STAT 2593 Final Examination: Quality Control in Manufacturing, Exams of Statistics

Download STAT 2593 Final Examination: Quality Control in Manufacturing and more Exams Statistics in PDF only on Docsity! Department of Mathematics & Statistics STAT 2593 Final Examination December 10, 1997 TIME: 3 hours. Total marks: 74. SHOW ALL WORK! 1. (a) During a 40 hour “run-in” period, samples consisting of 4 power supply units 16 marks were selected once each hour from an assembly line, and the high-voltage output from each unit was determined. The means and standard deviations of each of the 40 samples were determined and entered into columns of a Minitab work- sheet. Summary statistics for these columns were then calculated via the Minitab “DESCRIBE” command with the following results: MTB > desc c5 c6 Variable N Mean Median TrMean StDev SEMean xbar 40 2999.4 2999.4 2999.5 8.5 1.4 s 40 19.034 18.347 19.122 5.616 0.888 Variable Min Max Q1 Q3 xbar 2979.1 3018.1 2993.3 3004.1 s 6.754 30.424 15.859 24.597 Determine 3 SD control limits for (i) an X̄-chart and (ii) an S-chart for the process, assuming that the process was in statistical control during the the 40 hour run-in period. (b) Suppose that for a similar assembly line in a different company the control limits are UCL = 3296 volts and LCL = 3224 volts for the X̄-chart, and UCL = 50.15 volts and LCL = 0 volts for the S-chart. Three of the hourly samples that were selected from the assembly line were entered into columns of a Minitab worksheet and DESCribed, with the following results: MTB > desc c1-c3 Variable N Mean Median TrMean StDev SEMean sample.1 4 3197.1 3197.4 3197.1 21.9 11.0 sample.2 4 3263.0 3256.8 3263.0 23.5 11.8 sample.3 4 3277.2 3279.7 3277.2 80.3 40.1 Variable Min Max Q1 Q3 sample.1 3173.3 3220.2 3176.0 3217.8 sample.2 3242.6 3295.5 3244.3 3287.7 sample.3 3182.7 3367.0 3198.4 3353.7 . . . . . . (continued over page) For each of the three samples, determine if there is an indication that the system was out of statistical control with respect to either location or spread at the time the sample was collected, giving a brief reason for your answer. (c) Suppose that for the assembly line referred to in part (b) the process mean shifts from its “in control value” of 3260 to 3210. What is the probability that this change will be detected via the X̄-chart the next time that a sample is taken? (d) Suppose that the mean of the process changes in such a way that the probability that this change will be detected, via the X̄-chart, is 0.27 each time a sample is taken. (i) What is the probability that exactly 5 samples are taken before the change is detected? (I. e. the change is detected at the 5th sample after the change occurred.) (ii) How may samples do you expect to be taken until the change is detected? 2. In an experiment to compare two different methods (“MSI” and “SIB”) for determin- 10 marks ing residual chlorine content in sewage effluents, 8 specimens of water were measured using each of the methods. The measurements were recorded in a Minitab worksheet. Following there are two analyses of these data (parts of which have been obliterated). One of these analyses is correct; the other is incorrect. MTB > #---------------------------------------------------------- MTB > # Analysis number 1: MTB > twos c1 c2; SUBC> pool; SUBC> alte **. Twosample T for MSI vs SIB N Mean StDev SE Mean MSI 8 5.02 4.22 1.5 SIB 8 5.43 4.12 1.5 95% C.I. for mu MSI - mu SIB: ( ****, ****) T-Test mu MSI = mu SIB (vs *****): T= **** P=**** DF= ** Both use Pooled StDev = 4.17 MTB > #---------------------------------------------------------- . . . . . . (continued over page) 2 (a) Which is the correct analysis? Give a brief reason. (b) Using the correct analysis, (you will need to supply some of the obliterated items) find a 95% confidence interval for the population mean difference in frac- ture toughness between the two grades of steel. (c) Suppose it is claimed that the high-purity steel is 10 units tougher on average than