Stat 401B Exam 2 Fall 2017

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1 Stat 0B Exam Fall 07 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.

This exam uses parts of an air quality dataset of De Vito archived on the UC Irvine Machine Learning Repository ( http://archive.ics.uci.edu/ml/datasets/air+quality# ).

2 This exam uses parts of an air quality dataset of De Vito archived on the UC Irvine Machine Learning Repository ( ). Gas concentration measurements from both a metal oxide multi-sensor device and a (co-located) reference certified analyzer, together with temperature and humidity readings were recorded hourly on a roadside in Italy for a year. (We use cases from the dataset for AM readings that are not missing CO values.). Below are some summary statistics for CO measurements made by the certified analyzer when temperatures were in various ranges. C. to 0 C 0. to C. to 0 C 0. to C. C n = 0 n = 7 n = 77 n = n = 8 n = y =. y =. y =. y =.7 y =.9 y =.7 s =. s =. s =.77 s =.78 s =.7 s =.9 7 pts a) Give two-sided limits you are "9% sure" cover the difference in mean CO measurements when temperatures are C and when they are. C. (Use only data from those two temperature conditions in making your answer.) pts b) (Roughly speaking) what magnitude of (two-sided) p-value is indicated by your limits in a) for a test of equality of mean CO measurements when temperatures are. C? Explain. C and when they are 7 pts c) If one judges the CO measurements for days when the AM temperature is C to be normally distributed, give an upper limit that you are "9% sure" is bigger than at least 99% of the AM CO measurements for such days. (Plug in completely, but there is no need to simplify.)

$7 pts d) 7 of the n = 0 low-temperature AM CO measurements were below.0. Give two-sided 9% confidence limits for the long run fraction of such CO measurements that will be below.0. Below is a (-way) ANOVA table for the n = CO measurements from AM made by the certified analyzer, broken into the "temperature classes" as indicated on the previous page.$

3 7 pts d) 7 of the n = 0 low-temperature AM CO measurements were below.0. Give two-sided 9% confidence limits for the long run fraction of such CO measurements that will be below.0. Below is a (-way) ANOVA table for the n = CO measurements from AM made by the certified analyzer, broken into the "temperature classes" as indicated on the previous page. Source SS df MS F Temp Class Error.7 Total 0. 8 pts pts e) Finish filling in the table above. f) Find an approximate p -value for assessing whether there is clear evidence of differences in mean CO measurement across the different temperature classes. Carefully indicate how you got this. pts g) Find a standard error for the quantity ( y+ y + y) ( y + y + y) (that could be used to make confidence limits for ( μ+ μ + μ) ( μ + μ + μ) if so desired). (There is no need to simplify.). The balance of this exam concerns the relationship between AM CO measurements made using the (cheap) metal oxide multi-sensor device and other variables in the study, including AM CO measurements made by the (expensive) certified analyzer. Use the R output at the end of this exam in answering the following questions. Don't do hand calculations you don't need to do!

4 To begin, let y = CO measurement made by the (cheap) multi-sensor device x = CO measurement made by the (expensive) certified analyzer and consider SLR of y on x and inference under the normal SLR model. (By the way, the units of the two measurements are not the same.) pts a) What fraction of the raw variability in multi-sensor CO measurement can be accounted for using the certified analyzer measurement as a predictor? pts b) Give an estimate of the standard deviation of multi-sensor measurement for any fixed certified analyzer measurement. pts c) Give two-sided 9% confidence limits for the increase in mean multi-sensor measurement that accompanies a unit increase in analyzer measurement. (There is no need to simplify.) pts d) Give two-sided 9% confidence limits for the mean multi-sensor measurement for an analyzer measurement of.0. (Plug in completely, but there is no need to simplify.)

pts e) (Calibration) As it turns out, if y new is a new observation coming from an unknown value of x, the values of x with 90% prediction intervals covering the new observation is a 90% confidence

What are 90% confidence limits for the analyzer measurement corresponding to a new multisensor measurement of 00? (Mark the plot indicating your interval.

a function of temperature. What complications of interpretation does this likely introduce into the analysis of these data using the MLR model y = β + β x + β x + β x + β x + ε?

5 pts e) (Calibration) As it turns out, if y new is a new observation coming from an unknown value of x, the values of x with 90% prediction intervals covering the new observation is a 90% confidence interval for the x value that generated y new. Below is a plot of 90% prediction limits for y based on the dataset. What are 90% confidence limits for the analyzer measurement corresponding to a new multisensor measurement of 00? (Mark the plot indicating your interval.) pts Now consider also the additional environmental variables observed in the study x = temperature x = relative humdity x = absolute humidity f) For a fixed absolute humidity, relative humidity is a function of temperature. What complications of interpretation does this likely introduce into the analysis of these data using the MLR model y = β + β x + β x + β x + β x + ε? 0 pts g) What about the set of scatterplots on the printout suggests that at most one of the variables above will be useful in addition to x as a predictor of y? What about the MLR output for the model mentioned in f) above indicates that x = relative humdity is probably the best choice among x, x, and x for use as a predictor in addition to x? Scatterplots: MLR Output:

