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Chapter 8: Problem 27

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 mean pulse rate (in beats per minute) of adult males is 72 bpm. The hypothesis test results in a \(P\) -value of 0.0095.

Short Answer

Expert verified
Reject \(H_{0}\). The mean pulse rate of adult males is not 72 bpm.

Step by step solution

01

Identify Given Values

Given information: Significance level \( \alpha \)=0.05, p-value = 0.0095. Original claim: The mean pulse rate of adult males is 72 bpm.
02

Define Hypotheses

The null hypothesis \(H_{0}\) is that the mean pulse rate of adult males is 72 bpm. The alternative hypothesis \(H_{a}\) is that the mean pulse rate of adult males is not 72 bpm.
03

Compare the p-value with Significance Level

Compare the given p-value (0.0095) to the significance level \( \alpha \) (0.05).
04

Decision Rule

If the p-value is less than or equal to the significance level, reject the null hypothesis \(H_{0}\). If the p-value is greater than the significance level, fail to reject \H_{0}\.
05

State Conclusion About Null Hypothesis

Since the p-value (0.0095) is less than the significance level (0.05), we reject the null hypothesis \(H_{0}\).
06

State Final Conclusion in Simple Terms

Based on the test result, we have enough evidence to support the claim that the mean pulse rate of adult males is not 72 bpm.

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Key Concepts

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

Significance Level
In hypothesis testing, the significance level, denoted as \(\alpha\), is a threshold set by the researcher to determine whether to reject the null hypothesis.
This level reflects the probability of rejecting the null hypothesis when it is actually true, which is known as making a Type I error.
Commonly used significance levels include 0.05, 0.01, and 0.10.
When \(\alpha\) is set to 0.05, it means there is a 5% risk of concluding that a difference exists when there is no actual difference. In our exercise, the significance level \(\alpha\) is 0.05, indicating a 5% tolerance for Type I error. By comparing the p-value to this significance level, decisions are made regarding the null hypothesis.
p-value
The p-value is a crucial component in hypothesis testing. It measures the strength of evidence against the null hypothesis.
Specifically, it quantifies the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.
If the p-value is less than or equal to the significance level \(\alpha\), the null hypothesis is rejected.
This suggests that the observed data is statistically significant.
In our problem, the p-value is 0.0095, which is much smaller than the significance level of 0.05.
Thus, we have strong evidence to reject the null hypothesis.
Null Hypothesis
The null hypothesis, denoted as \(H_{0}\), is a statement of no effect or no difference. It represents a standard or status quo that researchers seek to challenge.
In hypothesis testing, the null hypothesis is what we aim to test, and we assume it to be true until we find enough evidence to reject it.
For example, our exercise states that the mean pulse rate of adult males is 72 bpm.
Thus, the null hypothesis \(H_{0}\) would be: The mean pulse rate of adult males is 72 bpm.
Rejecting the null hypothesis suggests that there is enough evidence to support the alternative hypothesis.
Alternative Hypothesis
The alternative hypothesis, denoted as \(H_{a}\), is a statement that contradicts the null hypothesis.
It represents the effect or difference that the researcher aims to provide evidence for.
The alternative hypothesis is accepted if the null hypothesis is rejected based on the p-value.
In the given exercise, if the null hypothesis states that the mean pulse rate of adult males is 72 bpm, then the alternative hypothesis would state:
The mean pulse rate of adult males is not 72 bpm.
This two-tailed test suggests that the mean pulse rate could be either higher or lower than 72 bpm.
The outcome of the hypothesis test in this scenario, based on the given p-value, supports the alternative hypothesis.

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Most popular questions from this chapter

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.

When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1. a. Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1. b. Use the \(P\) -value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1. c. Use the sample data to construct a \(95 \%\) confidence interval estimate of the proportion of zeros. What does the confidence interval suggest about the claim that the proportion of zeros equals 0.1? d. Compare the results from the critical value method, the \(P\) -value method, and the confidence interval method. Do they all lead to the same conclusion?

Type I and Type II Errors provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as \(p=0.1 .\) ) The proportion of people with blue eyes is equal to 0.35.

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?

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.
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