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Chapter 10: Problem 24

For which of the following \(P\) -values will the null hypothesis be rejected when performing a level \(.05\) test: a. 001 d. \(.047\) b. \(.021\) e. 148 c. \(.078\)

Short Answer

Expert verified
The null hypothesis will be rejected for P-values 0.001, 0.047 and 0.021.

Step by step solution

01

Rule of Hypothesis Testing

In hypothesis testing, if the p-value is less than the given significance level (in this case .05), then the null hypothesis is rejected. This means that the null hypothesis鈥攚hich typically proposes that there are no differences or effects鈥攊s likely incorrect.
02

Compare P-value with Level of Significance

From the set of p-values {0.001, 0.047, 0.021, 0.148, 0.078}, compare each of these values with the given level of significance .05. \n 1) 0.001 < 0.05, so reject the null hypothesis \n 2) 0.047 < 0.05, so reject the null hypothesis \n 3) 0.021 < 0.05, so reject the null hypothesis \n 4) 0.148 > 0.05, so do not reject the null hypothesis \n 5) 0.078 > 0.05, so do not reject the null hypothesis

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Key Concepts

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

Understanding the p-value
The p-value is a fundamental component of hypothesis testing. It helps us decide whether there is enough evidence to reject the null hypothesis. Essentially, the p-value measures how extreme the observed data is, assuming the null hypothesis is true.
When the p-value is small, it means that the observed data would be very unusual if the null hypothesis were true. This often leads us to reconsider the validity of the null hypothesis.
  • A p-value less than or equal to 0.05 is often considered significant.
  • This suggests that the null hypothesis may not be true.
By comparing the p-value to a pre-determined significance level, we can make informed decisions about our hypothesis. If the p-value is less than this level, we reject the null hypothesis.
The Level of Significance
The level of significance, often denoted as alpha (伪), is a threshold set by the researcher before conducting a hypothesis test. It's the probability of making a type I error, which is rejecting a true null hypothesis. In many fields, a common choice for the level of significance is 0.05.
  • A level of 0.05 means there is a 5% risk of rejecting the null hypothesis when it is actually true.
  • It serves as a cut-off point to decide whether the results are statistically significant.
Using the level of significance, we determine the critical region鈥攖he range of values where we would reject the null hypothesis. This approach ensures that our hypothesis testing maintains a consistent standard of evidence.
The Null Hypothesis
The null hypothesis, typically denoted as \( H_0 \), is a default statement predicting no effect or no difference. It forms the basis for hypothesis testing by providing a point of comparison.
  • Often, the null hypothesis assumes that any kind of change or difference you may observe is random and not significant.
  • The goal of the test is to evaluate if there is enough evidence to reject this hypothesis.
By setting up a null hypothesis, we create a framework where results can be interpreted clearly. If our observed p-value is smaller than our level of significance, we reject \( H_0 \). Rejection indicates that the evidence suggests a meaningful effect or difference.

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Most popular questions from this chapter

In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American society is a serious problem, and 1283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3,2002 ). Is there convincing evidence that more than three-quarters of U.S. adults believe that rudeness is a worsening problem? Test the relevant hypotheses using a significance level of \(.05\).

The state of Georgia's HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, \(53.2 \%\) had a GPA below \(3.0\) at the end of their first year ("Who Loses HOPE? Attrition from Georgia's College Scholarship Program," Southern Economic Journal [1999]: 379-390). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

Occasionally, warning flares of the type contained in most automobile emergency kits fail to ignite. A consumer advocacy group wants to investigate a claim against a manufacturer of flares brought by a person who claims that the proportion of defective flares is much higher than the value of \(.1\) claimed by the manufacturer. A large number of flares will be tested, and the results will be used to decide between \(H_{0}=\pi=.1\) and \(H_{a}: \bar{\pi}>.1\), where \(\pi\) represents the true proportion of defective flares made by this manufacturer. If \(H_{0}\) is rejected, charges of false advertising will be filed against the manufacturer. a. Explain why the alternative hypothesis was chosen to be \(H_{a}: \pi>.1\). b. In this context, describe Type I and Type II errors, and discuss the consequences of each.

The success of the U.S. census depends on people filling out and returning census forms. Despite extensive advertising, many Americans are skeptical about claims that the Census Bureau will guard the information it collects from other government agencies. In a USA Today poll (March 13,2000 ), only 432 of 1004 adults surveyed said that they believe the Census Bureau when it says the information you give about yourself is kept confidential. Is there convincing evidence that, despite the advertising campaign, fewer than half of U.S. adults believe the Census Bureau will keep information confidential? Use a significance level of \(.01\).

A number of initiatives on the topic of legalized gambling have appeared on state ballots. Suppose that a political candidate has decided to support legalization of casino gambling if he is convinced that more than twothirds of U.S. adults approve of casino gambling. USA Today (June 17,1999 ) reported the results of a Gallup poll in which 1523 adults (selected at random from households with telephones) were asked whether they approved of casino gambling. The number in the sample who approved was 1035 . Does the sample provide convincing evidence that more than two-thirds approve?
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