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Chapter 10: Problem 87

Suppose that you are an inspector for the Fish and Game Department and that you are given the task of determining whether to prohibit fishing along part of the Oregon coast. You will close an area to fishing if it is determined that more than \(3 \%\) of fish have unacceptably high mercury levels. a. Which of the following pairs of hypotheses would you test: $$ H_{0}: p=0.03 \text { versus } H_{a}: p>0.03 $$ or $$ H_{0}: p=0.03 \text { versus } H_{\dot{a}}: p<0.03 $$ Explain the reason for your choice. b. Would you use a significance level of 0.10 or 0.01 for your test? Explain.

Short Answer

Expert verified
a. We should choose the pair of hypotheses: $$ H_{0}: p=0.03 \text { versus } H_{a}: p>0.03 $$ This is because we are interested in whether the proportion of fish with high mercury levels is greater than \(3 \%\) and want to close the area if this condition is met. b. We should use a significance level of \(0.01\) for this test to minimize the chances of committing a Type I error (incorrectly closing the area) and have higher confidence in our decision.

Step by step solution

01

Part a: Identifying the null and alternative hypotheses

We have to determine if we should close a fishing area based on the condition that more than \(3 \%\) of fish have high mercury levels. This means we are interested in whether the proportion (p) of fish having high mercury levels is greater than \(0.03\). Therefore, we need to choose the pair of hypotheses: $$ H_{0}: p=0.03 \text { versus } H_{a}: p>0.03 $$ Explanation: The null hypothesis (\(H_{0}\)) assumes that the percentage of fish with high mercury levels is equal to \(3 \%\). On the other hand, the alternative hypothesis (\(H_{a}\)) considers the situation where the percentage of fish with high mercury levels is more than \(3 \%\). As we want to close the fishing area if the percentage of fish with high mercury levels is more than \(3 \%\), we choose this pair of hypotheses.
02

Part b: Choosing the significance level for the test

The two options for the significance level are \(0.10\) and \(0.01\). The significance level represents the probability of committing a Type I error. A lower significance level indicates lower chances of committing a Type I error, i.e., incorrectly rejecting the null hypothesis when it's actually true. In the context of this problem, a Type I error would mean closing the fishing area even if the percentage of fish with high mercury levels is actually \(3 \%\) or less. As closing the fishing area has economic consequences, we want to minimize the chances of incorrectly closing the fishing area when it isn't necessary. Therefore, a lower significance level (\(0.01\)) should be used for this test. Explanation: By choosing a significance level of \(0.01\), we are having higher confidence in our decision to close the fishing area, as there is a less probability of committing a Type I error (incorrectly closing the area). On the other hand, a significance level of \(0.10\) would have higher chances of a Type I error, leading to potential unnecessary closures.

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Key Concepts

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

Null Hypothesis
When conducting hypothesis testing in statistics, the null hypothesis, denoted as H0, serves as the default statement or the status quo that there is no effect or no difference. It is the proposition that there is no significant change or relationship. In the context of the fishing mercury level example, the null hypothesis asserts that the proportion of fish with high mercury levels is exactly 3%, which is represented as H0: p = 0.03.

This hypothesis is what we assume to be true until we have sufficient evidence to suggest otherwise. It is critical to define the null hypothesis clearly because it sets the stage for statistical testing, and the outcome of the test will determine whether this hypothesis can be retained or if we must consider an alternative scenario.
Alternative Hypothesis
Opposing the null hypothesis is the alternative hypothesis, denoted as Ha or H1, which represents a statement we believe to be true if the null hypothesis is rejected. In addressing mercury levels in fish, the alternative hypothesis posits that more than 3% of fish have high mercury levels (Ha: p > 0.03).

The alternative hypothesis is framed based on the context of the investigation. In our example, the conservation goal is to take action if mercury levels exceed the 3% threshold; thus, the alternative hypothesis is directional, indicating a concern only when the actual proportion surpasses the benchmark. If the evidence strongly supports Ha, we may decide to close the fishing area to protect the ecosystem and public health.
Significance Level
The significance level of a hypothesis test, often denoted by α, is the threshold for making a decision about the null hypothesis. It quantifies the risk of committing a Type I error, which means falsely rejecting H0 when it is actually correct. The significance level is pre-selected by the researcher, with common choices being 0.01, 0.05, or 0.10.

In the case of determining fishing restrictions, a more conservative significance level, such as 0.01, implies a lower risk of closing the area under false pretenses. A lower α means requiring more conclusive evidence against the null hypothesis before we take action. Considering the economic and social impacts of a closure, a stringent significance level is appropriate to avoid unjustified regulatory measures based on the evidence available.
Type I Error
A Type I error occurs when we reject the null hypothesis even though it's true. In simpler terms, it's a 'false alarm'. The probability of making this error is equal to the significance level, α, set by the researcher. In our example, a Type I error would lead us to close the fishing area based on an incorrect conclusion that the mercury levels are too high, which could have unnecessary economic repercussions.

To mitigate this risk, we choose a low significance level. However, this increases the chance of a Type II error, where we fail to reject a false null hypothesis, potentially leaving an area open that should be closed. The balance between Type I and Type II errors is crucial in decision-making processes where consequences are significant, as in environmental conservation efforts.

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Most popular questions from this chapter

Refer to the instructions given prior to Exercise \(10.47 .\) The article "iPhone Can Be Addicting, Says New Survey" (www.msnbc.com, March 8,2010 ) described a survey administered to 200 college students who owned an iPhone, One of the questions on the survey asked students if they slept with their iPhone in bed with them. You would like to use the data from this survey to determine if there is convincing evidence that a majority of college students with iPhones sleep with their phones.

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The article "Facebook Use and Academic Performance Among College Students" (Computers in Human Behavior \([2015]: 265-272)\) estimated that \(87 \%\) percent of students at a large public university in California who are Facebook users update their status at least two times a day. Suppose that you plan to select a random sample of 400 students at your college. You will ask each student in the sample if they are a Facebook user and if they update their status at least two times a day. You plan to use the resulting data to decide if there is evidence that the proportion for your college is different from the proportion reported in the article for the college in California. What hypotheses should you test?

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 1035 of the people in a random sample of 1523 American adults said they approved of casino gambling. Is there convincing evidence that more than two-thirds approve?

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?
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