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Chapter 10: Problem 6

In a hypothesis test, what does it mean to say that the null hypothesis was rejected?

Short Answer

Expert verified
Rejecting the null hypothesis in a hypothesis test means that we have found enough statistical evidence to support the alternative hypothesis and reject the assertion that the observed result is merely the result of random chance.

Step by step solution

01

Understanding the Hypothesis Test

The first step in understanding the meaning of the null hypothesis being rejected is understanding what a hypothesis test is. A hypothesis test is a systematic way to test claims or ideas about a group or population.
02

Understanding the Null Hypothesis

The next step is understanding what a null hypothesis is. In a statistical hypothesis test, the null hypothesis, denoted by \(H_0\), is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt. The null hypothesis assumes that any kind of difference or importance you see in a set of data is due to chance.
03

Rejecting the Null Hypothesis

Lastly, the term 'rejecting the null hypothesis' is understood. In hypothesis testing, if the observed data fall in the rejection region, we reject the null hypothesis, and conclude that we find evidence against the null hypothesis and in favor of the alternative hypothesis. Specifically, rejecting the null hypothesis means the probability of obtaining a result as extreme as, or more extreme than, the observed result is small enough to reject the assertion that the observed result is merely the result of random chance. In other words, we find enough statistical evidence to support the alternative hypothesis.

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Most popular questions from this chapter

The report "2007 Electronic Monitoring and Surveillance Survey: Many Companies Monitoring, Recording, Videotaping-and Firing-Employees" (American Management Association, 2007) summarized a survey of 304 U.S. businesses. Of these companies, 201 indicated that they monitor employees' web site visits. Assume that it is reasonable to regard this sample as representative of businesses in the United States. a. Is there sufficient evidence to conclude that more than \(75 \%\) of U.S. businesses monitor employees' web site visits? Test the appropriate hypotheses using a significance level of 0.01 . b. Is there sufficient evidence to conclude that a majority of U.S. businesses monitor employees' web site visits? Test the appropriate hypotheses using a significance level of 0.01 .

Suppose that for a particular hypothesis test, the consequences of a Type I error are very serious. Would you want to carry out the test using a small significance level \(\alpha\) (such as 0.01 ) or a larger significance level (such as 0.10 )? Explain the reason for your choice.

According to a Washington Post-ABC News poll, 331 of 502 randomly selected American adults said they would not be bothered if the National Security Agency collected records of personal telephone calls. The data were used to test \(H_{0}: p=0.5\) versus \(H_{a}: p>0.5,\) and the null hypothesis was rejected. a. Based on the hypothesis test, what can you conclude about the proportion of American adults who would not be bothered if the National Security Agency collected records of personal telephone calls? b. Is it reasonable to say that the data provide strong support for the alternative hypothesis? c. Is it reasonable to say that the data provide strong evidence against the null hypothesis?

A number of initiatives on the topic of legalized gambling have appeared on state ballots. A political candidate has decided to support legalization of casino gambling if he is convinced that more than two-thirds of American adults approve of casino gambling. Suppose that 1,035 of the people in a random sample of 1,523 American adults said they approved of casino gambling. Is there convincing evidence that more than two-thirds approve?

An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is expensive, so it will be done only if there is strong evidence that the proportion of defective installations is less for the robots than for human assemblers. Let \(p\) denote the actual proportion of defective installations for the robots. It is known that the proportion of defective installations for human assemblers is 0.02 . a. Which of the following pairs of hypotheses should the manufacturer test? $$H_{0}: p=0.02 \text { versus } H_{a}: p<0.02$$ or $$H_{0}: p=0.02 \text { versus } H_{a}: p>0.02$$ Explain your choice. b. In the context of this exercise, describe Type I and Type II errors. c. Would you prefer a test with \(\alpha=0.01\) or \(\alpha=0.10 ?\) Explain your reasoning.
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