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Chapter 10: Problem 22

One type of error in a hypothesis test is failing to reject a false null hypothesis. What is the other type of error that might occur when a hypothesis test is carried out?

Short Answer

Expert verified
The other type of error that might occur when a hypothesis test is carried out is Type I error, which is rejecting a true null hypothesis. This error occurs when the researcher finds evidence against the null hypothesis when it is actually true and can lead to misleading conclusions or incorrect decisions.

Step by step solution

01

Understand the two types of errors in hypothesis testing

In hypothesis testing, we compare a null hypothesis (H鈧�) with an alternative hypothesis (H鈧�). There are two types of errors that can occur while making a decision based on this test: 1. Type I error: Rejecting a true null hypothesis 2. Type II error: Failing to reject a false null hypothesis The exercise provides the definition of Type II error. Now, let's find the other type of error.
02

Identify the other type of error

The other type of error that might occur when a hypothesis test is carried out is Type I error, which is rejecting a true null hypothesis. This error occurs when the researcher finds evidence against the null hypothesis when it is actually true.
03

Understand the consequences of Type I error

Type I error can lead to falsely concluding that there is a significant effect or difference when there really is none. This can result in a misleading conclusion or incorrect decision, which can have real-life consequences, especially in fields like medicine or public policy. It is important for researchers to be aware of the potential for Type I error and to carefully design their studies to minimize the probability of making this type of error.

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Key Concepts

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

Type I Error
A Type I error is one of the two main errors that can occur in hypothesis testing. It happens when the null hypothesis, which is the default position that there is no effect or no difference, is incorrectly rejected. Imagine a fire alarm that goes off without a fire; that's akin to committing a Type I error in statistics - a false positive.

To put it in the context of a courtroom, a Type I error would be analogous to convicting an innocent person. The null hypothesis represents the assumption of innocence, and rejecting it unfairly would mean an innocent verdict was disregarded. Scientists usually set a threshold of significance denoted as \( \alpha \) before conducting a study to control the likelihood of making a Type I error. A common \( \alpha \) level is 0.05, indicating a 5% risk of committing a Type I error in any given test.
Type II Error
Conversely, a Type II error occurs when a false null hypothesis is not rejected. This is equivalent to a 'missed opportunity' to identify an actual effect or difference. Consider this as a security checkpoint missing a prohibited item in a bag 鈥� the error is a false negative.

In the judicial example, a Type II error would be like letting a guilty person go free. The consequence of this error could mean missing out on important discoveries or safety measures because the evidence wasn't deemed strong enough to challenge the status quo. Researchers strive to reduce the chances of committing Type II errors by increasing their study's power, which can be accomplished by increasing the sample size or choosing more sensitive measures, amongst other strategies.
Null Hypothesis
The null hypothesis (denoted as \( H_0 \) ) is a fundamental concept in hypothesis testing that serves as a starting presumption. It assumes there is no difference or effect regarding the subject matter of the research. For instance, if scientists were investigating a new drug, the null hypothesis would propose that the drug has no effect on patients compared to a placebo.

It is essential to understand that the null hypothesis is not a claim that seeks proof but rather a statement that's intended to be challenged by the alternative hypothesis. Researchers gather data through experimentation, and if the data strongly conflict with the null hypothesis, it may be rejected in favor of the alternative hypothesis.
Alternative Hypothesis
The alternative hypothesis (denoted as \( H_a \) or \( H_1 \) ) stands in opposition to the null hypothesis. It suggests that there is an effect, difference, or correlation. Following the new drug example, the alternative hypothesis would propose that the drug has a significant impact on patients' health.

In hypothesis testing, researchers aim to gather sufficient evidence to support the alternative hypothesis. However, for this to occur, they need to reach a level of significance that is unlikely to be due to chance alone, often below the \( \alpha \) threshold. The alternative hypothesis is what researchers typically hope to confirm, representing new discoveries or advancements in their field.

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Most popular questions from this chapter

"Most Like It Hot" is the title of a press release issued by the Pew Research Center (March 18, 2009, www.pewsocialtrends. org). The press release states that "by an overwhelming margin, Americans want to live in a sunny place." This statement is based on data from a nationally representative sample of 2260 adult Americans. Of those surveyed, 1288 indicated that they would prefer to live in a hot climate rather than a cold climate. Suppose that you want to determine if there is convincing evidence that a majority of all adult Americans prefer a hot climate over a cold climate. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is 0.000001 . What conclusion would you reach if \(\alpha=0.01 ?\)

The article "Euthanasia Still Acceptable to Solid Majority in U.S." (www.gallup.com, June \(24,2016,\) retrieved November 29,2016 ) summarized data from a survey of 1025 adult Americans. When asked if doctors should be able to end a terminally ill patient's life by painless means if requested to do so by the patient, 707 of those surveyed responded yes. For proposes of this exercise, assume that it is reasonable to regard this sample as a random sample of adult Americans. Suppose that you want to use the data from this survey to decide if there is convincing evidence that more than two-thirds of adult Americans believe that doctors should be able to end a terminally ill patient's life if requested to do so by the patient. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is \(0.058 .\) What conclusion would you reach if \(\alpha=0.05 ?\) c. Would you have reached a different conclusion if \(\alpha=0.10 ?\) Explain.

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 1 and Type II errors. c. Would you prefer a test with \(\alpha=0.01\) or \(\alpha=0.10 ?\) Explain your reasoning.

A county commissioner must vote on a resolution that would commit substantial resources to the construction of a sewer in an outlying residential area. Her fiscal decisions have been criticized in the past, so she decides to take a survey of residents in her district to find out if they favor spending money for a sewer system. She will vote to appropriate funds only if she can be reasonably sure that a majority of the people in her district favor the measure. What hypotheses should she test?

The paper "Bedtime Mobile Phone Use and Sleep in Adults" (Social Science and Medicine [2016]: \(93-101\) ) describes a study of 844 adults living in Belgium. Suppose that it is reasonable to regard this sample as a random sample of adults living in Belgium. You want to use the survey data to decide if there is evidence that a majority of adults living in Belgium take their cell phones to bed with them. Let \(p\) denote the population proportion of all adults living in Belgium who take their cell phones to bed with them. (Hint: See Example \(10.10 .)\) a. Describe the shape, center, and variability of the sampling distribution of \(\hat{p}\) for random samples of size 844 if the null hypothesis \(H_{0}: p=0.50\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.52\) for a sample of size 844 if the null hypothesis \(H_{0}: p=0.50\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.54\) for a sample of size 844 if the null hypothesis \(H_{0}: p=0.50\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.59 .\) Based on this sample proportion, is there convincing evidence that the null hypothesis \(H_{0}: p=\) 0.50 is not true, or is \(\hat{p}\) consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation. e. Do you think it would be reasonable to generalize the concusion of this test to adults living in the United States? Explain why or why not.
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