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Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a. In this context, what is a Type I error? b. In this context, what is a Type II error? c. Which type of error would the factory owner consider more serious? d. Which type of error might customers consider more serious?

A company is sued for job discrimination because only \(19 \%\) of the newly hired candidates were minorities when \(27 \%\) of all applicants were minorities. Is this strong evidence that the company's hiring practices are discriminatory? a. Is this a one-tailed or a two-tailed test? Why? b. In this context, what would a Type I error be? c. In this context, what would a Type II error be? d. In this context, what is meant by the power of the test? e. If the hypothesis is tested at the \(5 \%\) level of significance instead of \(1 \%\), how will this affect the power of the test? \(\mathrm{f}\). The lawsuit is based on the hiring of 37 employees. Is the power of the test higher than, lower than, or the same as it would be if it were based on 87 hires?

Highway safety engineers test new road signs, hoping that increased reflectivity will make them more visible to drivers. Volunteers drive through a test course with several of the new- and old-style signs and rate which kind shows up the best. a. Is this a one-tailed or a two-tailed test? Why? b. In this context, what would a Type I error be? c. In this context, what would a Type II error be? d. In this context, what is meant by the power of the test? e. If the hypothesis is tested at the \(1 \%\) level of significance instead of \(5 \%\), how will this affect the power of the test? f. The engineers hoped to base their decision on the reactions of 50 drivers, but time and budget constraints may force them to cut back to 20 . How would this affect the power of the test? Explain.

A statistics professor has observed that for several years students score an average of 105 points out of 150 on the semester exam. A salesman suggests that he try a statistics software package that gets students more involved with computers, predicting that it will increase students' scores. The software is expensive, and the salesman offers to let the professor use it for a semester to see if the scores on the final exam increase significantly. The professor will have to pay for the software only if he chooses to continue using it. a. Is this a one-tailed or two-tailed test? Explain. b. Write the null and alternative hypotheses. c. In this context, explain what would happen if the professor makes a Type I error. d. In this context, explain what would happen if the professor makes a Type II error. e. What is meant by the power of this test?

A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than \(20 \%\) of the residents of the city have heard the ad and recognize the company's product. The radio station conducts a random phone survey of 400 people. a. What are the hypotheses? b. The station plans to conduct this test using a \(10 \%\) level of significance, but the company wants the significance level lowered to \(5 \%\). Why? c. What is meant by the power of this test? d. For which level of significance will the power of this test be higher? Why? e. They finally agree to use \(\alpha=0.05,\) but the company proposes that the station call 600 people instead of the 400 initially proposed. Will that make the risk of Type II error higher or lower? Explain.

In a drawer are two coins. They look the same, but one coin produces heads \(90 \%\) of the time when spun while the other one produces heads only \(30 \%\) of the time. You select one of the coins. You are allowed to spin it once and then must decide whether the coin is the \(90 \%\) - or the \(30 \%\) -head coin. Your null hypothesis is that your coin produces \(90 \%\) heads. a. What is the alternative hypothesis? b. Given that the outcome of your spin is tails, what would you decide? What if it were heads? c. How large is \(\alpha\) in this case? d. How large is the power of this test? (Hint: How many possibilities are in the alternative hypothesis?) e. How could you lower the probability of a Type I error and increase the power of the test at the same time?

A basketball player with a poor foul-shot record practices intensively during the off-season. He tells the coach that he has raised his proficiency from \(60 \%\) to \(80 \%\). Dubious, the coach asks him to take 10 shots, and is surprised when the player hits 9 out of 10. Did the player prove that he has improved? a. Suppose the player really is no better than before-still a \(60 \%\) shooter. What's the probability he can hit at least 9 of 10 shots anyway? (Hint: Use a Binomial model.) b. If that is what happened, now the coach thinks the player has improved when he has not. Which type of error is that? c. If the player really can hit \(80 \%\) now, and it takes at least 9 out of 10 successful shots to convince the coach, what's the power of the test? d. List two ways the coach and player could increase the power to detect any improvement.

An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About \(40 \%\) break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. a. Suppose the new, expensive clay really is no better than her usual clay. What's the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.) b. If she decides to switch to the new clay and it is no better, what kind of error did she commit? c. If the new clay really can reduce breakage to only \(20 \%,\) what's the probability that her test will not detect the improvement? d. How can she improve the power of her test? Offer at least two suggestions.