/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 Errors in the courtroom Consider... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 47

Errors in the courtroom 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 the consequence of a Type I error. c. Explain in context the conclusion of the test if you fail to reject \(\mathrm{H}_{0}\) d. Describe the consequence of a Type II error.

Short Answer

Expert verified
Rejecting 4120 implies concluding guilt; a Type I error wrongly convicts the innocent, while a Type II error fails to convict the guilty.

Step by step solution

01

Understanding the Hypotheses

The null hypothesis (4120) represents the claim that the defendant is not guilty. The alternative hypothesis (4121) represents the claim that the defendant is guilty. When conducting hypothesis testing, we assess whether there is enough evidence to reject the null hypothesis in favor of the alternative.
02

Conclusion if 4120 is Rejected

If the null hypothesis (4120) is rejected, the conclusion in the courtroom is that there is sufficient evidence to support the claim that the defendant is guilty. This implies that the alternative hypothesis (4121), which states that the defendant is guilty, is considered to be true based on the evidence presented.
03

Consequences of a Type I Error

A Type I error occurs if we reject the null hypothesis when it is actually true. In the context of the courtroom, this means declaring the defendant guilty when they are, in fact, not guilty. This results in an innocent person being wrongly convicted.
04

Conclusion if Fail to Reject 4120

If we fail to reject the null hypothesis (4120), the conclusion is that there is not enough evidence to support the claim that the defendant is guilty. It does not necessarily mean the defendant is innocent, only that the evidence is insufficient to prove guilt beyond a reasonable doubt.
05

Consequences of a Type II Error

A Type II error occurs if we fail to reject the null hypothesis when the alternative hypothesis is true. In this context, it means not convicting the defendant when they are actually guilty, resulting in a guilty person being wrongly acquitted.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Type I Error
In the realm of hypothesis testing, a Type I error occurs when we reject the null hypothesis, even though it is actually true. This type of error is akin to a false positive in many scenarios. In the context of a courtroom, consider a trial where the null hypothesis (
  • \( H_0 \): The defendant is not guilty
) is wrongly rejected in favor of the alternative hypothesis (
  • \( H_a \): The defendant is guilty
).

When a Type I error is made, it results in the conviction of an innocent defendant. This is because, despite there being a lack of concrete evidence for guilt, the conclusion reached is one of guilt.
This can have severe consequences, as the falsely convicted person unfairly faces the repercussions of a guilty verdict. To minimize the risk of this error, courts often require evidence beyond a reasonable doubt.
Having a carefully defined significance level (denoted by \( \alpha \)) is crucial as it represents the probability of committing a Type I error.
Type II Error
A Type II error occurs when we fail to reject the null hypothesis when it is actually false. This is like a false negative, where one overlooks something that is indeed present. In the courtroom context, this translates to failing to convict a defendant who is truly guilty.

If the null hypothesis (
  • \( H_0 \): The defendant is not guilty
) is not rejected due to insufficient evidence, yet the defendant is guilty, a Type II error has been made. The outcome is the unintended acquittal of a guilty person, who then evades justice. This allows the defendant to remain free, potentially risking further harm to society.

The probability of making a Type II error is denoted by \( \beta \). Efforts to reduce \( \beta \) include increasing the sample size or using stronger evidence, which can improve the power of a test and help ensure that guilty parties are rightfully convicted.
Null Hypothesis
The concept of a null hypothesis is central to hypothesis testing. It is a statement we aim to test and commonly represents a position of no effect or no difference. In a courtroom setting, the null hypothesis typically states that the defendant is not guilty. This sets the baseline for the trial.
  • \( H_0 \): The defendant is not guilty
Holding the null hypothesis stable implies we need substantial evidence to conclude otherwise.
In practical terms, not rejecting the null hypothesis does not prove innocence; it merely indicates that the evidence is insufficient to prove guilt beyond reasonable doubt.

Judicial systems abide by the principle of "innocent until proven guilty," which aligns with the logic of favoring the null hypothesis unless evidence certainly supports the alternative. This requires meticulous evaluation of evidence and adherence to standards that protect the innocent from false condemnation while aiming to identify true guilt efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Examples of hypotheses Give an example of a null hypothesis and an alternative hypothesis about a (a) population proportion and (b) population mean.

Interpret medical research studies a. An advertisement by Schering Corp. in 1999 for the allergy drug Claritin mentioned that in a clinical trial, the proportion who showed symptoms of nervousness was not significantly greater for patients taking Claritin than for patients taking placebo. Does this mean that the population proportion having nervous symptoms is exactly the same using Claritin and using placebo? How would you explain this to someone who has not studied statistics? b. An article in the medical journal \(B M J\) (by M. Petticrew et al., published November 2002 ) found no evidence to back the commonly held belief that a positive attitude can lengthen the lives of cancer patients. The authors noted that the studies that had indicated a benefit from some coping strategies tended to be smaller studies with weaker designs. Using this example and the text discussion, explain why you need to have some skepticism when you hear that new research suggests that some therapy or drug has an impact in treating a disease.

Low-carbohydrate diet A study plans to have a sample of obese adults follow a proposed low-carbohydrate diet for three months. The diet imposes limited eating of starches (such as bread and pasta) and sweets, but otherwise no limit on calorie intake. Consider the hypothesis, The population mean of the values of weight change \((=\) weight at start of study \(-\) weight at end of study) is a positive number. a. Is this a null or an alternative hypothesis? Explain your reasoning. b. Define a relevant parameter, and express the hypothesis that the diet has no effect in terms of that parameter. Is it a null or alternative hypothesis?

Which \(t\) has P-value \(=0.05 ?\) A \(t\) test for a mean uses a sample of 15 observations. Find the \(t\) test statistic value that has a P-value of 0.05 when the alternative hypothesis is (a) \(\mathrm{H}_{a}: \mu \neq 0,\) (b) \(\mathrm{H}_{a}: \mu>0,\) and (c) \(\mathrm{H}_{a}: \mu<0 ?\)

Another test of astrology Examples \(1,3,\) and 5 referred to a study about astrology. Another part of the study used the following experiment: Professional astrologers prepared horoscopes for 83 adults. Each adult was shown three horoscopes, one of which was the one an astrologer prepared for them and the other two were randomly chosen from ones prepared for other subjects in the study. Each adult had to guess which of the three was theirs. Of the 83 subjects, 28 guessed correctly. a. Defining notation, set up hypotheses to test that the probability of a correct prediction is \(1 / 3\) against the astrologers' claim that it exceeds \(1 / 3\). b. Show that the sample proportion \(=0.337,\) the standard error of the sample proportion for the test is \(0.052,\) and the test statistic is \(z=0.08\). c. Find the P-value. Would you conclude that people are more likely to select their horoscope than if they were randomly guessing, or are results consistent with random guessing?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.