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Process Quality Engineering HOMEWORK PROBLEMS AND SOLUTIONS Readings: Basic Statistics: Tools for Continuous Improvement, Kiemele and Schmidt, Sections 6.1 - 6.5 Exercises: K&S Problems 6.1, 6.10, 6.14, Practice Exam Questions Contents Problem Page Exercise 6.1 2 Exercise 6.10 3 Exercise 6.14 5 Practice Exam Questions 9 HW5.DOC, May 31, 1995, page 1 Process Quality Engineering Exercise 6.1, page 6-63 The trailing edge of a turbine blade is to be robotically machined to a target of 6 mm. A simple random sample of 50 blades are inspected and found to have a mean of 5.98 mm with a standard deviation of .047 mm. Test at the .01 level whether the process is centered at the target value. SOLUTION One sample hypothesis test of the mean (). Step 1 State the hypotheses to be tested. H 0 : 6 mm H1 : 6 mm Step 2 Determine a planning value for . = .01 Step 3 Take a sample of size n 30 and compute y and s2. y 5.98 mm s 0.047 mm n 50 Compute the standardized test statistic: When n 30 the z dist. is a good y 0 5.98 6 0.02 0 z 3.01 approx. to the s 0.047 0.00665 t dist. n 50 Step 4 Using a standardized normal distribution, compute the area in the tail beyond z 0 . Pz z0 Pz 3.01 1 Pz 3.01 1 .9987 0.0013 Step 5 Compute p as 2 times the area found in Step 4. p = 2(.0013) = 0.0026 HW5.DOC, May 31, 1995, page 2 Process Quality Engineering Step 6 Given that p (= .0026) < (= 0.01), we reject H 0 0 with (1 - p)% = (1 - .0026)% = 99.74% confidence. We do not believe that the process is centered at the target value (6 mm). Exercise 6.10, page 6-65 A medical researcher is testing a new drug to see if it slows the growth a malignant tumors in rats. The control group of 40 rats is given none of the drug and the test group of 40 rats is given a dose every day for two weeks. At the end of the test, the following statistics were calculated. Control Rats Test Rats Average Growth of Tumor 6.2 grams 4.1 grams Standard Deviation 3.1 grams 2.8 grams Conduct the appropriate hypothesis test to determine if the new drug appears to slow the mean rate of tumor growth. SOLUTION Two Samples Hypothesis Test of the Mean Control Rats(y1 ) Test Rats(y2 ) Average Growth of Tumor 6.2 grams 4.1 grams Standard Deviation 3.1 grams 2.8 grams Step 1 State the hypothesis H 0 1 2 H1 1 2 Step 2 Select = 0.01 HW5.DOC, May 31, 1995, page 3 Process Quality Engineering Step 3 Obtain n1, n2 , y1, y2, s2 , s2 1 2 n1 n2 40 y1 6.2 y2 4.1 s1 3.1 s2 2.8 t0 y1 y2 sp n1 1s12 n 2 1s2 2 1 1 n1 n2 1 sp n1 n 2 393.1 392.8 2 2 sp 40 40 2 680.55 2.953811 78 y1 y 2 6.2 4.1 2.1 t0 3.18 1 1 2 0.66049 sp 2.953811 n 1 n2 40 Or, if n1 , n2 30 use y1 y 2 6.2 4.1 2.1 z0 3.18 s1 s2 2 3.12 2.82 0.66049 2 n1 n 2 40 40 Step 4 If n1 , n2 30 use z-distribution to estimate area beyond z 0 . Pz z 0 Pz 3.18 1 Pz 3.18 1 .9993 0.0007 Step 5 Find p (one-tailed test) p Pz z 0 0.0007 Step 6 Conclusion Since p .0007 0.01, we reject H 0 1 2 with 1 p% 1 .0007% 99.93% confidence. 1 2 or the new drug appears to slow the mean rate of tumor growth. HW5.DOC, May 31, 1995, page 4 Process Quality Engineering Exercise 6.14, page 6-67 A marketing company wants to determine if a new product will sell better on the east coast or west coast. Fifty individuals on each coast are randomly selected to test the new product. The following table shows the results of the market test: Estimated Market Share with New Product East coast 18% (i.e., 9 of 50 strongly in favor) West coast 26% (i.e., 13 of 50 strongly in favor a) The company will first market the product in the area which shows the most potential for high sales. Are the market shares of the two coasts significantly different? b) The company feels that the new market share must be at least 15% if the product is to be successful. Test each sample to see if they exceed this level. SOLUTION a) Two sample test of