Chapter 10: Problem 27
Describe the two types of errors that might be made when a hypothesis test is carried out.
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A survey of 1,000 adult Americans ("Military Draft Study," AP-Ipsos, June 2005 ) included the following question: "If the military draft were reinstated, would you favor or oppose drafting women as well as men?" Forty-three percent responded that they would favor drafting women if the draft were reinstated. Using the five-step process for hypothesis testing \(\left(\mathrm{HMC}^{3}\right)\) and a 0.05 significance level, determine if there is convincing evidence that less than half of adult Americansp favor drafting women.
Every year on Groundhog Day (February 2), the famous groundhog Punxsutawney Phil tries to predict whether there will be 6 more weeks of winter. The article "Groundhog Has Been Off Target" (USA Today, Feb. 1,2011 ) states that "based on weather data, there is no predictive skill for the groundhog." Suppose that you plan to take a random sample of 20 years and use weather data to determine the proportion of these years the groundhog's prediction was correct. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for samples of size 20 if the groundhog has only a \(50-50\) chance of making a correct prediction. b. Based on your answer to Part (a), what sample proportion values would convince you that the groundhog's predictions have a better than \(50-50\) chance of being correct?
Step 5 of the five-step process for hypothesis testing is communication of results. What is involved in completing this step?
In a representative sample of 2,013 American adults, 1,590 indicated that lack of respect and courtesy in American society is a serious problem (Associated Press, April 3,2002 ). Is there convincing evidence that more than three- quarters of American adults believe that lack of respect and courtesy is a serious problem? Test the relevant hypotheses using a significance level of 0.05 .
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?
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