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Chapter 9: Problem 52

Which error, Type I or Type II, would usually be considered more serious for decisions in the following tests? Explain why. a. A trial to test a murder defendant's claimed innocence, when conviction results in the death penalty. b. A medical diagnostic procedure, such as a mammogram.

Short Answer

Expert verified
a. Type I error is more serious; b. Type II error is more serious.

Step by step solution

01

Understanding Type I and Type II Errors

Type I error occurs when a true null hypothesis is incorrectly rejected, which is a 'false positive'. Type II error occurs when a false null hypothesis is not rejected, which is a 'false negative'.
02

Analyzing Case (a): Trial for a Murder Defendant

In a murder trial where conviction results in the death penalty, a Type I error would mean convicting an innocent person, while a Type II error would mean acquitting a guilty person. Convicting an innocent person is generally seen as more serious because it results in an irreversible penalty, the death of an innocent.
03

Analyzing Case (b): Medical Diagnostic Procedure

In a medical diagnostic procedure like a mammogram, a Type I error would mean diagnosing cancer in a healthy person (false positive), while a Type II error would mean failing to diagnose cancer in a patient (false negative). The more serious error is usually a Type II, as failing to diagnose cancer can delay treatment and lead to worse health outcomes.
04

Conclusion on Error Severity

In both cases, the severity of errors depends on the consequences. In the legal case, convicting an innocent person (Type I) is more severe, while in the medical case, missing a diagnosis (Type II) is more critical.

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Key Concepts

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

Type I Error
When engaging in hypothesis testing, a Type I Error is a crucial concept to grasp. It happens when the null hypothesis, which is actually true, is rejected erroneously. This type of mistake is often referred to as a 'false positive'. For example, imagine a scenario in a court trial concerning a murder case. If a Type I Error occurs, this would mean that someone who is innocent is judged guilty. Let's consider why this is important:
  • The implication of a Type I Error in critical situations, such as legal trials, is serious because it leads to wrongful punishment.
  • Such an error poses significant ethical and moral concerns, as the consequences might be life-altering, like an innocent person receiving the death penalty.
The goal is to minimize Type I Errors in critical situations to prevent irreversible damage.
Type II Error
On the other hand, a Type II Error occurs when the null hypothesis is false, but we fail to reject it. This is often known as a 'false negative'. In practical terms, this would mean that there is an actual effect or condition present that goes undetected. For instance, in medical diagnostics, such as a mammogram, a Type II Error could lead to missing a cancer diagnosis.
  • In medical tests, a Type II Error can prevent necessary treatment, leading to worsened health conditions or delayed diagnosis.
  • It is crucial to balance the risk of Type I and Type II Errors, as overly cautious testing could result in missing significant medical issues.
The severity of a Type II Error is context-dependent but can carry significant risks, especially when early treatment is vital.
Hypothesis Testing
Hypothesis testing is a method used in statistics to make inferences about a population based on sample data. It plays a fundamental role in decision-making processes across various fields. Let's break down the core concepts:
  • A null hypothesis (\( H_0 \)) assumes that there is no effect or no difference in the context of the test. This is the hypothesis that we aim to test.
  • An alternative hypothesis (\( H_1 \)) suggests that there is an effect or a difference.
  • During testing, a decision is made either to reject the null hypothesis or not. This decision is influenced by the data and statistical analysis.
In hypothesis testing, a key challenge is to control the error rates (Type I and Type II Errors). The test must be designed to minimize the likelihood of these errors, balancing the risk sensibly according to the context. Understanding these principles helps make informed decisions whether in a courtroom or medical examination room, enhancing the validity and reliability of conclusions.

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Most popular questions from this chapter

Consider the test of \(\mathrm{H}_{0}:\) The defendant is not guilty against \(\mathrm{H}_{a}:\) The defendant is guilty. a. Explain, in context, the conclusion of the test if \(\mathrm{H}_{0}\) is rejected. b. Describe, in context, a Type I error. c. Explain, in context, the conclusion of the test if you fail to reject \(\mathrm{H}_{0}\) d. Describe, in context, a Type II error.

Results of \(99 \%\) confidence intervals are consistent with results of two- sided tests with which significance level? Explain the connection.

An ankle-foot orthosis (AFO) is a specially designed brace to support and improve the function of the foot and ankle. A 2016 study on the treatment of knee osteoarthritis investigated the biomechanical effects of the Agilium Freestep AFO on the lever arm of the ground reaction force (GRF) in a gait analysis lab. Results show that the lever arm of the GRF was significantly reduced by \(14 \%\) with the Agilium Freestep AFO (www.oandp.org). Statistical analyses were conducted using the Student's \(t\) -test with a power of \(80 \%\). a. What should be the null and the alternative hypotheses in this study? b. How should the power of \(80 \%\) be interpreted? c. In context, what is a Type II error for this test?

Example 8 tested a therapy for anorexia, using hypotheses \(\mathrm{H}_{0}: \mu=0\) and \(\mathrm{H}_{a}: \mu \neq 0\) about the population mean weight change \(\mu .\) In the words of that example, what would be (a) a Type I error and (b) a Type II error?

\(\begin{array}{ll}\ & \mathbf{H}_{0} \text { or } \mathbf{H}_{a} \text { ? For parts a and } \mathrm{b} \text { , is the statement a null }\end{array}\) hypothesis, or an alternative hypothesis? a. In Canada, the proportion of adults who favor legalized gambling equals 0.50 . b. The proportion of all Canadian college students who are regular smokers is less than \(0.24,\) the value it was 10 years ago. c. Introducing notation for a parameter, state the hypotheses in parts a and b in terms of the parameter values.
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