Department of Mathematics & Statistics STAT 2593 Final Examination 17 April, 2000

Transcription

1 Department of Mathematics & Statistics STAT 2593 Final Examination 17 April, 2000 TIME: 3 hours. Total marks: 80. (Marks are indicated in margin.) Remember that estimate means to give an interval estimate. Use the 95% confidence level unless otherwise specified. SHOW ALL WORK! 1. Suppose that 80% of trees near a certain busy highway have a lead content in their 6 marks bark of at least 920 micro grams per gram dry weight. Suppose that a random sample of 10 trees is taken. What is the probability that at most 7 of these trees have a bark lead content of 920 or more? What is the probability that exactly 7 of these 10 trees have a bark lead content of 920 or more? Now suppose that a random sample of 100 trees is taken. What is the probability that at most 70 of these trees have a bark lead content of 920 or more? (Hint: If you use appropriate tables you will have to do very little calculation in any of parts,, or.) 2. Let X be a discrete random variable taking only integer (whole number) values. 10 marks If P (X 6) = 0.4, what is the value of P (X 5)? Let X be a continuous random variable. If P (X 6) = 0.45 what is the value of P (X 6)? Let X be a discrete random variable taking only integer (whole number) values. If P (X 6) = 0.4, and P (X <5) = 0.3, what is the value of P (X =5)? (d) Let X be a continuous random variable. If P (X >6) = 0.25 what is the value of P (X <6)? (e) Let X be a discrete random variable. If P (X 6) = 0.4 andp (X 6) = 0.8 what is the value of P (X =6)?

2 3. In a study on the feeding habits of bats, 22 bats (11 male and 11 female) were tagged 6 marks and tracked by radio. The researchers wished to determine if male and female bats fly a different distance between feedings on average. The distances flown (in meters) between feedings were observed for each of the bats. The resulting data were entered into a Minitab worksheet; it may be assumed that all relevant data have a normal distribution. Two analyses of the data are displayed below, one of which is correct and one of which is incorrect: MTB > # Analysis number 1: MTB > twos c1 c2 Two sample T for Female vs Male N Mean StDev SE Mean Female Male % CI for mu Female mu Male: ( 37, 105) TTest mu Female = mu Male (vs not =): T = 0.99 P = 0.33 DF = 19 MTB > # Analysis number 2: MTB > let c3 = c1c2 MTB > name c3 F M MTB > ttest F M Test of mu = 0.0 vs mu not = 0.0 Variable N Mean StDev SE Mean T P F M MTB > tint F M Variable N Mean StDev SE Mean 95.0 % CI F M ( 49.5, 117.1) Which analysis is correct? Give a brief reason for your answer. Make, at the 0.05 significance level, the appropriate decision about the hypothesis being tested. (Make use of any relevant information appearing in the correct Minitab analysis.) Express in plain words what your decision about the hypothesis test actually means. 2

3 4. In a famous experiment to assess the efficacy of aspirin in preventing heart attacks, 8 marks 22,000 healthy middleaged men were divided into two equal groups, one of which was given a daily dose of aspirin and one of which was given a placebo that looked and tasted identical to the aspirin. At the time that the experiment was halted, 104 members of the aspirin group and 189 members of the placebo group had had heart attacks. (d) State clearly, in symbols, appropriate null and alternative hypotheses about the efficacy of aspirin in preventing heart attacks. State clearly the rejection region for conducting the test at the 0.01 significance level. Calculate the value of the test statistic. State your decision about the test (at the 0.01 significance level) and express in plain words what your decision actually means. 5. In order to establish control charts on the shear strength of spot welds, 30 successive 8 marks samples of size 4 were obtained, and sample means x i and sample standard deviations s i were obtained. Suppose that 30 i=1 x i =12, 660 and 30 i=1 s i = 537. Assume that the process was in statistical control while these 30 samples were collected. Determine the centre line and upper and lower 3sigma limits for an Xchart (for samples of size n =4). Determine the centre line and upper and lower 3sigma limits for an Schart (for samples of size n =4). For a similar welding process, at a different company, the Xchart has been established with control limits UCL = 500 and LCL = 410. Suppose that this process goes out of control; the process mean shifts downward by 25 units. What is the probability that this shift will be detected from the next sample taken after the downward shift occurs? (d) In yet another process, at a third company, the process goes out of control in such a way that the probability of detecting that it is out of control, from any particular sample, is What is the probability that the problem will be detected from the first five samples (i.e. at or before the fifth sample) taken after the process goes out of control? How many samples do you expect to be taken before the problem is detected? 3