hypothesis for proportions n1 n 2 30 Step 1 State hypothesis H 0 : 1 2 H1 : 1 2 Step 2 Determine = .05 Step 3 x1 x2 Compute proportions p1 n , p2 n 1 2 9 p1 = 9 of 50 strongly in favor = = 0.18 (East coast) 50 13 p2 = 13 of 50 strongly in favor = 0.26 (West coast) 50 HW5.DOC, May 31, 1995, page 5 Process Quality Engineering p1 p 2 0.18 0.26 z0 x1 x 2 x1 x 2 1 1 9 13 9 13 2 1 1 n 1 n2 n 1 n2 n1 n 2 50 50 50 50 50 0.08 0.966 0.22.78.04 Step 4 Use z-distribution to determine area beyond z 0 . Pz z0 Pz 0.966 1 Pz 0.966 By interpolating: 0.96 .8315 x .8315 d 0.006 d 0.01 0.966 x 0.0025 d 0.006 0.97 .8340 0.0025 0.01 0.006 d 0.0025 0.0015 0.01 x 0.8315 0.0015 0.8330 Pz 0.966 Pz 0.966 1 0.8330 0.167 Step 5 Compute p (two-tailed test) p 2Pz 0.966 20.1670 0.3340 Step 6 Conclusion: Since p 0.3340 0.05, we fail to reject H 0 1 2 ,with 1 p% 1 0.3340% 66.6% confidence. There is no significant difference in market share between either coast. b) Perform one sample test of hypothesis for proportions on each sample from each coast. Step 1 State hypothesis: East West 1 0.15 2 0.15 1 0.15 2 0.15 HW5.DOC, May 31, 1995, page 6 Process Quality Engineering Step 2 Determine = 0.05 Step 3 Compute proportion p1 , p2 , then compute p i 0 z0 0 0 1 n East West 9 0.18 p1 50 p2 13 50 0.26 0.18 0.15 0.26 0.15 z0 z0 0.150.85 0.150.85 50 50 0.03 0.11 0.0505 0.0505 0.5940 2.18 Step 4 Use z-distribution to compute area beyond z0 (upper tailed test). East West Pz z 0 Pz 0.59 Pz z0 Pz 2.18 1 Pz 0.59 1 Pz 2.18 1 0.7224 1 0.9854 0.2776 0.0146 Step 5 Compute p (one tailed test) East West p 0.2776 p 0.0146 HW5.DOC, May 31, 1995, page 7 Process Quality Engineering Step 6 Conclusion: East West Since p (= 0.2776) > (= 0.05), we Since p (= 0.0146) < (=0.05), we failed to reject H0 ( 1 0.15) with reject H0 ( 2 0.15) with (1 - p)% = 72.24% confidence. (1 - p)% = 98.54% confidence. East coast sample does not West coast sample does exceed exceed 15% (market share). 15% (market share). Practice Exam Questions 1. You have been asked to determine whether your company’s “lightning”batteries have longer life than your competition’s “durable”batteries. Call the types “L” and “D”. a) Describe the null and alternative hypotheses that you would use to test the relation between L and D . b) Describe IN PLAIN ENGLISH what you would tell your boss if the hypothesis was rejected. c) Describe IN PLAIN ENGLISH what you would tell your boss if the hypothesis was accepted. 2. Suppose that you are testing the hypothesis H 0 : 1000 vs. H1 : 1000 and your set of 16 data values produces X = 989 and s = 12. a) What is the p-value for the test? b) What is your conclusion, statistically speaking? c) What is your conclusion, IN PLAIN ENGLISH? HW5.DOC, May 31, 1995, page 8 Process Quality Engineering 3. You are conducting a two-sample test of H0 : A B vs. H 1 : A B You have 11 observations of population A with sample mean 12.5 and s2 4 . A You have 21 observations of population B with sample mean 12.0 and s2 3 . B a) What is the test statistic that you would use? Compute it and the resulting p-value. b) What do you conclude? 4. You are investigating whether your new "Bountiful" paper towels are stronger with a new manufacturing process, compared with the existing process. You want to decide whether to shut down the factory and switch to the new processing machines. You have decided to use a hypothesis test for mean strength to help you decide. You feel comfortable with a = .05. You have collected test data and found the strength (in pounds) for 11 New towels to average 2 XNew = 13.9, with s New = 4 , and for 21 Existing towels you had XExi st ing = 12.0, and s2 st ing = 3. Exi a) Set up the appropriate hypotheses: H0 : vs H1 : b) Compute the appropriate test statistic (hint: sP = 10 30 ) c) Do you accept/reject H0 ? d) Explain the result IN PLAIN ENGLISH HW5.DOC, May 31, 1995, page 9