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Chapter 8: Problem 26

Use a significance level of \(\alpha=0.05\) and use the given information for the following: a. State a conclusion about the null hypothesis. (Reject \(H_{0}\) or fail to reject \(H_{0 .}\) ) b. Without using technical terms or symbols, state a final conclusion that addresses the original claim. Original claim: Fewer than \(90 \%\) of adults have a cell phone. The hypothesis test results in a \(P\) -value of 0.0003.

Short Answer

Expert verified
Reject Ho; fewer than 90% of adults have a cell phone.

Step by step solution

01

Understanding the Hypotheses

First, define the null and alternative hypotheses. The null hypothesis (Ho) is that 90% or more of adults have a cell phone. The alternative hypothesis (Ha) is that fewer than 90% of adults have a cell phone.
02

Determine the P-value Decision Rule

Compare the given P-value with the significance level, \(\alpha=0.05\). If the P-value is less than \(\alpha\), reject Ho; otherwise, fail to reject Ho.
03

Compare P-value and Significance Level

Given that the P-value is 0.0003, compare it with the significance level 0.05. Since 0.0003 is much smaller than 0.05, we have enough evidence to reject the null hypothesis.
04

Draw a Conclusion About Ho

Since 0.0003 < 0.05, reject the null hypothesis. This means we do not support the notion that 90% or more adults have a cell phone.
05

State a Final Conclusion in Simple Terms

Based on the given P-value and significance level, conclude that there's strong evidence to support the claim that fewer than 90% of adults have a cell phone.

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Key Concepts

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

null hypothesis
The null hypothesis, often denoted as \(H_0\), is a statement that assumes there is no effect or no difference in the context of a scientific study. It serves as a starting point for our hypothesis test. In this example, the null hypothesis claims that 90% or more of adults have a cell phone. We assume this to be true until we find evidence strong enough to support an alternative claim. Essentially, the null hypothesis represents the status quo or the default position.
alternative hypothesis
The alternative hypothesis, indicated as \(H_a\), is a statement that contradicts the null hypothesis. This hypothesis suggests there is an effect or difference. In our case, the alternative hypothesis claims that fewer than 90% of adults have a cell phone. If we find enough evidence against the null hypothesis, we will accept the alternative hypothesis. The alternative hypothesis represents what we aim to prove in our study.
significance level
The significance level, denoted by \( \alpha \), is a threshold chosen by researchers to determine whether to reject the null hypothesis. This level indicates the probability of rejecting the null hypothesis when it is actually true. In our example, the significance level is set at 0.05. This means there is a 5% risk of concluding that the alternative hypothesis is true when it is not. Selecting an appropriate significance level is essential, as it affects the reliability and validity of our test results.
P-value
The P-value represents the probability of observing results as extreme as, or more extreme than, the ones obtained, given that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis. In our example, the P-value is 0.0003. Since this value is much smaller than our significance level of 0.05, we have strong evidence against the null hypothesis. A smaller P-value indicates stronger evidence in favor of the alternative hypothesis.
statistical conclusion
The statistical conclusion is derived by comparing the P-value with the significance level. If the P-value is less than the significance level, we reject the null hypothesis. In our case, since 0.0003 < 0.05, we reject the null hypothesis. This leads us to conclude that fewer than 90% of adults have a cell phone. Our final conclusion, simplified, is that the data strongly supports the original claim that less than 90% of adults are cell phone users.

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Most popular questions from this chapter

Chantix (varenicline) tablets are used as an aid to help people stop smoking. In a clinical trial, 129 subjects were treated with Chantix twice a day for 12 weeks, and 16 subjects experienced abdominal pain (based on data from Pfizer, Inc.). If someone claims that more than \(8 \%\) of Chantix users experience abdominal pain, that claim is supported with a hypothesis test conducted with a 0.05 significance level. Using 0.18 as an alternative value of \(p,\) the power of the test is \(0.96 .\) Interpret this value of the power of the test.

The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analyzed with a formal hypothesis test with the alternative hypothesis of \(p>0.5,\) which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is greater than 0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer: \(0.999,0.5,0.95,0.05,0.01,0.001 ?\) Why?

Use a significance level of \(\alpha=0.05\) and use the given information for the following: a. State a conclusion about the null hypothesis. (Reject \(H_{0}\) or fail to reject \(H_{0 .}\) ) b. Without using technical terms or symbols, state a final conclusion that addresses the original claim. Original claim: The standard deviation of pulse rates of adult males is more than 11 bpm. The hypothesis test results in a \(P\) -value of 0.3045.

Cans of Coke Data Set 26 "Cola Weights and Volumes" in Appendix B includes volumes (oz) of regular Coke. Based on that data set, assume that the cans are produced so that the volumes have a standard deviation of 0.115 oz. A new filling process is being tested for filling cans of cola, and a random sample of volumes is listed below. The sample has these summary statistics: \(n=10\) \(\bar{x}=12.0004\) oz, \(s=0.2684\) oz. If we want to use the sample data to test the claim that the sample is from a population with a standard deviation equal to 0.115 oz, what requirements must be satisfied? How does the normality requirement for a hypothesis test of a claim about a standard deviation differ from the normality requirement for a hypothesis test of a claim about a mean? $$\begin{array}{llllllllll} 12.078 & 11.851 & 12.108 & 11.760 & 12.142 & 11.779 & 12.397 & 11.504 & 12.147 & 12.238 \end{array}$$

Use these results from a USA Today survey in which 510 people chose to respond to this question that was posted on the USA Today website: "Should Americans replace passwords with biometric security (fingerprints, etc)?" Among the respondents, 53\% said "yes." We want to test the claim that more than half of the population believes that passwords should be replaced with biometric security. If we use the same significance level to conduct the hypothesis test using the \(P\) -value method, the critical value method, and a confidence interval, which method is not equivalent to the other two?
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