the commercial-purity steel. On the basis of your confidence interval do you think this claim is true? 5. In a clinical trial on the effectiveness of two drugs (ticrynafen and hydrochlorithiazide) 9 marks in the treatment of high blood pressure each of the two drugs was given at either a high or a low dosage level to separate groups of subjects for 6 weeks. The response was recorded as drop (baseline minus final value) in systolic blood pressure (mm Hg) for each subject. The data were analyzed in Minitab, producing the following output, parts of which have been obliterated: MTB > retr ’f:drugs’ MTB > aovo c1-c4 Analysis of Variance Source DF SS MS F p Factor (i) (iv) 222.9 (vi) ***** Error (ii) 2119.6 (v) Total 27 2788.4 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----+---------+---------+---------+-- Tic-lo 8 13.762 8.221 (------*------) Tic-hi 7 20.629 6.480 (-------*------) Hyd-lo (iii) 5.620 7.909 (--------*-------) Hyd-hi 8 15.450 12.786 (-----*------) ----+---------+---------+---------+-- Pooled StDev = 9.398 0 10 20 30 (a) State (clearly) the hypotheses being tested. (b) Supply the missing numbers from the locations labelled (i), (ii), . . . (vi) in the above Minitab output. (c) State your decision about the test at the 0.05 and 0.01 significance levels. (d) Explain briefly in words what your decisions mean. 5 6. In a study on the the silver content of galena crystals grown in a closed hydrother- 10 marks mal system over a range of temperatures, data were collected on the crystallization temperature in ◦C and Ag2S content in mol % ×100. A regression model was fitted in Minitab with the following results: MTB > gstd MTB > plot c2 c1 Character Plot - Ag2S - - * * - 50+ - * - - * - * * * 25+ * ** * - * * * - * * * - - 0+ ** * - - --------+---------+---------+---------+---------+--------cr.temp 300 360 420 480 540 MTB > regr c2 1 c1; SUBC> pred 375; SUBC> pred 425. The regression equation is Ag2S = - 15.4 + 0.102 cr.temp Predictor Coef Stdev t-ratio p Constant -15.38 12.78 -1.20 0.244 cr.temp 0.10223 0.03167 3.23 0.005 s = 14.09 R-sq = 36.7% R-sq(adj) = 33.1% Analysis of Variance SOURCE DF SS MS F p Regression 1 2068.9 2068.9 10.42 0.005 Error 18 3573.9 198.5 Total 19 5642.8 Fit Stdev.Fit 95.0% C.I. 95.0% P.I. ***** 3.19 ( *****, *****) ( *****, *****) Fit Stdev.Fit 95.0% C.I. 95.0% P.I. 28.07 3.33 ( 21.07, 35.06) ( -2.36, 58.49) . . . . . . (continued over page) 6 6. (Continued.) (a) Is the intercept (constant term) of the true regression line equal to 0? Explain briefly. (b) Find a 95% confidence interval for the amount of increase in mean Ag2S content when the crystallization temperature is increased by 1◦C. (Hint: What quantity, given in the Minitab output, tells you the estimated increase in mean Ag2S content when the crystallization temperature is increased by 1◦C?) (c) Find a 95% prediction interval for an individual observation of Ag2S content when the crystallization temperature is 375 ◦C. (d) Suppose that a Professor of Metalurgy tells you that the mean Ag2S content, when the crystallization temperature is 425 ◦C, is 40 units. Would you believe this assertion? (e) Suppose that your lab partner tells you that at a crystallization temperature of 425 ◦C, she observed an Ag2S content of 40 units. Would you believe her? (f) Suppose the temperatures had been recorded in ◦F rather than ◦C. What would the fitted regression equation be? (Hint: To convert degrees Farenheit to degrees Celsius, use C = 5/9× (F − 32)). 7. The lifetime T of a mechanical component, in thousands of hours has cumulative 10 marks distribution function F (t) =  c× (1/10− 1/t) 10 ≤ t ≤ 100 0 t ≤ 10 1 t ≥ 100 (a) What value must c have? (b) Write down an expression for the probability density function of T . (c) Find the mean, µ, of T (i. e. find E(T )). (d) What is the probability that the component lasts more than 50 thousand hours? (e) What is the probability that the component lasts more than 50 thousand hours given that it has lasted more than 30 thousand hours? 7