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Pairs of \(P\) -values and significance levels, \(\alpha,\) are given. For each pair, state whether the observed \(P\) -value leads to rejection of \(H_{0}\) at the given significance level. a. \(\quad P\) -value \(=.084, \alpha=.05\) b. \(\quad P\) -value \(=.003, \alpha=.001\) c. \(P\) -value \(=.498, \alpha=.05\) d. \(\quad P\) -value \(=.084, \alpha=.10\) e. \(\quad P\) -value \(=.039, \alpha=.01\) f. \(P\) -value \(=.218, \alpha=.10\)

Short Answer

Expert verified
a. Fail to reject \(H_{0}\) \n b. Fail to reject \(H_{0}\) \n c. Fail to reject \(H_{0}\) \n d. Reject \(H_{0}\) \n e. Fail to reject \(H_{0}\) \n f. Fail to reject \(H_{0}\)

Step by step solution

01

Case a: Comparing P-value and Significance Level

Given that P-value = 0.084 and \(\alpha\) = 0.05. Since 0.084 is greater than 0.05, we fail to reject the null hypothesis \(H_{0}\).
02

Case b: Comparing P-value and Significance Level

Given that P-value = 0.003 and \(\alpha\) = 0.001. Since 0.003 is greater than 0.001, we fail to reject the null hypothesis \(H_{0}\).
03

Case c: Comparing P-value and Significance Level

Given that P-value = 0.498 and \(\alpha\) = 0.05. Since 0.498 is greater than 0.05, we fail to reject the null hypothesis \(H_{0}\).
04

Case d: Comparing P-value and Significance Level

Given that P-value = 0.084 and \(\alpha\) = 0.1. Since 0.084 is less than 0.1, we reject the null hypothesis \(H_{0}\).
05

Case e: Comparing P-value and Significance Level

Given that P-value = 0.039 and \(\alpha\) = 0.01. Since 0.039 is greater than 0.01, we fail to reject the null hypothesis \(H_{0}\).
06

Case f: Comparing P-value and Significance Level

Given that P-value = 0.218 and \(\alpha\) = 0.1. Since 0.218 is greater than 0.1, we fail to reject the null hypothesis \(H_{0}\).

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Key Concepts

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

Understanding the P-Value
The p-value is a crucial concept in hypothesis testing. It helps us make decisions about the null hypothesis. So, what exactly is a p-value? To put it simply, the p-value indicates the probability of obtaining test results at least as extreme as the observed data, assuming that the null hypothesis is true.
For example, if we have a p-value of 0.084, it means there is an 8.4% chance of observing data as extreme as what we have if the null hypothesis \(H_0\) is true.
  • A small p-value (typically ≤ 0.05) suggests that the observed data is unlikely under the null hypothesis, leading us to consider rejecting \(H_0\).
  • A large p-value (> 0.05) suggests that the observed data is consistent with the null hypothesis, meaning we fail to reject \(H_0\).
Significance Level Explored
The significance level, often denoted as \( \alpha \), is a threshold researchers set for determining whether the p-value is small enough to reject the null hypothesis. It's the risk level one is willing to take to reject \(H_0\) when it is actually true. Typically, common values for \( \alpha \) are 0.05, 0.01, or 0.10.
This means that:
  • An \( \alpha\) of 0.05 indicates a 5% risk of incorrectly rejecting the null hypothesis.
  • If our p-value is less than \( \alpha \), it suggests strong evidence against \(H_0\), so we reject it.
  • If our p-value is greater than \( \alpha \), we do not have sufficient evidence to reject \(H_0\), so we fail to reject it.
The choice of \( \alpha\) depends on the field of study or specific experimental conditions. For high-stakes decisions, like drug approvals, researchers may choose a very low \( \alpha\), such as 0.01, to minimize the risk of incorrect rejection of \(H_0\).
The Role of the Null Hypothesis
The null hypothesis, denoted as \(H_0\), is the starting assumption for any hypothesis test. It often represents a position of no effect or no difference. For instance, if you're testing whether a new drug is effective, \(H_0\) might state that the drug has no effect.

In hypothesis testing, our goal is typically to see if we have enough evidence to reject \(H_0\) in favor of an alternative hypothesis \(H_1\), which represents an effect or difference. But we need substantial evidence to make such a decision.
  • We always assume \(H_0\) is true until evidence suggests otherwise.
  • The p-value shows how compatible your data is with \(H_0\).
  • Failing to reject \(H_0\) doesn't prove \(H_0\) is true; it only shows that we lack strong evidence against it given our data.
It's important to remember that not rejecting \(H_0\) because of a higher p-value might simply mean we need more data or a different testing setup to make definitive conclusions.

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Most popular questions from this chapter

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 \(H_{d}: \mu>\) \(150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

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 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data supports 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?

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose that the specifications state that the mean strength of welds should exceed \(100 \mathrm{lb} / \mathrm{in}^{2}\). The inspection team decides to test \(H_{0}: \mu=100\) versus \(H_{a}: \mu>100 .\) Explain why this alternative hypothesis was chosen rather than \(\mu<100\).

The report "2007 Electronic Monitoring \& Surveillance Survey: Many Companies Monitoring. Recording, Videotaping-and Firing-Employees" (American Management Association, 2007 ) summarized the results of a survey of 304 U.S. businesses. Of these companies, 201 indicated that they monitor employees' web site visits. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of businesses in the United States. a. Is there sufficient evidence to conclude that more than \(60 \%\) of U.S. businesses monitor employees' web site visits? Test the appropriate hypotheses using a significance level of .01 . b. Is there sufficient evidence to conclude that a majority of U.S. businesses monitor employees' web site visits? Test the appropriate hypotheses using a significance level of .01

An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is an expensive process, so it will be undertaken only if there is strong evidence that the proportion of defective installations is lower for the robots than for human assemblers. Let \(p\) denote the proportion of defective installations for the robots. It is known that human assemblers have a defect proportion of .02 . a. Which of the following pairs of hypotheses should the manufacturer test: \(H_{0}: p=.02\) versus \(H_{a}: p<.02\) or \(H_{0}: p=.02\) versus \(H_{a}: p>.02\) Explain your answer. b. In the context of this exercise, describe Type I and Type II errors. c. Would you prefer a test with \(\alpha=.01\) or \(\alpha=.1 ?\) Explain your reasoning.

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