Stat 401B Exam 2 Fall 2015

Transcription

1 Stat 401B Exam Fall 015 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed ATTENTION! Incorrect numerical answers unaccompanied by supporting reasoning will receive NO partial credit. Correct numerical answers to difficult questions unaccompanied by supporting reasoning may not receive full credit. SHOW YOUR WORK/EXPLAIN YOURSELF! Completely absurd answers (that fail basic sanity checks but that you don't identify as clearly incorrect) may receive negative credit. 1

2 1. Below are some data (and corresponding summary statistics) taken from a paper by Leigh and Taylor that appeared in the Ceramic Bulletin in They concern measured densities (in g/cc) of crushed T-61 tabular alumina powder under r 4 different measurement protocols. Protocol 1 Protocol Protocol 3 Protocol 4.13,.15,.15, 1.96,.01,1.91,.3,.19,.18, 1.88,1.90,1.87,.19, ,.00.1,. 1.89,1.89 n1 5 n 5 n3 5 n4 5 y1.164 y y3.06 y s1.030 s.040 s3.01 s4.011 Initially, consider only data from Protocol 1. 6 pts a) Give two-sided limits that you are 95% sure would contain the next measured density produced under Protocol 1. (Plug in completely, but you need not simplify.) b) A lab manager wishes to announce with 95% confidence that the "measurement capability" (defined as ) for Protocol 1 is no worse than some number (say, #). Provide an appropriate number, #, for this person based on the data above. Now consider data from Protocols 1 and 4 only. c) Do the two samples provide definitive indication that the two measurement protocols have different precisions (different associated variabilities)? Compute some appropriate statistic and use an appropriate reference distribution. (Say exactly what reference distribution you are considering and support a "Yes" or a "No" answer.)

3 6 pts d) Give 95% two-sided confidence limits for the difference in mean densities produced by Protocols 1 and 4. (Plug in completely, but you need not simplify.) Now consider data from all 4 protocols. 7 pts e) Find a single-number estimate of the standard deviation of measured density for any fixed protocol under the one-way normal model. 4 pts f) Below is a normal plot of 0 values yij yi. Say what it indicates about the reliability of inferences based on the one-way normal model in this context. (The line on the plot has intercept 0 and slope 1/ s.) P 3

4 8 pts g) As it turns out, the grand sample variance of the 0 measured densities recorded on page is Use this fact and your answer to part e) above to complete the ANOVA table below. (If you were unable to do part e), you may use the incorrect value of.00 here.) SOURCE SS df MS F 6 pts h) Protocols 1 and 3 were in fact carried out using 6-mesh material while Protocols and 4 were carried out using 60-mesh material. Compare the average of 6-mesh mean densities to the average of 60-mesh mean densities using two-sided 95% confidence limits and your value of s P from the ANOVA table in part g). (Plug in completely, but you need not simplify.) 6 pts i) Suppose that at some later time, 30 of 50 measurements made using Protocol produce values less than.00 g/cc. Give a lower 95% confidence bound for the fraction of all Protocol measurements less than.00 g/cc. (Plug in completely, but you need not simplify.) 4

5 . A data set in Probability and Statistics With R for Engineers & Scientists by M. Akritas concerns heat produced during hardening of cement as related to the composition of the cement. Available were y measured heat produced (calories/gm) x % tricalcium aluminate x x 1 3 % tricalcium silicate % tetracalcium alumino ferrite x4 % dicalcium silicate values for n 13 cement batches. There is some R code and output based on these data at the end of this exam. Use it as appropriate in the rest of the exam. a) On what basis would you suggest that x 4 is the best single predictor of y (from among the predictors available here)? (What about the printout suggests this?) Consider first a simple linear regression of y on x until further notice. b) Give 95% two-sided confidence limits for the standard deviation of measured heat produced at a fixed tricalcium silicate percentage. (Plug in completely, but there is no need to simplify.) c) Give 95% two-sided confidence limits for the rate of change of mean heat produced with respect to % tricalcium silicate (in the units of the data). (Plug in completely, but there is no need to simplify.) d) For what percentage of tricalcium silicate do these data provide the best information about mean heat produced? Explain. 5

6 Now consider the other predictor variables (not just x ). e) In a model that includes only predictors x1 and x give 95% two-sided limits for the rate of change of mean heat measurement (in cal/g) with respect to % tricalcium silicate. 7 pts f) Under the model that includes only predictors x1 and x give limits that you are 95% sure will contain a next heat measurement under the conditions that x1 7and x 6. g) What fraction of the raw variability in heat produced is accounted for by fitting an equation involving all of x1, x, and x 3? h) Give and interpret the p -value for testing the hypothesis that together the three predictor variables x, x, and x fail to be useful in modeling heat produced. 1 3 i) In the presence of x1 and x, does x 3 add (statistically) significantly to one's ability to model heat produced? Give a p -value and say what hypothesis is being tested in what model. 6

7 R Code and OutPut > CementVS y x1 x x3 x > cor(cementvs) y x1 x x3 x4 y x x x x > plot(cementvs) 7

8 > cement.out1<-lm(y~x,data = CementVS) > summary(cement.out1) Call: lm(formula = y ~ x, data = CementVS) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-05 *** x *** Residual standard error: on 11 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 1.96 on 1 and 11 DF, p-value: > anova(cement.out1) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x *** Residuals > predict(cement.out1,se.fit=true,interval="confidence",level=.95) $fit fit lwr upr $se.fit [1] [9] $df [1] 11 $residual.scale [1] > 8

9 > cement.out<-lm(y~x1+x,data = CementVS) > summary(cement.out) Call: lm(formula = y ~ x1 + x, data = CementVS) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-10 *** x e-07 *** x e-08 *** Residual standard error:.406 on 10 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 9.5 on and 10 DF, p-value: 4.407e-09 > anova(cement.out) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x e-08 *** x e-08 *** Residuals > predict(cement.out,se.fit=true,interval="confidence",level=.95) $fit fit lwr upr $se.fit [1] [8] $df [1] 10 $residual.scale [1] > 9

10 > cement.out3<-lm(y~x1+x+x3,data = CementVS) > summary(cement.out3) Call: lm(formula = y ~ x1 + x + x3, data = CementVS) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-07 *** x e-05 *** x e-07 *** x Residual standard error:.31 on 9 degrees of freedom Multiple R-squared: 0.983, Adjusted R-squared: F-statistic: on 3 and 9 DF, p-value: 3.367e-08 > anova(cement.out3) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x e-08 *** x e-07 *** x Residuals