Stat 231 Final Exam. Consider first only the measurements made on housing number 1.

Transcription

1 December 16, 1997 Stat 231 Final Exam Professor Vardeman 1. The first page of printout attached to this exam summarizes some data (collected by a student group) on the diameters of holes bored in certain steel housing covers. A member of the group measured the diameters of <œ* parts 7œ3 times each. Data values were recorded in units of "! $ mm above the nominal (or ideal) diameter. Consider first only the measurements made on housing number 1. (a) Give a 90% two-sided confidence interval for the long run mean measured diameter for housing 1. (No need to simplify.) (b) Give a 90% two-sided prediction interval for the next measured diameter for housing number 1. (No need to simplify.) (c) Give a 90% upper confidence bound for the long run standard deviation of measured diameters for housing number 1. Now consider the measurements made on both housings 1 and 2. (d) Test the hypothesis that there is no difference between the long run mean measured diameters for housings 1 and 2 using! œþ!". (Show all 7 steps.) Finally, consider the measurements made on all 9 housings. -1-

2 (e) What model assumptions must be made in order to do inference for comparing these 9 mean long run measured diameters? (f) Under the assumptions of part (e), what :-value is associated with a test of H À. œ. œ â œ.? With! œ Þ!& should this hypothesis be rejected? Explain.! " # * :-value œ Reject? yes no (circle one) Explanation: (g) Based on the model assumptions of part (e) give a 90% two-sided confidence interval for the difference in long run mean measured diameters for housings 1 and 2. (No need to simplify.) (h) Suppose that the parts measured by these students were in fact a random sample from a large lot of such housings and there is interest in estimating how much the holes bored in these housings actually vary in diameter (where this is to be assessed without the corruption of measurement error). Give an ANOVA-based estimate of the long run standard deviation among housing mean measurements. 2. Stat 231 concerns both PROBABILITY and STATISTICS. Define and contrast these two subjects. 3. Pages 2 through 4 of the printout attached to this exam concern a study of the effectiveness of armor plating. Armor-piercing bullets were fired at an angle of 40 against armor plate of -2-

3 thickness B" (in Þ!!" inches) and Brinell hardness number B#, and the resulting "ballistic limit," C (in feet/second), was measured. (a) There are two simple linear regressions included on the printout. Which of these seems to be the most effective summarization of the data? Explain. Find the value of for this model. G : Best SLR Model: Explanation: G œ : (b) Using the second of the two simple linear regressions, give a 90% two-sided confidence interval for the increase in mean ballistic limit that accompanies a 1 unit increase in Brinell hardness number. (No need to simplify.) Henceforth use the MLR analysis beginning in the middle of page 3 of the printout. (c) Give and interpret a :-value for testing whether B" and B# together provide some important ability to explain or predict C. :-value œ Interpretation: (d) Give the value of and degrees of freedom for an F statistic used to test the hypothesis that after accounting for B, B provides no important additional ability to predict C. # ".0 œ, 0 œ (e) Tomorrow I plan to test a piece of armor plating Þ#&$ inch thick and with Brinell hardness number %!(. Give me an interval that is in some sense "90% sure" to contain tomorrow's ballistic limit. 4. In the manufacture of a certain glass product, defects of three different types (A, B and C) can occur on the product. These differ in both importance and frequency of occurrence. Suppose -3-

4 that on one item, \ A, \ B and \ C are the numbers of occurrences of these defects and that these can be modeled as independent Poisson random variables. (a) Suppose that œ E\ œ Þ#Þ Find the probability that \ 2. - A A A (b) I decide to assign "demerits" to an inspected item according to the expression Hœ\ + #\ %\. A B C Suppose that -A œ E\ A œ Þ#, -B œ E\ B œ Þ1 and that -C œ E\ C œ Þ05. Find the mean and variance of H. (c) For a different product (having a different product grading scheme) items have a mean number of demerits of Þ& with a corresponding standard deviation of demerits of "Þ!. Approximate the probability that a sample of "!! of these items has a total of at least 48 demerits. (Hint: >9>+6./7/<3>= %) is the same as B %Þ).) 5. End-of-the-line inspection is done on 100 items manufactured on a particular production line with the results below (in terms of actual Good/Defective character and inspection result). Inspection Result Pass Fail Actual Part Good 72 8 Condition Defective 4 16 Consider first the random selection of a single item from the lot. (a) What it the probability that the item is actually "Good" or has a "Pass" inspection result? (b) What is the conditional probability that it is "Good" given that it "Passes" final inspection? -4-