Henceforth consider analysis based on the MLR model y = β + β x + β x + ε 0 pts h) Interpret the coefficient β in the -predictor model in

pts i) Give an estimate of the standard deviation of multi-sensor measurement for any fixed combination of certified analyzer measurement

pts j) After accounting for the certified analyzer measurement, does relative humidity add (statistically) significantly to one's ability

6 Henceforth consider analysis based on the MLR model y = β + β x + β x + ε 0 pts h) Interpret the coefficient β in the -predictor model in the context of this study. pts i) Give an estimate of the standard deviation of multi-sensor measurement for any fixed combination of certified analyzer measurement and relative humidity. How does it compare to your answer in b)? Why is this relationship reasonable? pts j) After accounting for the certified analyzer measurement, does relative humidity add (statistically) significantly to one's ability to explain or model the multi-sensor measurement? Explain using an appropriate p -value directly from the R output. pts k) Give 9% two-sided prediction limits for an additional multi-sensor reading for the conditions x =.7 and x =.. (Plug in completely, but you need not simplify.)

7 R Code and Output #The head() function prints just the first few lines of what would otherwise be #a long list or table of values head(airquality) Date TempGroup CO.Analyzer. CO.Multi. Temp RelH AbsH summary(airquality) Date Min. :.00 TempGroup CO.Analyzer. :0 Min. :0.00 CO.Multi. Min. : 8. Temp Min. : 0.0 st Qu.: 79. :7 st Qu.:.00 st Qu.: 9. st Qu.:0. Median :7.0 Mean :7.0 :77 : Median :.00 Mean :.77 Median :08.9 Mean :07. Median :. Mean :. rd Qu.:.7 :8 rd Qu.:.00 rd Qu.:.9 rd Qu.:.0 Max. :.00 : Max. :.900 Max. :7. Max. :9.80 RelH AbsH Min. :.00 Min. :0.90 st Qu.:.9 Median :.0 st Qu.:0.78 Median :.00 Mean :.9 Mean :.0 rd Qu.:.0 Max. :8.0 rd Qu.:.97 Max. :.9 var(airquality[,]) [] 0.77 var(airquality[,]) [] 0.8 slr.out<-lm(co.multi.~co.analyzer.,data=airquality) summary(slr.out) Call: lm(formula = CO.Multi. ~ CO.Analyzer., data = Airquality) Residuals: Min Q Median Q Max Coefficients: Estimate Std. Error t value Pr(t) (Intercept) <e- *** CO.Analyzer <e- *** Signif. codes: 0 *** 0.00 ** 0.0 * Residual standard error: 99. on degrees of freedom Multiple R-squared: 0.0, Adjusted R-squared: 0.09 F-statistic: 89. on and DF, p-value: <.e- aov(slr.out) Call: aov(formula = slr.out) Terms: CO.Analyzer. Residuals Sum of Squares Deg. of Freedom Residual standard error: Estimated effects may be unbalanced 7

slrpred<-predict.lm(slr.out,se.fit=true,interval="prediction",level=.90) Warning message: In predict.lm(slr.out, se.fit = TRUE, interval = "prediction", level = 0.

8 slrpred<-predict.lm(slr.out,se.fit=true,interval="prediction",level=.90) Warning message: In predict.lm(slr.out, se.fit = TRUE, interval = "prediction", level = 0.9) : predictions on current data refer to _future_ responses plot(airquality$co.analyzer.,airquality$co.multi.,xlab="analyzer",ylab="multisensor") abline(slr.out,col="red") points(airquality$co.analyzer.,slrpred$fit[,],type="l",col="blue") points(airquality$co.analyzer.,slrpred$fit[,],type="l",col="blue") grid(lwd=,col="black") #This plot is in the body of the exam pairs(airquality) summary(lm(co.multi.~co.analyzer.+temp+relh+absh,data=airquality)) Call: lm(formula = CO.Multi. ~ CO.Analyzer. + Temp + RelH + AbsH, data = Airquality) Residuals: Min Q Median Q Max Coefficients: Estimate Std. Error t value Pr(t) (Intercept) < e- *** CO.Analyzer < e- *** Temp RelH ** AbsH Signif. codes: 0 *** 0.00 ** 0.0 * Residual standard error: 9. on 09 degrees of freedom Multiple R-squared: 0.7, Adjusted R-squared: 0.9 F-statistic: 8. on and 09 DF, p-value: <.e- 8

9 mlr.out<-lm(co.multi.~co.analyzer.+relh,data=airquality) summary(mlr.out) Call: lm(formula = CO.Multi. ~ CO.Analyzer. + RelH, data = Airquality) Residuals: Min Q Median Q Max Coefficients: Estimate Std. Error t value Pr(t) (Intercept) < e- *** CO.Analyzer < e- *** RelH e-08 *** --- Signif. codes: 0 *** 0.00 ** 0.0 * Residual standard error: 9. on degrees of freedom Multiple R-squared: 0.7, Adjusted R-squared: 0. F-statistic: 8.7 on and DF, p-value: <.e- head(airquality) Date TempGroup CO.Analyzer. CO.Multi. Temp RelH AbsH head(predict.lm(mlr.out,se.fit=true,interval="confidence",level=.9)$fit) fit lwr upr head(predict.lm(mlr.out,se.fit=true,interval="confidence",level=.9)$se.fit) []

Stat 401B Exam 2 Fall 2015

Stat 401B Exam 2 Fall 2015 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