Page 1 of 8 STOR 155 Section 1 Final Examination Thursday April 30 2009 Name Solution Pledge I have neither given nor received aid on this examination Signature Instructions Do not do any actual numerical calculations Answers in a form that you would type into an Excel field such as 28 SQRT 82 2 with a working answer are expected 1 A company makes 20 of its cars at factory A and the rest at factory B Factory A produces 1 lemons and Factory B produces 2 lemons A car is chosen at random What is the probability that a It came from Factory B P B 1 P not B 1 P A 1 0 2 0 8 b It is a lemon if it came from Factory B P L B 0 02 c It is a lemon from Factory B P L B P L B P B 0 02 0 8 0 016 d It is a lemon P L P L A or L B P L A P L B P L A L B P L A P A P L B 0 0 01 0 2 0 02 0 8 0 002 0 016 0 018 e It came from Factory B if it is a lemon P B L P L B P L 0 02 0 8 0 01 0 2 0 02 0 8 0 016 0 018 5 5 5 5 5 2 A survey of 2000 student loan borrowers found that 200 had loans totaling more than a Give a 99 best guess Confidence Interval for the proportion of all loans totaling more 40 000 than 40 000 Page 2 of 8 X Binom 2000 p where p proportion 40k phat X n 200 2000 0 1 margin of error is NORMINV 0 995 0 SQRT 0 1 0 9 2000 Left CI 0 1 NORMINV 0 995 0 SQRT 0 1 0 9 2000 Right CI 0 1 NORMINV 0 995 0 SQRT 0 1 0 9 2000 or CONFIDENCE 0 01 SQRT 0 1 0 9 2000 b Give an Excel expression for the exact p value for concluding that the proportion of all loans more than 40 000 is at least 5 H0 p 0 05 H1 p 0 05 p val P X 200 p 0 05 1 P X 199 1 BINOMDIST 199 2000 0 05 true c Use the Normal approximation to give an alternate answer to b p val P X 200 p 0 05 1 P X 200 1 NORMDIST 200 2000 0 05 SQRT 2000 0 05 1 0 05 true 1 NORMDIST 200 100 SQRT 50 true d Why is the approximation used in c appropriate n p 2000 0 5 1000 10 n 1 p 2000 1 0 5 1000 10 e What is the 98 conservative margin of error in estimating the proportion of all loans over 40 000 NORMINV 0 99 0 SQRT 0 5 0 5 2000 CONFIDENCE 0 02 SQRT 0 5 0 5 2000 5 5 5 5 5 3 Scores on tests for a class were Page 3 of 8 a Assuming each column represents one student give a formula for the p value to show that scores on the 1st exam are significantly higher than those on the 2nd exam H0 1 2 H1 1 2 p val P X1bar X2bar calculated value 1 2 TTEST B1 E1 B2 E2 1 1 5 5 5 5 5 b Again assuming each column represents one student give an 80 Confidence Interval for the difference between the mean scores Dbar TINV 0 2 n 1 sd Dbar AVERAGE B3 E3 TINV 0 2 3 STDEV B3 E3 SQRT 4 c Assuming the exam scores come from two different classes give a formula for the p value to assess whether exam scores are significantly different between the two exams H0 1 2 H1 1 2 p val P X1bar X2bar calculated value 1 2 TTEST B1 E1 B2 E2 2 3 d Write the equation of the least squares regression line of the 2nd score as a function of the 1st score in terms of Excel commands Y SLOPE B2 E2 B1 E1 INTERCEPT B2 E2 B1 E1 e Write an Excel command to calculate the correlation between exam scores Will the answer be positive 0 or negative CORREL B2 E2 B1 E1 positive 4 For a random variable with distribution Find a P 1 Y 2 f 0 f 1 0 2 0 2 0 4 Page 4 of 8 0 1 y 3 f y 0 5 0 2 0 2 0 1 1 5 5 5 5 5 b P Y 1 Y 0 P Y 1 Y 0 P Y 0 P Y 1 P Y 0 f 1 f 1 f 3 0 2 0 2 0 1 2 3 c P Y 1 Y 0 0 can t happen at same time d The expected value of Y EX sum X prob of X 1 f 1 0 f 0 1 f 1 3 f 3 1 0 5 0 0 2 1 0 2 3 0 1 0 5 0 2 0 3 e The standard deviation of Y SD x SQRT sum X EX 2 prob of X SQRT 1 2 0 5 0 2 0 2 1 2 0 2 3 2 0 1 Page 5 of 8 5 A TV ad claims that at most 30 of people prefer Brand X Suppose that 6 out of 10 randomly selected people prefer Brand X a Give an exact p value to decide whether or not we should dispute the claim H0 p 0 3 H1 p 0 3 let X prefer Binom 10 p p val P X 6 p 0 3 1 P X 5 1 BINOMDIST 5 10 0 3 true b If the p value in part a turns out to be 0 03 give a yes no conclusion Have strong evidence claim is wrong c If the p value in part a turns out to be 0 03 give a gray level conclusion Moderately strong evidence Significant but not overwhelming d Give a 98 conservative confidence interval for the proportion of people that prefer Brand X 0 6 CONFIDENCE 0 02 SQRT 0 5 1 0 5 10 e How large a sample in the best guess sense is needed so that with probability 90 the estimated proportion of people that prefer Brand X is within 0 01 of the actual number NORMINV 0 95 0 1 m 2 p 1 p NORMINV 0 95 0 1 0 01 2 0 6 1 0 6 OR NORMINV 0 95 0 1 0 01 2 0 3 1 0 3 5 5 5 5 5 6 A set of 4 Normal Quantile plots in scrambled order watch the labels are Page 6 of 8 3 3 b a i Which most likely is from a data set of IQ scores with histogram ii Which most likely is from a data set of 21 car mileages of 2 seater cars iii …
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