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Midterm I Spring 2020 Solutions A Barbara Kuzmak 3/3/2020 Questions 1 – 3 Donut Problem #Donuts A donuts <- read.csv("~/Documents/STAT 5303 Spring 2020/Data/donuts.csv") attach(donuts) model.1<-with(donuts,aov(amt.sugar~topping.type*sugar.type)) summary(model.1) ## Df Sum Sq Mean Sq F value Pr(>F) ## topping.type 4 4890 1222.6 795.267 < 2e-16 *** ## sugar.type 1 13 12.8 8.318 0.00578 ** ## topping.type:sugar.type 4 19 4.8 3.151 0.02188 * ## Residuals 50 77 1.5 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 plot(model.1,which=1) 10 15 20 25 30 4 2 0 2 Fitted values R es id ua ls aov(amt.sugar ~ topping.type * sugar.type) Residuals vs Fitted 33 3 5 1 plot(model.1,which=2) 2 1 0 1 2 3 2 1 0 1 2 Theoretical Quantiles St an da rd ize d re sid ua ls aov(amt.sugar ~ topping.type * sugar.type) Normal Q Q 33 3 5 car::outlierTest(model.1) ## rstudent unadjusted p-value Bonferroni p ## 33 -3.754959 0.0004606 0.027636 library(car) ## Loading required package: carData boxCox(model.1) 2 2 1 0 1 2 17 0 15 0 13 0 λ lo g lik e lih oo d 95% Question 1 The data does not look like it meets the constant variance assumption. The data looks like it is normally distributed. Observation 33 may be an outlier. I tried two different transformations on amt.sugar since 0 and 0.5 are in the 95% CI. model.1a<-with(donuts,aov(log(amt.sugar)~topping.type*sugar.type)) summary(model.1a) ## Df Sum Sq Mean Sq F value Pr(>F) ## topping.type 4 13.450 3.362 527.289 < 2e-16 *** ## sugar.type 1 0.068 0.068 10.669 0.00197 ** ## topping.type:sugar.type 4 0.113 0.028 4.422 0.00388 ** ## Residuals 50 0.319 0.006 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 plot(model.1a,which=1) 3 2.2 2.4 2.6 2.8 3.0 3.2 3.4 0. 1 0. 0 0. 1 0. 2 Fitted values R es id ua ls aov(log(amt.sugar) ~ topping.type * sugar.type) Residuals vs Fitted 12 49 54 plot(model.1a,which=2) 2 1 0 1 2 2 1 0 1 2 Theoretical Quantiles St an da rd ize d re sid ua ls aov(log(amt.sugar) ~ topping.type * sugar.type) Normal Q Q 12 49 54 car::outlierTest(model.1a) ## No Studentized residuals with Bonferroni p < 0.05 ## Largest |rstudent|: 4 ## rstudent unadjusted p-value Bonferroni p ## 12 2.332992 0.023798 NA model.1b<-with(donuts,aov(amt.sugar^0.5~topping.type+sugar.type + topping.type:sugar.type)) summary(model.1b) ## Df Sum Sq Mean Sq F value Pr(>F) ## topping.type 4 61.96 15.491 718.190 < 2e-16 *** ## sugar.type 1 0.23 0.227 10.546 0.00208 ** ## topping.type:sugar.type 4 0.37 0.092 4.271 0.00474 ** ## Residuals 50 1.08 0.022 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 plot(model.1b,which=1) 3.0 3.5 4.0 4.5 5.0 5.5 0. 4 0. 2 0. 0 0. 2 Fitted values R es id ua ls aov(amt.sugar^0.5 ~ topping.type + sugar.type + topping.type:sugar.type) Residuals vs Fitted 33 49 54 plot(model.1b,which=2) 5 2 1 0 1 2 2 1 0 1 2 Theoretical Quantiles St an da rd ize d re sid ua ls aov(amt.sugar^0.5 ~ topping.type + sugar.type + topping.type:sugar.type) Normal Q Q 33 49 54 car::outlierTest(model.1b) ## No Studentized residuals with Bonferroni p < 0.05 ## Largest |rstudent|: ## rstudent unadjusted p-value Bonferroni p ## 33 -2.675555 0.010115 0.60689 The square root transformation seems to stabilize the variance