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Chapter 10: Problem 28

Give an example of a situation where you would want to select a small significance level.

Short Answer

Expert verified
A small significance level is desirable in cases where the cost or risk of a Type I error is high. For example, in pharmaceutical testing, where falsely declaring a safe drug as harmful can have significant consequences, a lower significance level like 1% would be chosen to minimize the risk of this error.

Step by step solution

01

Understand the Concept of Significance Level

In statistical hypothesis testing, the significance level (伪) is the probability of rejecting the null hypothesis when it is true. This error is known as a Type I error. A smaller significance level would mean that we are less likely to make a Type I error, i.e., less likely to reject a true null hypothesis. When a lower significance level is chosen, it means that more substantial evidence is required to reject the null hypothesis.
02

Identify Cases Where a Small Significance Level is Preferable

A small significance level is preferable in situations where the cost or risk of a false positive (Type I error) is high. For instance, in drug testing, where a Type I error would mean declaring a harmful drug to be safe.
03

Formulate a Concrete Example

Consider a pharmaceutical company testing a new drug. The null hypothesis could be that the drug is safe. A Type I error in this case would mean rejecting the null hypothesis when it is true, that is, declaring the drug to be harmful when it is, in fact, safe. As this scenario has potentially severe consequences, namely halting the production and distribution of a safe drug, it is crucial to minimize the probability of making this type of error. Therefore, a smaller significance level, like 0.01 or 1%, would be a good choice in this scenario to reduce the risk of falsely identifying the drug as harmful.

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Most popular questions from this chapter

In a survey of 1,000 women ages 22 to 35 who work full-time, 540 indicated that they would be willing to give up some personal time in order to make more money (USA Today, March 4, 2010). The sample was selected to be representative of women in the targeted age group. a. Do the sample data provide convincing evidence that a majority of women ages 22 to 35 who work fulltime would be willing to give up some personal time for more money? Test the relevant hypotheses using \(\alpha=0.01\) b. Would it be reasonable to generalize the conclusion from Part (a) to all working women? Explain why or why not.

USA Today (March 4, 2010) described a survey of 1,000 women ages 22 to 35 who work full time. Each woman who participated in the survey was asked if she would be willing to give up some personal time in order to make more money. To determine if the resulting data provided convincing evidence that the majority of women ages 22 to 35 who work full time would be willing to give up some personal time for more money, what hypotheses should you test?

In a survey conducted by Yahoo Small Business, 1,432 of 1,813 adults surveyed said that they would alter their shopping habits if gas prices remain high (Associated Press, November 30,2005\() .\) The article did not say how the sample was selected, but for purposes of this exercise, assume that the sample is representative of adult Americans. Based on the survey data, is it reasonable to conclude that more than threequarters of adult Americans would alter their shopping habits if gas prices remain high?

A television station has been providing live coverage of a sensational criminal trial. The station's program director wants to know if more than half of potential viewers prefer a return to regular daytime programming. A survey of randomly selected viewers is conducted. With \(p\) representing the proportion of all viewers who prefer regular daytime programming, what hypotheses should the program director test?

The article "Most Customers OK with New Bulbs" (USA Today, Feb. 18,2011 ) describes a survey of 1,016 randomly selected adult Americans. Each person in the sample was asked if they have replaced standard light bulbs in their home with the more energy efficient compact fluorescent (CFL) bulbs. Suppose you want to use the survey data to determine if there is evidence that more than \(70 \%\) of adult Americans have replaced standard bulbs with CFL bulbs. Let \(p\) denote the proportion of all adult Americans who have replaced standard bulbs with CFL bulbs. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for random samples of size 1,016 if the null hypothesis \(H_{0}: p=0.70\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.72\) for a sample of size 1,016 if the null hypothesis \(H_{0}: p=0.70\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.75\) for a sample of size 1,016 if the null hypothesis \(H_{0}: p=0.70\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.71\). Based on this sample proportion, is there convincing evidence that more than \(70 \%\) have replaced standard bulbs with CFL bulbs, or is this sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.
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