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Chapter 10: Problem 44

For which of the following combinations of \(P\) -value and significance level would the null hypothesis be rejected? a. \(P\) -value \(=0.426 \quad \alpha=0.05\) b. \(P\) -value \(=0.033 \quad \alpha=0.01\) c. \(P\) -value \(=0.046 \quad \alpha=0.10\) d. \(P\) -value \(=0.026 \quad \alpha=0.05\) e. \(P\) -value \(=0.004 \quad \alpha=0.01\)

Short Answer

Expert verified
Based on the comparison of P-values with their respective alpha levels, the null hypothesis would be rejected only in cases c, d, and e.

Step by step solution

01

Compare P-value and Alpha for Case a

In case a, P-value is equal to 0.426 and alpha is 0.05. Since the P-value is larger than alpha, the null hypothesis would not be rejected.
02

Compare P-value and Alpha for Case b

In case b, P-value is equal to 0.033 and alpha is 0.01. Again the P-value is larger than alpha, so the null hypothesis would not be rejected.
03

Compare P-value and Alpha for Case c

In case c, P-value is 0.046 and alpha is 0.10. Now, the P-value is smaller than alpha, so the null hypothesis would be rejected in this case.
04

Compare P-value and Alpha for Case d

In case d, P-value is 0.026 and alpha is 0.05. Here, the P-value is smaller than alpha so the null hypothesis would be rejected.
05

Compare P-value and Alpha for Case e

Finally, in case e, P-value is 0.004 and alpha is 0.01. In this case as well, the P-value is smaller than alpha, hence the null hypothesis would be rejected.

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Most popular questions from this chapter

Explain why you would not reject the null hypothesis if the \(P\) -value were 0.37 .

Explain why failing to reject the null hypothesis in a hypoth- b. esis test does not mean there is convincing evidence that the null hypothesis is true.

Suppose that for a particular hypothesis test, the consequences of a Type I error are very serious. Would you want to carry out the test using a small significance level \(\alpha\) (such as 0.01 ) or a larger significance level (such as 0.10 )? Explain the reason for your choice.

The National Cancer Institute conducted a 2-year study to determine whether cancer death rates for areas near nuclear power plants are higher than for areas without nuclear facilities (San Luis Obispo Telegram-Tribune, September 17,1990 ). A spokesperson for the Cancer Institute said, "From the data at hand, there was no convincing evidence of any increased risk of death from any of the cancers surveyed due to living near nuclear facilities. However, no study can prove the absence of an effect." a. Suppose \(p\) denotes the true proportion of the population in areas near nuclear power plants who die of cancer during a given year. The researchers at the Cancer Institute might have considered two rival hypotheses of the form \(H_{0}: p\) is equal to the corresponding value for areas without nuclear facilities \(H_{a}: p\) is greater than the corresponding value for areas without nuclear facilities Did the researchers reject \(H_{0}\) or fail to reject \(H_{0} ?\) b. If the Cancer Institute researchers are incorrect in their conclusion that there is no evidence of increased risk of death from cancer associated with living near a nuclear power plant, are they making a Type I or a Type II error? Explain. c. Comment on the spokesperson's last statement that no study can prove the absence of an effect. Do you agree with this statement?

The article "Poll Finds Most Oppose Return to Draft, Wouldn't Encourage Children to Enlist" (Associated Press, December 18,2005 ) reports that in a random sample of 1,000 American adults, 700 indicated that they oppose the reinstatement of a military draft. Suppose you want to use this information to decide if there is convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than two-thirds. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is \(0.013 .\) What conclusion would you reach if \(\alpha=0.05 ?\)
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