slightly better than the log, therefore we will use the square root transformation. Outliers do not seem to be an issue now. 6 Question 2 summary(model.1a) ## Df Sum Sq Mean Sq F value Pr(>F) ## topping.type 4 13.450 3.362 527.289 < 2e-16 *** ## sugar.type 1 0.068 0.068 10.669 0.00197 ** ## topping.type:sugar.type 4 0.113 0.028 4.422 0.00388 ** ## Residuals 50 0.319 0.006 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 The results indicate that both main effects, topping.type and sugar.type, and their two-way interaction are statistically significant. Let’s look at the interaction. library(lattice) library(cfcdae) ## Registered S3 method overwritten by 'DoE.base': ## method from ## factorize.factor conf.design ## Registered S3 method overwritten by 'xts': ## method from ## as.zoo.xts zoo ## Registered S3 methods overwritten by 'lme4': ## method from ## cooks.distance.influence.merMod car ## influence.merMod car ## dfbeta.influence.merMod car ## dfbetas.influence.merMod car interactplot(topping.type,sugar.type,(amt.sugar)^0.5, confidence=0.95,cex.axis=.5) 7 11 1 1 13.0 3. 5 4. 0 4. 5 5. 0 5. 5 6. 0 topping.type M ea n of (a mt .su ga r)^ 0.5 2 2 2 2 2 Frosted Frosted n Sprinkles Glazed Plain Powdered sugar.type 2 1 Regular Low Sugar The interaction plot is parallel for the most part, except at frosted. This is where the interaction is. The lines widen for frosted regular vs. frosted low sugar. The 95% confidence lines do not overlap at frosted regular vs. frosted low sugar. The lines are parallel for all other comparisons for a given topping.type for the two different sugar types and the confidence intervals overlap. #Examine the differences between topping.type and sugar.type tapply((amt.sugar)^0.5,list(topping.type,sugar.type),mean) ## Low Sugar Regular ## Frosted 3.440242 3.875808 ## Frosted n Sprinkles 5.705032 5.767231 ## Glazed 3.062236 3.107329 ## Plain 3.066201 3.088848 ## Powdered 3.163003 3.213226 TukeyHSD(model.1b,"topping.type:sugar.type") ## Tukey multiple comparisons of means ## 95% family-wise confidence level ## ## Fit: aov(formula = amt.sugar^0.5 ~ topping.type + sugar.type + topping.type:sugar.type) ## ## $`topping.type:sugar.type` ## diff ## Frosted n Sprinkles:Low Sugar-Frosted:Low Sugar 2.264789921 ## Glazed:Low Sugar-Frosted:Low Sugar -0.378006294 ## Plain:Low Sugar-Frosted:Low Sugar -0.374040994 ## Powdered:Low Sugar-Frosted:Low Sugar -0.277238986 ## Frosted:Regular-Frosted:Low Sugar 0.435565678 ## Frosted n Sprinkles:Regular-Frosted:Low Sugar 2.326989566 ## Glazed:Regular-Frosted:Low Sugar -0.332913012 ## Plain:Regular-Frosted:Low Sugar -0.351393424 8 ## Powdered:Regular-Frosted:Low Sugar -0.227016074 ## Glazed:Low Sugar-Frosted n Sprinkles:Low Sugar -2.642796215 ## Plain:Low Sugar-Frosted n Sprinkles:Low Sugar -2.638830914 ## Powdered:Low Sugar-Frosted n Sprinkles:Low Sugar -2.542028907 ## Frosted:Regular-Frosted n Sprinkles:Low Sugar -1.829224243 ## Frosted n Sprinkles:Regular-Frosted n Sprinkles:Low Sugar 0.062199645 ## Glazed:Regular-Frosted n Sprinkles:Low Sugar -2.597702932 ## Plain:Regular-Frosted n Sprinkles:Low Sugar -2.616183345 ## Powdered:Regular-Frosted n Sprinkles:Low Sugar -2.491805995 ## Plain:Low Sugar-Glazed:Low Sugar 0.003965301 ## Powdered:Low Sugar-Glazed:Low Sugar 0.100767308 ## Frosted:Regular-Glazed:Low Sugar 0.813571972 ## Frosted n Sprinkles:Regular-Glazed:Low Sugar 2.704995860 ## Glazed:Regular-Glazed:Low Sugar 0.045093282 ## Plain:Regular-Glazed:Low Sugar 0.026612870 ## Powdered:Regular-Glazed:Low Sugar 0.150990220 ## Powdered:Low Sugar-Plain:Low Sugar 0.096802007 ## Frosted:Regular-Plain:Low Sugar 0.809606671 ## Frosted n Sprinkles:Regular-Plain:Low Sugar 2.701030560 ## Glazed:Regular-Plain:Low Sugar 0.041127982 ## Plain:Regular-Plain:Low Sugar 0.022647569 ## Powdered:Regular-Plain:Low Sugar 0.147024920 ## Frosted:Regular-Powdered:Low Sugar 0.712804664 ## Frosted n Sprinkles:Regular-Powdered:Low Sugar 2.604228552 ## Glazed:Regular-Powdered:Low Sugar -0.055674025 ## Plain:Regular-Powdered:Low Sugar -0.074154438 ## Powdered:Regular-Powdered:Low Sugar 0.050222912 ## Frosted n Sprinkles:Regular-Frosted:Regular 1.891423888 ## Glazed:Regular-Frosted:Regular -0.768478689 ## Plain:Regular-Frosted:Regular -0.786959102 ## Powdered:Regular-Frosted:Regular -0.662581752 ## Glazed:Regular-Frosted n Sprinkles:Regular -2.659902578 ## Plain:Regular-Frosted n Sprinkles:Regular -2.678382990 ## Powdered:Regular-Frosted n Sprinkles:Regular -2.554005640 ## Plain:Regular-Glazed:Regular -0.018480413 ## Powdered:Regular-Glazed:Regular 0.105896938 ## Powdered:Regular-Plain:Regular 0.124377350 ## lwr ## Frosted n Sprinkles:Low Sugar-Frosted:Low Sugar 1.9841037 ## Glazed:Low Sugar-Frosted:Low Sugar -0.6586926 ## Plain:Low Sugar-Frosted:Low Sugar -0.6547273 ## Powdered:Low Sugar-Frosted:Low Sugar -0.5579253 ## Frosted:Regular-Frosted:Low Sugar 0.1548794 ## Frosted n Sprinkles:Regular-Frosted:Low Sugar 2.0463033 ## Glazed:Regular-Frosted:Low Sugar -0.6135993 ## Plain:Regular-Frosted:Low Sugar -0.6320797 ## Powdered:Regular-Frosted:Low Sugar -0.5077023 ## Glazed:Low Sugar-Frosted n Sprinkles:Low Sugar -2.9234825 ## Plain:Low Sugar-Frosted n Sprinkles:Low Sugar -2.9195172 ## Powdered:Low Sugar-Frosted n Sprinkles:Low Sugar -2.8227152 ## Frosted:Regular-Frosted n Sprinkles:Low Sugar -2.1099105 ## Frosted n Sprinkles:Regular-Frosted n Sprinkles:Low Sugar -0.2184866 ## Glazed:Regular-Frosted n Sprinkles:Low Sugar -2.8783892 ## Plain:Regular-Frosted n Sprinkles:Low Sugar -2.8968696 9 ## Powdered:Regular-Frosted n Sprinkles:Low Sugar -2.7724923 ## Plain:Low Sugar-Glazed:Low Sugar -0.2767210 ## Powdered:Low Sugar-Glazed:Low Sugar -0.1799190 ## Frosted:Regular-Glazed:Low Sugar 0.5328857 ## Frosted n Sprinkles:Regular-Glazed:Low Sugar 2.4243096 ## Glazed:Regular-Glazed:Low Sugar -0.2355930 ## Plain:Regular-Glazed:Low Sugar -0.2540734 ## Powdered:Regular-Glazed:Low Sugar -0.1296961 ## Powdered:Low Sugar-Plain:Low Sugar -0.1838843 ## Frosted:Regular-Plain:Low Sugar 0.5289204 ## Frosted n Sprinkles:Regular-Plain:Low Sugar 2.4203443 ## Glazed:Regular-Plain:Low Sugar -0.2395583 ## Plain:Regular-Plain:Low Sugar -0.2580387 ## Powdered:Regular-Plain:Low Sugar -0.1336614 ## Frosted:Regular-Powdered:Low Sugar 0.4321184 ## Frosted n Sprinkles:Regular-Powdered:Low Sugar 2.3235423 ## Glazed:Regular-Powdered:Low Sugar -0.3363603 ## Plain:Regular-Powdered:Low Sugar -0.3548407 ## Powdered:Regular-Powdered:Low Sugar -0.2304634 ## Frosted n