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Chapter 20: Problem 8

More errors For each of the following situations, state whether a Type I, a Type II, or neither error has been made. a) A test of \(\mathrm{H}_{0}: p=0.8\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.8\) fails to reject the null hypothesis. Later it is discovered that \(p=0.9\) b) A test of \(\mathrm{H}_{0}: p=0.5\) vs. \(\mathrm{H}_{\mathrm{A}}: p \neq 0.5\) rejects the null hypothesis. Later is it discovered that \(p=0.65\) c) A test of \(\mathrm{H}_{0}: p=0.7\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.7\) fails to reject the null hypothesis. Later is it discovered that \(p=0.6\)

Short Answer

Expert verified
A: Type II error, B: Neither error, C: Type II error.

Step by step solution

01

Understanding Errors

A Type I error occurs when the null hypothesis is true but we reject it. A Type II error happens when the null hypothesis is false, but we fail to reject it.
02

Analyze Situation A

In situation A, the null hypothesis was not rejected. The null hypothesis states that \(p = 0.8\), but later it is found that \(p = 0.9\). Since the null hypothesis is false (\(p eq 0.8\)) and it was not rejected, a Type II error has been made.
03

Analyze Situation B

For situation B, the null hypothesis was rejected in favor of the alternative that \(p eq 0.5\). Later it is discovered that \(p = 0.65\), which makes the null hypothesis indeed false, so rejecting it was the correct decision. Neither error was made in this case.
04

Analyze Situation C

In situation C, the null hypothesis that \(p = 0.7\) was not rejected. Upon discovering that \(p = 0.6\), which opposes the null hypothesis (since \(p < 0.7\)), this indicates a false negative and thus a Type II error has been made.

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Key Concepts

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

Type I Error
In hypothesis testing, a Type I error occurs when the null hypothesis is true, but we wrongly reject it. This error type is also known as a "false positive." In other words, it happens when we claim there is an effect or a difference when in fact, there isn鈥檛 one.
For instance, if you were testing whether a drug has an effect on reducing symptoms and concluded that it does, when in fact it doesn鈥檛, a Type I error may have occurred.
  • Null Hypothesis ( H_{0} ): There is no effect or difference.
  • Type I Error: Incorrectly rejecting H_{0} .
The probability of committing a Type I error is denoted by the symbol alpha , which is also known as the significance level of the test. Commonly, alpha is set to 0.05, implying a 5% risk of concluding that a difference exists when there is none.
Type II Error
A Type II error occurs in hypothesis testing when the null hypothesis is false, but we fail to reject it. This error is also referred to as a "false negative." It's like saying there is no effect or difference when, in reality, there is one.
Consider a medical test aimed at detecting a disease; if the test falsely indicates a person is healthy when they actually have the disease, this is a Type II error.
  • Null Hypothesis ( H_{0} ): There is no effect or difference.
  • Type II Error: Incorrectly failing to reject H_{0} .
The probability of making a Type II error is represented by the symbol beta . It's important to minimize beta to increase the power of a test, which is 1 - beta , reflecting the likelihood of correctly rejecting the null hypothesis when it is indeed false.
Null Hypothesis
The null hypothesis, often denoted as H_{0} , is a central concept in hypothesis testing. It represents the hypothesis that there is no effect, difference, or relationship in the context of a study.
This is essentially the default or "no change" position that a test seeks to challenge or find evidence against. The null hypothesis remains as our assumption unless the test data indicates strong evidence for an alternative hypothesis ( H_{A} ).
  • H_{0} Example: A new teaching method does not affect student performance compared to the standard method.
  • H_{A} Example: A new teaching method improves student performance compared to the standard method.
The goal is to determine if data provides enough evidence to reject H_{0} in favor of H_{A} . If the data is inconclusive or insufficient, H_{0} is not rejected, and it is assumed correct within the context of the test.

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Most popular questions from this chapter

Equal opportunity? A company is sued for job discrimination because only \(19 \%\) of the newly hired candidates were minorities when \(27 \%\) of all applicants were minorities. Is this strong evidence that the company's hiring practices are discriminatory? a) Is this a one-tailed or a two-tailed test? Why? b) In this context, what would a Type I error be? c) In this context, what would a Type II error be? d) In this context, what is meant by the power of the test? e) If the hypothesis is tested at the \(5 \%\) level of significance instead of \(1 \%,\) how will this affect the power of the test? f) The lawsuit is based on the hiring of 37 employees. Is the power of the test higher than, lower than, or the same as it would be if it were based on 87 hires?

Quality control Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a) In this context, what is a Type I error? b) In this context, what is a Type II error? c) Which type of error would the factory owner consider more serious? d) Which type of error might customers consider more serious?

Ads A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than \(20 \%\) of the residents of the city have heard the ad and recognize the company's product. The radio station conducts a random phone survey of 400 people. a) What are the hypotheses? b) The station plans to conduct this test using a \(10 \%\) leve of significance, but the company wants the significance level lowered to \(5 \%\). Why? c) What is meant by the power of this test? d) For which level of significance will the power of this test be higher? Why? e) They finally agree to use \(\alpha=0.05,\) but the company proposes that the station call 600 people instead of the 400 initially proposed. Will that make the risk of Type II error higher or lower? Explain.

Dropouts A Statistics professor has observed that for several years about \(13 \%\) of the students who initially enroll in his Introductory Statistics course withdraw before the end of the semester. A salesman suggests that he try a statistics software package that gets students more involved with computers, predicting that it will cut the dropout rate. The software is expensive, and the salesman offers to let the professor use it for a semester to see if the dropout rate goes down significantly. The professor will have to pay for the software only if he chooses to continue using it. a) Is this a one-tailed or two-tailed test? Explain. b) Write the null and alternative hypotheses. c) In this context, explain what would happen if the professor makes a Type I error. d) In this context, explain what would happen if the professor makes a Type II error. e) What is meant by the power of this test?

One sided or two? In each of the following situations, is the alternative hypothesis one-sided or two-sided? What are the hypotheses? a) A business student conducts a taste test to see whether students prefer Diet Coke or Diet Pepsi. b) PepsiCo recently reformulated Diet Pepsi in an attempt to appeal to teenagers. The company runs a taste test to see if the new formula appeals to more teenagers than the standard formula. c) A budget override in a small town requires a two thirds majority to pass. A local newspaper conducts a poll to see if there's evidence it will pass. d) One financial theory states that the stock market will go up or down with equal probability. A student collects data over several years to test the theory.
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