time and legal systems allow for corrections in the case of false accusations.

A Type II error, conversely, happens when we fail to reject the null hypothesis when the alternative hypothesis is true. This is like missing a silent alarm when there's an actual threat — a false negative. In our scenario, this would mean that a child suffering from abuse is not correctly identified, which could leave them in a potentially dangerous environment.

Consequences of Type II Error

When it comes to child safety, a Type II error is more severe as it may put the child, or other children in the family, at continued risk of harm. This is precisely why, from the perspective of medical professionals and authorities, preventing Type II errors is often given higher priority.

Null Hypothesis

The null hypothesis, symbolized by \( H_{0} \), is a statement that indicates no effect or no difference. It's the assumption that any kind of pattern or difference you see in a set of data occurs purely by chance. In other contexts, it's like assuming 'innocence until proven guilty.'

In the given medical example, the null hypothesis is that a child's symptoms are due to child abuse. It's a starting point for investigation and it's the hypothesis that the test aims to challenge. If enough evidence accumulates to disprove it, only then would a doctor move towards rejecting the null hypothesis.

• Presumption of Status Quo: \( H_{0} \) presumes that the current condition, that symptoms are indicative of abuse, is the norm.
• Standard of Evidence: It requires strong evidence against it to be rejected, helping to safeguard against false accusations.

Setting the null hypothesis is critical, as it defines the direction of statistical testing and thus influences the consequences related to Type I and Type II errors.

Alternative Hypothesis

The alternative hypothesis, denoted by \( H_{a} \), represents a statement that contradicts the null hypothesis. It’s what you might begin to believe is true if you have sufficient evidence to doubt the validity of the null hypothesis. Formulating it is an important step in designing a hypothesis test.

In our medical example, the alternative hypothesis is that the symptoms are not due to child abuse, which implies another cause such as a disease or natural causes. If a doctor gathers enough evidence, they might lean towards this hypothesis instead.

• Indication of a Possible Different Reality: The alternative hypothesis opens the possibility that the symptoms have a different explanation other than child abuse.
• Directing the Investigation: If evidence begins to accumulate supporting \( H_{a} \), it will guide the doctor towards further tests or treatments that align with a different cause.

The nature of the alternative hypothesis has a profound effect on what constitutes a screw-up in the testing process — it determines when a Type II error could occur.