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Chapter 10: Problem 76

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?

Short Answer

Expert verified
The null hypothesis (H0) is that 50% or less of the residents support the spending money on a sewer system. The alternative hypothesis (H1) is that more than 50% of residents support the spending money on a sewer system.

Step by step solution

01

- Identify the Null Hypothesis

This is the default assumption unless proven otherwise. The null hypothesis (H0) in this case is: '50% or less than the residents in her district favor the spending money for a sewer system'.
02

- Identify the Alternative Hypothesis

This is what the commissioner will accept if the null hypothesis is rejected based on the results of the survey. The alternative hypothesis (H1) in this case is: 'More than 50% of the residents in her district favor the spending money for a sewer system.'
03

- Conduct the Survey and Test the Hypotheses

The commissioner needs to carry out the survey sampling a representative part of her district's population. Then, she will calculate the percentage of respondents supporting the measure and use appropriate hypothesis testing methods to determine if the null hypothesis can be rejected or not.

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Most popular questions from this chapter

In a representative sample of 1,000 adult Americans, only 430 could name at least one justice who was currently serving on the U.S. Supreme Court (Ipsos, January 10,2006 ). Using a significance level of \(0.01,\) determine if there is convincing evidence in support of the claim that less than half of adult Americans can name at least one justice currently serving on the Supreme Court.

The authors of the article "Perceived Risks of Heart Disease and Cancer Among Cigarette Smokers" (Journal of the American Medical Association [1999]: \(1019-1021\) ) expressed the concern that a majority of smokers do not view themselves as being at increased risk of heart disease or cancer. A study of 737 current smokers found that only 295 believe they have a higher than average risk of cancer. Do these data suggest that \(p,\) the proportion of all smokers who view themselves as being at increased risk of cancer, is less than \(0.5,\) as claimed by the authors of the paper? For purposes of this exercise, assume that this sample is representative of the population of smokers. Test the relevant hypotheses using \(\alpha=0.05\)

Every year on Groundhog Day (February 2), the famous groundhog Punxsutawney Phil tries to predict whether there will be 6 more weeks of winter. The article "Groundhog Has Been Off Target" (USA Today, Feb. 1,2011 ) states that "based on weather data, there is no predictive skill for the groundhog." Suppose that you plan to take a random sample of 20 years and use weather data to determine the proportion of these years the groundhog's prediction was correct. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for samples of size 20 if the groundhog has only a \(50-50\) chance of making a correct prediction. b. Based on your answer to Part (a), what sample proportion values would convince you that the groundhog's predictions have a better than \(50-50\) chance of being correct?

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Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=0.0003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=0.350\)
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