Stat 231 Final Exam Fall 2011

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1 Stat 3 Final Exam Fall 0 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed

2 . An experiment was run to compare the fracture toughness of high purity 8% Ni-maraging steel to that of commercial purity steel of the same type. Summary statistics for the measured toughness of several specimens of each type are below. (Units are MPa m.) High Purity Commercial Purity n = 0 specimens tested n = 8 specimens tested x = 65.6 x = 59.8 s =.8 s =. 0 pts a) Give a one-sided lower 95% confidence bound for the ratio of standard deviations of toughness for specimens of this type, σ/ σ. (Plug in completely, but you need not simplify.) 5 pts b) Because it is expensive, an engineer wants to specify High Purity steel for use in a project only if its mean fracture toughness exceeds that of Commercial Purity steel by more than 4 MPa m. With α =.0 do the data summarized above force the conclusion that High Purity steel should be specified? (Show the whole 7-step hypothesis testing format/process.)

3 0 pts c) Give 95% two-sided confidence limits for the difference in mean fracture toughness for specimens of these two types of steel (High Purity minus Commercial Purity). (Plug in completely, but you need not simplify.) 0 pts d) Assuming that fracture toughness measurements of specimens of Commercial Purity steel are normally distributed, give two-sided limits that you are 95% sure contain 99% of such measurements. (Plug in completely, but you need not simplify.) 3

4 . Company officials at a meat processor are concerned with the fraction of packages of sausage filled by a particular machine that contain more than 6. oz of sausage. They plan to weigh the net contents of a sample of packages filled by the machine. 0 pts a) If 5 packages are to be weighed and roughly 30% of all packages actually contain more than 6. oz of sausage, find the probability that 8 or more in the sample hold more than 6. oz of sausage. 7 pts b) Find a sample size that would guarantee that a 95% confidence interval for the fraction of all packages filled to more than 6. oz will have width no more than.0. (The interval will be of form ˆp ±Δ for Δ no larger than.0.) 8 pts c) Company officials eventually settle on a sample size of 400 and find 50 (of 400) packages inspected contain more than 6. oz of sausage. Give 95% two-sided confidence limits for the fraction of all packages filled by this machine that contain more than 6. oz of meat (assuming that it is behaving in a physically stable manner). 4

5 3. One measure of air pollution is the amount of beta radioactivity in the air (e.g. measured in 3 μ Ci / m ). Below are summary statistics from 5 pollution measurement stations (all located within 0 feet of each other at a given site) at 4 different Ames locations on a single summer day. Location Location Location 3 Location 4 n = n = 5 n 3 = 5 n 4 = 5 5 x = 3. x =.9 x 3 = 3.4 x 4 = 3.0 s =. s =.3 s 3 =. s 4 =. 3 pts a) Give -sided 95% confidence limits for the standard deviation of pollution measurements at a fixed location in Ames, based on the values above (and the assumption that measurements at a given location are normal with a standard deviation that is common across locations). Use information from ALL 4 locations in the calculation. pts b) Locations and are at schools in residential areas, and locations 3 and 4 are in commercial districts. What is a "margin of error" for ( x3+ x4) ( x+ x) based on 95% confidence limits for the quantity ( μ3 + μ4) ( μ+ μ) (that might be taken as a measure of increase in pollution in commercial areas over that in residential ones)? 5

6 4. A QC inspector periodically inspects 9 in 9 in floor tiles produced by a particular production line in the person's facility, looking for pock marks above a critical size. The inspector has come to the conclusion that the number of such blemishes on a single tile can be modeled as Poisson with mean λ =.5 blemishes. 0 pts a) Evaluate the probability that a given tile has at least one pock mark that is above the critical size. 0 pts b) Approximate the probability that 00 tiles have from 45 to 55 pock marks total using the central limit theorem. (This is the probability of an average of.45 to.55 pock marks per tile.) 6

7 5. Attached at the end of this exam is some JMP output that concerns the analysis of a classical data set (of N.H. Prater) concerned with the yield of gasoline in the refining of crude oil. The response variable is y = yield expressed as a percent of crude oil converted to gasoline after distillation and fractionation and predictors include the process variable x = the "endpoint" or temperature ( F) at which all the gasoline is vaporized and crude oil properties x = 0% point ASTM (the temperature at which 0% of the crude oil has become vapor) x3 = the crude oil gravity (degrees API), and x = the crude oil vapor pressure (lbf/in ) 4 First consider a simple linear regression analysis of y based on x alone. 5 pts a) What fraction of the raw variability in y is accounted for by a linear equation in x? (Report a number.) 5 pts b) What are 95% confidence limits for the increase in mean yield that accompanies a Fincrease in "endpoint" in this model? (Report numerical limits.) 7 pts c) What are 95% prediction limits for the next yield of this process run at an endpoint of 300 F? (Plug in completely, but you need not simplify.) Now consider analyses of y as potentially depending upon x, x, x3, and x 4 represented in the JMP reports. 7

8 0 pts d) After accounting for endpoint ( x ), do the crude oil properties ( x, x 3, and x 4 ) add detectable ability to explain/model/predict the yield ( y )? Provide an observed value of an F test statistic, degrees of freedom, and an answer for α =.05. F = df.. =, Yes/No (circle the correct one) 5 pts e) If you had to choose a model for y based on some or all of the predictors using only the information provided on the JMP reports, which predictor(s) would you employ? WHY? (Explain your choice!) predictor(s) you would employ: Explanation: 5 pts f) In a model including all predictors, give 95% two-sided confidence limits for the increase in mean yield that accompanies a Fincrease in "endpoint" with properties of the crude oil held fixed. (Plug in completely, but you need not simplify.) 8 pts g) In a model including all predictors, a standard error of prediction for the conditions of the first set of predictors in the data is.93. Give 95% confidence limits for the mean yield under that set of conditions. (Plug in completely, but you need not simplify.) 8

9 6. Bert and Ernie are both applying for summer work at the Large Corporation. There are 3 positions open and a total of 0 applicants including Bert and Ernie. Bert is not eligible for job A as his uncle is the supervisor for that position. Otherwise, all applicants are eligible for all 3 jobs. 7 pts a) In how many different ways can the 3 jobs A,B, and C be filled by the 0 applicants? 7 pts b) Suppose that all applicants are equally qualified, so that hiring is done essentially at random (except for the restriction that Bert cannot have job A). What is the probability that neither Bert nor Ernie is hired? 6 pts 7. Events A, B are independent. Suppose P( A) =.3 and P( B) =.4. What is ( not ( or )) c set notation this is P ( A B).) ( ) P A B. (In 9

10 8. A manufacturing process produces axels whose diameters may be modeled as continuous random variables. The diameter (in inches) of a single axel, X, may be described by a distribution with probability density (pictured below) for.99 < x <.0 f ( x) =.0 0 otherwise X Axels are (exactly) 0 inches long, so the volume of an axel is V = 0 π = 5πX. 0 pts a) Find EV, the expected volume of the axel. 0 pts b) Find VarV, the variance of the volume of the axel. You must set up completely a correct expression in terms of definite integrals, but you need not evaluate. (Hint: VarV EV ( EV) =.) 0

11 JMP Reports for Problem 5

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Stat 231 Final Exam Fall 2013 Slightly Edited Version

Stat 231 Final Exam Fall 2013 Slightly Edited Version Stat 31 Final Exam Fall 013 Slightly Edited Version I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed 1 1. An IE 361 project group studied the operation