4 6. Ten pregnant women were given an injection of pitocin to induce labour. Their systolic 6 marks blood pressures, immediately before and after the injection, were Patient Before After These data were entered into a Minitab worksheet; it may be assumed that all relevant data have a normal distribution. Two analyses of the data are displayed below, one of which is correct and one of which is incorrect: MTB > # Analysis number 1: MTB > twos c1 c2 Two sample T for before vs after N Mean StDev SE Mean before after % CI for mu before mu after: ( 9.8, 3.6) TTest mu before = mu after (vs not =): T = 0.98 P = 0.34 DF = 17 MTB > # Analysis number 2: MTB > let c3 = c2 c1 MTB > name c3 aft bef MTB > tint aft bef Variable N Mean StDev SE Mean 95.0 % CI aft be ( 0.09, 6.11) Which analysis is correct? Give a brief reason for your answer. Estimate the difference between the before and after population means. (Make use of any relevant information appearing in the correct Minitab analysis.) On the basis of this estimate, does it appear that there really is a difference between these means? Explain briefly. 4

5 7. The decline of water supplies in many areas of the world has created the need for 12 marks an increased understanding of relationships between economic factors such as crop yield, and hydrologic and soil factors. A paper published in Water Resources Bulletin contained data on grain sorghum yield (in g/mrow) and distance upslope (in metres), on a sloping watershed. These data have been analyzed in Minitab; the following output (parts of which have been obliterated) was produced: MTB > plot c2 c * yield * 500+ * * * * * * * 400+ * * * 300+ * * upslope MTB > regr c2 1 c1; SUBC> pred 60; SUBC> pred 90. The regression equation is yield = upslope Predictor Coef StDev T P Constant upslope S = RSq = 61.6% RSq(adj) = 58.4% Analysis of Variance Source DF SS MS F P Regression Residual Error Total (continued over page) 5

6 Predicted Values Fit StDev Fit 95.0% CI 95.0% PI ( *****, *****) ( *****, *****) Fit StDev Fit 95.0% CI 95.0% PI ( 388.0, 452.1) ( 296.4, 543.7) On the basis of the foregoing Minitab output answer the following questions: Estimate the mean yield when the distance upslope is 0 metres. Test, at the 0.05 significance level, the hypothesis that mean yield decreases by 1 gm/mrow for each metre of distance upslope. Express your decision about the test clearly. Calculate the fitted value ŷ of mean yield when the distance upslope x is equal to 25 meters, and 150 meters, respectively. Plot the corresponding points on the Minitab scatter plot of the data on the previous page and hence draw the fitted line. (d) Calculate a 95% prediction interval for an observation of yield at a distance 60 metres upslope. (e) A hydrologist tells you that she once observed a yield of 550 gm/mrow at a distance of 60 metres upslope. Should you believe her? Explain briefly. (f) The Professor of Hydrology at Waterloo tells you that the mean yield at a distance of 90 metres upslope is 400 gm/mrow? Should you believe this assertion? Explain briefly. 8. In a study of the ability of a certain polymer to remove toxic wastes from water, 8 marks experiments were conducted at three different temperatures. Data on the percentage of impurities that were removed by the polymer at each of the three temperatures were analyzed in Minitab, with the following results: MTB > Oneway Percent Temp ; SUBC> Tukey 5. Analysis of Variance for Percent Source DF SS MS F P Temp Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +++ High (*) Low (*) Med (*) +++ Pooled StDev = (continued over page) 6

7 Tukey s pairwise comparisons Family error rate = Intervals for (column level mean) (row level mean) High Low Low Med Does the polymer perform equally well at all three temperatures? State the appropriate null and alternative hypotheses clearly. Complete the hypothesis test at the 0.05 level, using relevant information from the Minitab output. Express your decision about the test clearly. If the data indicate that there are differences amongst the three temperatures, use the foregoing Minitab output to determine where these differences lie. If the data indicate that there are no differences amongst the three temperatures, estimate the mean percentage of toxic waste removed by the polymer. (Treat the data as a single random sample.) 9. Counts of traffic accidents on highways in New Brunswick during 1997, classified by 10 marks the day of the week on which they occurred, were as follows: Day: Mon Tue Wed Thu Fri Sat Sun Total Count: (d) Does it appear that accidents are equally likely to happen on any day of the week? State clearly the null and alternative hypotheses to be tested. The test statistic for this hypothesis test is a sum of terms each of which corresponds to a day of the week. How much do Tuesdays contribute to the sum? The value of the test statistic is ; state the table value with which to compare this test statistic for the 0.01 significance level and express clearly your decision about the test at the 0.01 significance level. Counts of traffic accidents on highways in New Brunswick classified by the day of the week on which they occurred, are also available for 1998 and It is desired to know if the pattern of accident occurrence with respect to day of the week changes from year to year. To this end, the data were analyzed in Minitab with the following results:...(continued over page) 7

8 MTB > # Row numbers 1 to 7 correspond to days of the week MTB > # Monday to Sunday respectively. MTB > chis c1c3 Expected counts are printed below observed counts Total (i) Total ChiSq = (ii) (iii) = DF = (iv), PValue = ***** (e) Supply the values of items (i), (ii), (iii) and (iv) which have been obliterated from the foregoing output. Express clearly your decision about the hypothesis being tested in part (d), at the 0.05 level. 10. Suppose that the time in days required to perform a certain type of repair on a laptop 6 marks computer is a random variable Y having probability density function { a + by 2 0 y 1 f(y) = 0 otherwise Given that the mean time required to perform the repair is 0.6 days, determine the values of a and b. 8