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Unformatted text preview: AMS572.01 Midterm Exam Fall, 2010 Name ___________________________ID ________________Signature____________________ AMS Major? ______ Instruction: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please enter “Yes” or “No” for “AMS Major”. Please provide complete solutions for full credit. The exam goes 12:50-2:10pm. Good luck! 1. ( for all students ) In order to test the accuracy of speedometers purchased from a subcontractor, the purchasing department of an automaker orders a test of a sample of speedometers at a controlled speed of 55 mph. At this speed, it is estimated that the variance of the readings is 1. (a) Set up the hypotheses to detect if the speedometers have any bias. (b) How many speedometers need to be tested to have a 95% power to detect a bias of 0.5 mph or greater using a 0.01 level test? (c) A sample of the size determined in (b) has a mean of 55.2 and standard deviation of 0.8. Can you conclude that the speedometers have a bias? (d) Calculate the power of the test if 50 speedometers are tested and the actual bias is 0.5 mph. Assume a population standard deviation of 0.8. SOLUTION: This is basically #7.10 in your homework # 3 . (a) The appropriate hypotheses are: 55 : = μ H vs. 55 : ≠ μ a H (b) It is estimated that 1 = σ . To assure 95% power for detecting a bias of 0.5 mph or greater, use β =1-Power=0.05. Then, from equation (7.11), 27 . 71 ] 5 . 1 * ) 645 . 1 576 . 2 ( [ ] ) ( [ 2 2 2 = + ≈ + ≈ δ σ β α z z n Therefore, 72 speedometers should be tested. (c) The test statistic is 121 . 2 72 8 . . 55 2 . 55 =- =- = n s x t μ Since 576 . 2 | | 005 ,. 71 2 , 1 = =- t t t n α , our conclusion is to not reject H at level 01 . = α . There is not sufficient evidence that the speedometers have a bias. Note: since the sample size is large, and we did not mention explicating that the population is normal. Therefore, it is also OK to use the approximate Z-test here. (d) Since the population variance is given, so the suitable test here is the Z-test. If the bias is 0.5 mph, the power for this 2-sided test is ) ) 5 . 55 ( ( ) ) 5 . 55 ( ( 2 2 σ μ σ μ α α n z n z Power- +- Φ +- +- Φ = 967 . 967 . ) 843 . 1 ( ) 995 . 6 ( ) 8 . 50 ) . 55 5 . 55 ( 576 . 2 ( ) 8 . 50 ) 5 . 55 . 55 ( 576 . 2 ( = + = Φ +- Φ =- +- Φ +- +- Φ = 1 *** The two-sided test is more suitable here, however, the one-sided version given below will also be given full credit. (*It was really a careless mistake on my part as I was typing the solutions to your practice midterm 2010 in the middle of the night. Thus you will be given full credit here – but please learn from this mistake, and the next time around, for the same problem, we should all use the two-sided test. Live and learn – and never type the solutions in the middle of the night ☺test....
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This note was uploaded on 01/31/2011 for the course AMS 572 taught by Professor Weizhu during the Fall '10 term at SUNY Stony Brook.
- Fall '10