5 Consider now the random selection (without replacement) of two items from the lot. Define random variables \œthe number of "Good&Pass" items in the sample, and ]œthe number of "Defective&Fail" items in the sample. (c) Below is a partially complete table giving the joint probability function for \ and ]. Finish filling it in. Cœ# Bœ! Bœ" Bœ# C œ " Þ!$)) C œ! Þ!"$$ Þ"(%& Þ&"'% (d) ^œ\ ] is the number of items in the sample that are correctly classified by the inspection process. Using your answer to (c) find TÒ^ œ #Ó. 6. A commonly used model in reliability work is that with [œthe life of a device of interest, ]œln Ð[Ñhas a normal distribution with some mean. and some standard deviation 5. Suppose that the guarantee period for a device is.5 year and that its actual lifetime follows a model like this where. œ! and 5 œþ(. Find the probability that the device needs to be replaced during its warranty period. -5-

6 MTB > Oneway 'diameter' 'part'. One-Way Analysis of Variance Analysis Source part Error Total Level N Pooled stoev = of Variance on OF SS Mean diameter MS stoev F 6.73 p Individual 95% CIs For Mean Based on Pooled StDev (----*---) (---*----) (----*----) (---*----) (----*----) (---*----) (---*----) (----*---) (---*----)

7 MTB > print c1-c3 Data Display Row y xl x MTB > Regress 'y' 1 'xl'; SUBC> Constant. Regression Analysis The regression equation is y = xl Predictor Coef Stdev t-ratio p Constant xl s = R~q 11.5% R-sq(adj) = 6.6% Analysis of Variance SOURCE OF SS MS F P Regression Error Total

8 MTB > Regress 'y' 1 'x2'; SUBC> Constant. Regression Analysis The regression equation is y = x2 Predictor Coef Stdev t-ratio p Constant x s = R-sq 50.4% R-sq(adj) = 47.6% Analysis of variance SOURCE Regression Error Total OF SS MS F p Unusual Obs Observatir':1s x2 Y Fit Stdev.Fit Residual St.Resid 0.54 X -2.16R R denotes an obs. with a large st. resid. X denotes an obs. whose X value gives it large influence. MTB > MTB > SUBC> SUBC> SUBC> SUBC> SUBC> Name c4 = 'SRES1' c5 = 'FITS1' Regress 'y' 2 'xl' 'x2'; SResiduals 'SRES1'; Fits 'FITS1'; Constant; Pure; Predict 'xl' 'x2'. Regression Analysis The regression equation is y = xl x2 Predictor Coef Stdev t-ratio p Constant xl x s = R-sq 60.1% R-sq(adj) = 55.4% Analysis of Variance SOURCE Regression Error Total OF SS MS F p

9 SOURCE xl x2 OF 1 1 SEQ SS Fit Stdev.Fit 95.0% C.!. 95.0% P.! ( 985.7, ) ( 797.0, ) ( , ) ( 839.8, ) ( , ) ( , ) ( , ) ( 843.2, ) ( , ) ( 913.4, ) ( , ) ( 869.3, ) ( , ) ( 953.9, ) ( , ) ( , ) ( , ) ( 962.3, ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( 679.6, 989.8) ( 529.7, ) ( 849.1, ) ( 674.3, ) ( 893.1, ) ( 718.8, ) ( , ) ( 812.6, ) ( , ) ( 907.9, ) ( , ) ( 906.2, ) Cannot do pure error test -4-

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