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Chapter 10: Problem 9

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 constituents to find out whether they favor spending money for a sewer system. She will vote to appropriate funds only if she can be fairly certain that a majority of the people in her district favor the measure. What hypotheses should she test?

Short Answer

Expert verified
The hypotheses she should test are the null hypothesis \(H_0\): p 鈮� 0.5 implying that 50% or fewer people favor spending money for the sewer system, and the alternative hypothesis \(H_1\): p > 0.5 indicating that more than 50% favor it. She ought to vote for the sewer system if the survey result supports the alternative hypothesis and vote against it if the result leans towards the null hypothesis.

Step by step solution

01

Identifying the Hypotheses

From the scenario, the commissioner needs to make a decision about committing resources to construct a sewer based on whether a majority of people in her district favor the measure. Therefore, she needs to conduct a hypothesis test with two possibilities - a majority favor the measure (more than 50%) or they don't (50% or less). So, her hypotheses should be: \(H_0\): p 鈮� 0.5 (The majority of the people do not favor the measure, this is the so-called null hypothesis) \(H_1\): p > 0.5 (The majority of people favor the measure, this is the alternative hypothesis)
02

Interpreting the Hypotheses

In the context of the scenario provided, the null hypothesis \(H_0\): p 鈮� 0.5 implies that 50% or fewer people in the district favor spending money on a new sewer system. The alternative hypothesis \(H_1\): p > 0.5 implies that more than 50% of the people in her district favor spending money on the sewer system.
03

Making the Decision

If the result of the survey supports the alternative hypothesis (p > 0.5), then the commissioner can vote to allocate funds for the sewer system since the majority of her constituents favor the measure. If the survey results support the null hypothesis (p 鈮� 0.5), then she could decide against funding the sewer system.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis is a statement that assumes no effect or no difference. It represents a default position that there is nothing new happening, and anything observed is due to chance.
For the commissioner in the exercise, the null hypothesis (\(H_0\): p \leq 0.5) means that 50% or fewer of the constituents favor the construction of the sewer system. This suggests that there isn't a majority that supports the project.
This hypothesis is crucial because it acts as a baseline. The commissioner will test against this to see if opposition is the correct assumption, providing a standard for comparison.
You often see the null hypothesis stated as "less than or equal to" in cases where determining the non-existence of an effect is crucial. Rejecting it implies that there is enough evidence to support that the default position might not be true.
Alternative Hypothesis
The alternative hypothesis is the statement that goes against the null hypothesis, suggesting that there is an effect or difference.
In the commissioner's scenario, the alternative hypothesis (\(H_1\): p > 0.5) indicates that more than 50% of constituents favor constructing the sewer system.
This hypothesis is what the commissioner hopes to substantiate with her survey. It suggests there is a majority in favor, thus justifying the allocation of funds.
It鈥檚 essential to formulate this hypothesis because it defines the specific outcome you are testing for. Proving or disproving the alternative hypothesis can guide decision-making and provide new insights.
The alternative hypothesis is a positive statement ("p > 0.5") indicating there is enough of a majority to support the measure.
Decision Making in Statistics
Statistical decision-making involves choosing between the null and alternative hypotheses based on data.
In the context of the exercise, after conducting the survey, the commissioner will use statistical methods to analyze the data and decide whether to accept the null hypothesis (no majority support) or reject it for the alternative hypothesis (majority support).
The decision is often made with a significance level, commonly set at 0.05, which measures how certain we are about our results.
- A probability (p-value) is computed from the data to decide whether to reject the null hypothesis. If this p-value is less than the significance level, it suggests rejecting the null hypothesis.
- If the p-value is greater than 0.05, it implies there's not enough evidence to safely reject the null hypothesis.
This process allows the commissioner to make an informed decision, minimizing the chances of making an error based on the data. This logical method supports decisions, being critical when resources and community interests are at stake.

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Most popular questions from this chapter

Optical fibers are used in telecommunications to transmit light. Current technology allows production of fibers that transmit light about \(50 \mathrm{~km}\) (Research at Rensselaer, 1984 ). Researchers are trying to develop a new type of glass fiber that will increase this distance. In evaluating a new fiber, it is of interest to test \(H_{0}: \mu=50\) versus \(H_{a}: \mu>50\), with \(\mu\) denoting the true average transmission distance for the new optical fiber. a. Assuming \(\sigma=10\) and \(n=10\), use Appendix Table 5 to find \(\beta\), the probability of a Type II error, for each of the given alternative values of \(\mu\) when a level \(.05\) test is employed: \(\begin{array}{llll}\text { i. } 52 & \text { ii. } 55 & \text { iii. } 60 & \text { iv. } 70\end{array}\) b. What happens to \(\beta\) in each of the cases in Part (a) if \(\sigma\) is actually larger than \(10 ?\) Explain your reasoning.

A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least \(10 \mathrm{hr}\). A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data support the use of a one-sample \(t\) test. The relevant hypotheses are \(H_{0}: \mu=10\) versus \(H_{a}: \mu<10\). a. If \(t=-2.3\) and \(\alpha=.05\) is selected, what conclusion is appropriate? b. If \(t=-1.83\) and \(\alpha=.01\) is selected, what conclusion is appropriate? c. If \(t=0.47\), what conclusion is appropriate?

Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations ("Mammogram Cancer Scares More Frequent than Thought," USA Today, April 16,1998 ). Discussing the benefits and downsides of the screening process, the article states that, although the rate of false-positives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of \(H_{0}:\) no cancer is present and an alternative hypothesis of \(H_{0}:\) cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. What aspect of the relationship between the probability of Type I and Type II errors is being described by the statement in the article that if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise?

Duck hunting in populated areas faces opposition on the basis of safety and environmental issues. The San Luis Obispo Telegram-Tribune (June 18,1991 ) reported the results of a survey to assess public opinion regarding duck hunting on Morro Bay (located along the central coast of California). A random sample of 750 local residents included 560 who strongly opposed hunting on the bay. Does this sample provide sufficient evidence to conclude that the majority of local residents oppose hunting on Morro Bay? Test the relevant hypotheses using \(\alpha=.01\).

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ} \mathrm{F}\), researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ} \mathrm{F}\) versus \(H_{a}: \mu>150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.
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