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Error probabilities and power You read that a significance test at the $\mathrm{Î\pm }=0.01$

significance level has probability $0.14$of making a Type II error when a specific alternative is true.

a. What is the power of the test against this alternative?

b. What’s the probability of making a Type I error?

Power and error A scientist calculates that a test at the α=0.05 significance level has probability 0.23 of making a Type II error when a specific alternative is true.

a. What is the power of the test against this alternative?

b. What’s the probability of making a Type I error?

Do you have ESP? A researcher looking for evidence of extrasensory perception (ESP) tests $500$ subjects. Four of these subjects do significantly better $(P<0.01)$ than random guessing.

a. Is it proper to conclude that these four people have ESP? Explain your answer.

b. What should the researcher now do to test whether any of these four subjects have ESP?

Preventing colds A medical experiment investigated whether taking the herb echinacea could help prevent colds. The study measured $50$ different response variables usually associated with colds, such as low-grade fever, congestion, frequency of coughing, and so on. At the end of the study, those taking echinacea displayed significantly better responses at the$\mathrm{Î\pm }=0.05$ level than those taking a placebo for $3$ of the $50$ response variables studied. Should we be convinced that echinacea helps prevent colds? Why or why not?

Improving SAT scores A national chain of SAT-preparation schools wants to know if using a smartphone app in addition to its regular program will help increase student scores more than using just the regular program. On average, the students in the regular program increase their scores by $128$ points during the $3-$month class. To investigate using the smartphone app, the prep schools have $5000$ students use the app along with the regular

program and measure their improvement. Then the schools will test the following hypotheses: $H_0:\mathrm{μ}=128$ versus $H_a:\mathrm{μ}>128$ , where$\mathrm{μ}$ is the true

mean improvement in the SAT score for students who attend these prep schools. After 3 months, the average improvement was $\overline{x}=130$ with a standard deviation of $s_x=65$ The standardized test statistic is $t=2.18$ with a $P-$value of $0.0148$ Explain why this result is statistically significant, but not practically important.

Music and mazes A researcher wishes to determine if people are able to complete a certain pencil and paper maze more quickly while listening to classical music. Suppose previous research has established that the mean time needed for people to complete a certain maze (without music) is 40 seconds. The researcher decides to test the hypotheses $H_0:\mathrm{μ}=40$ versus $H_a:\mathrm{μ}<40$ where $\mathrm{μ}=$ the time in seconds to complete the maze while listening to classical music. To do so, the researcher has $10,000$ people complete the maze with classical music playing. The mean time for these people is $\overline{x}=39.92$ seconds, and the P-value of his significance test is $0.0002$ Explain why this result is statistically significant, but not practically important.

Sampling shoppers A marketing consultant observes $50$ consecutive shoppers at a supermarket, recording how much each shopper spends in the store. Explain why it would not be wise to use these data to carry out a significant test about the mean amount spent by all shoppers at this supermarket.

Stating hypotheses State the appropriate null and alternative hypotheses in each of the following settings. Explain why the sample data give some evidence for $H_a$ in each case.

a. The average height of $18-$year-old American women is $64.2$ inches. You wonder whether the mean height of this year’s female graduates from a large local high school differs from the national average. You measure an SRS of $48$ female graduates and find that $\overline{x}=63.5$ inches and $s_x=3.7$ inches.

b. Rob once read that one-quarter of all people have played/danced in the rain at some point in their lives. His friend Justin thinks that the proportion is higher than $0.25$ for their high school. To settle their dispute, they ask a random sample of $80$ students in their school and find out that $28$ have played/danced in the rain.

Checking conditions Refer to Exercise R9.1. Identify the appropriate test to

perform in each setting and show that the conditions for carrying out the test are met.

Calculations and conclusions Refer to Exercise R9.1. Find the standardized test statistic and $P$-value in each setting, and make an appropriate conclusion.