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1. In Japan, a study established x  23:3 to x  24:7 cm as cuto s for accepting the average women’s shoe size of   24:0 cm. Assuming a standard deviation of 3.2 cm and a sample size of 75, (a) What is the probability of Type I error? (b) What is the probability of Type II error if  shifts to 23.0 cm? (c) What is the power of the test in part (b)? Page 2 2. (a) Given a hypothesis testing problem, how do we determine which statistical table to use? (b) Suppose through a shoe store’s electronic trackers the average number of customers passing through is 212 per hour and a random sample produced the following data: n is 30 one-hour intervals, x is 231 customers per hour, and s is 82 customers per hour. i. Test at a level of significance of .10 if the number of customers passing through the store is no more or no less than 212. ii. Suppose only ¯x is now 250 customers per hour. Test at a level of significance of .05 if the number of customers passing through the store is greater than or equal to 212 customers per hour. Page 3 3. (a) Given a hypothesis testing problem, when do we use the sample size to calculate the degrees of freedom? (b) Professor La claims he can browse a shoe store on average of 11.7 minutes and figure out the best deals in any given store. His students, in complete disbelief test his claim by randomly clocking him at 6 di erent stores yielding the following information: n is 6 stores, x is 13.57 minutes, and s is 3.2 minutes [assuming a normal population]. i. At a level of significance of .05, test the claim the average time it takes Professor La can perform his deal analysis is at most 11.7 minutes. ii. Suppose ¯x is now 15.1 minutes. Test at a level of significance of .05 browsing a shoe store takes no more or no less than 11.7 minutes. Page 4 4. (a) Describe in your own words, what it means to find a “confidence interval around ”? (b) Suppose we took a preliminary sample of 45 students at FIT who work, and found their sample average to be $32.40 per hour with a sample standard deviation of $8.62. i. Construct a 88% confidence interval for . ii. Including the preliminary sample, what total sample size is needed to be 98% confidant of , the true average hourly rate is within $0.50 of x, our sample average? i.e., How many students are necessary to find the total appropriate sample size? (c) Suppose we took another sample of only 17 di erent students and found their average hourly rate to be $24.50 and the sample standard deviation to be $4.80. Construct a 96% confidence interval for . Assume a normal population. Page 5 5. Students slept immediately after learning the material for the entire eight hours prior to an exam, then woke up and took the exam. The group scored as follows [0 - 33 scale]. Assume a normal population: 24 30 26 24 28 30 27 28 30 24 26 (a) Calculate the sample mean and standard deviation, calculator function is okay. (b) State the Central Limit Theorem in your own words. (c) At  :05, test  ¤ 24:2. (d) Construct a 95% confidence interval for . Page 6

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xbar lower 23.3
xbar upper 24.7
µ 24
n 75
σ 3.2
zlower -1.89
zupper 1.89
α/2 0.029
α 0.058
Probability of Type I error 0.058...

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