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1 DEPARTMENT OF MATHEMATICS & STATISTICS STAT 2593 FINAL EXAMINATION DECEMBER 1994 TIME: 3 Hours MARKS Show all work. (4) (5) 1. When asked to make a decision, estimate, or inference from a set of data, a statistician proceeds in one of two ways. What are they? (Remember your answer to this question when working the rest of the exam!) 2. The resistance of a circuit is normally distributed with a mean of 200 ohms and a standard deviation of 10 ohms. What is the probability that the resistance of one of these circuits selected at random is between 190 and 220 ohms? (8) 3. The time to failure (in years) of a particular type of ball bearing follows a Weibull distribution with α = 1andβ = 4. By what time do we expect 80% of all such bearings to fail? (8) 4. Suppose that the probability that an experimental aircraft will succeed (i.e. be rated A+) in its first flight is What is the probability that, in 250 such aircraft, no more than 160 will succeed in their first flights? (8) 5. The service time for a person in a grocery line is distributed according to an exponential distribution with parameter λ = 5 customers per hour. If 3 customers are in front of you, what is the probability that you must wait more than 1 2 hour before it is your turn to be served? (Hint: Don t integrate!) (17) 6. The data in column C1 are measurements on load-carrying capacity (in pounds) of steel bolts used to attach street-light posts to the holding brackets at their base. (The data have been sorted.) MTB > print c1 capacity (Problem 6 continued on page 2)

2 MTB > describe c1 N MEAN MEDIAN TRMEAN STDEV SEMEAN capacity MIN MAX Q1 Q3 capacity (a) (b) (c) (d) Estimate the mean load-carrying capacity of these bolts. Estimate the standard deviation of load-carrying capacity of these bolts. Local safety regulations specify that the mean must be 500 pounds, and the standard deviation no more than 75 pounds. Do these bolts satisfy local safety regulations? (Hint: Look at your answers to (a) and (b).) The manufacturer wishes to export the bolts to a country where safety regulations specify that at least 90% of bolts have load-carrying capacity between 400 and 600 lbs. Are these bolts suitable for export? (12) 7. Two gunpowder manufacturers are competing for our business. We want assurance that their products are not markedly different in quality. One of the primary measures of quality is the muzzle velocity that a gunpowder produces. Testing produced the data presented below (speed measured in ft/sec.). MTB > describe c1 c2 N MEAN MEDIAN TRMEAN STDEV SEMEAN manuf_ manuf_ MIN MAX Q1 Q3 manuf_ manuf_ (a) (b) (c) Is there a difference in mean muzzle velocity for the two companies? Is there a difference in variability of muzzle velocity for the two companies? Which company would you choose, and why? (One or two sentences.) (5) 8. The purity of a certain chemical catalyst is very important when used in a gene-splicing procedure. Twenty samples of the catalyst from a synthetic production process and 20 samples from the traditional organic process are analyzed for impurities. The following data (total impurities in parts per million) have been obtained from this analyis. Is there a significant difference in the amount of impurities found in the two catalysts? (Explain which part of the MINITAB output you used to answer this question and why.) (Problem 8 continued on page 3) 2

3 MTB > twosample c1 c2 TWOSAMPLE T FOR synthetic VS organic N MEAN STDEV SE MEAN synthetic organic PCT CI FOR MU synthetic - MU organic: (-0.72, 2.52) MTB > let c3=c1-c2 MTB > tinterval c3 N MEAN STDEV SE MEAN 95.0 PERCENT C.I. difference ( 0.399, 1.401) (5) 9. There are two standard methods of measuring mercury levels in fish: selective reduction and the permanganate method. The mercury in each of 15 juvenile black marlin was measured by both techniques. The 15 measurements for each method (in ppm of mercury) are recorded in columns C1, C2 below. Do the two methods give comparable readings (on average)? (Explain which part of the MINITAB output you used to answer this question and why.) MTB > print c1 c2 Fish selective permanganate (Problem 9 continued on page 4) 3

4 MTB > twosample c1 c2 TWOSAMPLE T FOR selective VS permanganate N MEAN STDEV SE MEAN selective permanganate PCT CI FOR MU selective - MU permanganate: (-0.012, 0.136) MTB > let c3=c1-c2 MTB > tinterval c3 N MEAN STDEV SE MEAN 95.0 PERCENT C.I. difference ( , ) (6) 10. A quality control engineer inspects nickel plating on incoming shipments of armatures. Twelve such armatures are inspected. If the nickel thickness is less than 4.1 mm on two (or more) armatures, then the shipment will be rejected. Suppose that, for this shipment, nickel plating thickness actually follows a normal distribution with mean 4.2 mm and standard deviation.05 mm. What is the probability that the shipment will be rejected? (12) 11. A market research team conducted a survey to investigate the relationship of personality to attitude toward small cars. A sample of 299 adults in a metropolitan area were asked to fill out a 16-item self-perception questionnaire, on the basis of which they were classified into three types: cautious conservative, middle-of-the-roader, and confident explorer. They were then asked to give their overall opinion of small cars: favorable, neutral, or unfavorable. MTB > print c1-c3 ATTITUDE Cautious Midroad Explorer favourable neutral unfavourable (Problem 11 continued on page 5) 4

5 MTB > chisquare test of data in c1-c3 Expected counts are printed below observed counts Cautious Midroad Explorer Total favourable neutral unfavourable Total ChiSq = = (a) (b) (c) What is being tested, and what is the conclusion of the test? Estimate the proportion of all adults who look unfavourably on small cars. Does the estimate of part (b) change for the restricted population of cautious adults? (Hint: Use your answer to (a).) (10) 12. The data below were obtained from a study of the bismuth I-II transition pressure as a function of temperature. MTB > print c1 c2 ROW temp press (Problem 12 continued on page 6) 5

6 MTB > regress c2 on 1 predictor in c1; SUBC> predict 22. The regression equation is press = temp Predictor Coef Stdev t-ratio p Constant temp s = R-sq = 84.5% R-sq(adj) = 83.4% Analysis of Variance SOURCE DF SS MS F p Regression Error Total Fit Stdev.Fit 95% C.I. 95% P.I ( , ) ( , ) (a) Plot the data on the paper provided. (Be sure to write your name and student number on the graph paper!) (b) Sketch the regression line onto your graph of part (a). (c) Circle the point with largest absolute residual. (d) If temperature is increased by 1 C, by how much would you expect mean pressure to change? (e) Predict the value of a new pressure reading at 22 C. THE END!! **Remembertohandingraphpaper. ** 6

7 STUDENT S NAME: I.D #: STAT 2593 FINAL EXAMINATION DECEMBER 1994 QUESTION 12 7