Sprinkles:Regular-Frosted:Regular 1.6107376 ## Glazed:Regular-Frosted:Regular -1.0491650 ## Plain:Regular-Frosted:Regular -1.0676454 ## Powdered:Regular-Frosted:Regular -0.9432680 ## Glazed:Regular-Frosted n Sprinkles:Regular -2.9405888 ## Plain:Regular-Frosted n Sprinkles:Regular -2.9590693 ## Powdered:Regular-Frosted n Sprinkles:Regular -2.8346919 ## Plain:Regular-Glazed:Regular -0.2991667 ## Powdered:Regular-Glazed:Regular -0.1747893 ## Powdered:Regular-Plain:Regular -0.1563089 ## upr ## Frosted n Sprinkles:Low Sugar-Frosted:Low Sugar 2.545476191 ## Glazed:Low Sugar-Frosted:Low Sugar -0.097320024 ## Plain:Low Sugar-Frosted:Low Sugar -0.093354723 ## Powdered:Low Sugar-Frosted:Low Sugar 0.003447284 ## Frosted:Regular-Frosted:Low Sugar 0.716251948 ## Frosted n Sprinkles:Regular-Frosted:Low Sugar 2.607675837 ## Glazed:Regular-Frosted:Low Sugar -0.052226741 ## Plain:Regular-Frosted:Low Sugar -0.070707154 ## Powdered:Regular-Frosted:Low Sugar 0.053670197 ## Glazed:Low Sugar-Frosted n Sprinkles:Low Sugar -2.362109944 ## Plain:Low Sugar-Frosted n Sprinkles:Low Sugar -2.358144644 ## Powdered:Low Sugar-Frosted n Sprinkles:Low Sugar -2.261342636 ## Frosted:Regular-Frosted n Sprinkles:Low Sugar -1.548537973 ## Frosted n Sprinkles:Regular-Frosted n Sprinkles:Low Sugar 0.342885916 ## Glazed:Regular-Frosted n Sprinkles:Low Sugar -2.317016662 ## Plain:Regular-Frosted n Sprinkles:Low Sugar -2.335497074 ## Powdered:Regular-Frosted n Sprinkles:Low Sugar -2.211119724 ## Plain:Low Sugar-Glazed:Low Sugar 0.284651571 ## Powdered:Low Sugar-Glazed:Low Sugar 0.381453578 ## Frosted:Regular-Glazed:Low Sugar 1.094258242 ## Frosted n Sprinkles:Regular-Glazed:Low Sugar 2.985682131 ## Glazed:Regular-Glazed:Low Sugar 0.325779553 ## Plain:Regular-Glazed:Low Sugar 0.307299140 ## Powdered:Regular-Glazed:Low Sugar 0.431676491 10 ## Powdered:Low Sugar-Plain:Low Sugar 0.377488278 ## Frosted:Regular-Plain:Low Sugar 1.090292942 ## Frosted n Sprinkles:Regular-Plain:Low Sugar 2.981716830 ## Glazed:Regular-Plain:Low Sugar 0.321814252 ## Plain:Regular-Plain:Low Sugar 0.303333840 ## Powdered:Regular-Plain:Low Sugar 0.427711190 ## Frosted:Regular-Powdered:Low Sugar 0.993490934 ## Frosted n Sprinkles:Regular-Powdered:Low Sugar 2.884914823 ## Glazed:Regular-Powdered:Low Sugar 0.225012245 ## Plain:Regular-Powdered:Low Sugar 0.206531833 ## Powdered:Regular-Powdered:Low Sugar 0.330909183 ## Frosted n Sprinkles:Regular-Frosted:Regular 2.172110159 ## Glazed:Regular-Frosted:Regular -0.487792419 ## Plain:Regular-Frosted:Regular -0.506272831 ## Powdered:Regular-Frosted:Regular -0.381895481 ## Glazed:Regular-Frosted n Sprinkles:Regular -2.379216307 ## Plain:Regular-Frosted n Sprinkles:Regular -2.397696720 ## Powdered:Regular-Frosted n Sprinkles:Regular -2.273319369 ## Plain:Regular-Glazed:Regular 0.262205858 ## Powdered:Regular-Glazed:Regular 0.386583208 ## Powdered:Regular-Plain:Regular 0.405063621 ## p adj ## Frosted n Sprinkles:Low Sugar-Frosted:Low Sugar 0.0000000 ## Glazed:Low Sugar-Frosted:Low Sugar 0.0017645 ## Plain:Low Sugar-Frosted:Low Sugar 0.0020482 ## Powdered:Low Sugar-Frosted:Low Sugar 0.0554341 ## Frosted:Regular-Frosted:Low Sugar 0.0001868 ## Frosted n Sprinkles:Regular-Frosted:Low Sugar 0.0000000 ## Glazed:Regular-Frosted:Low Sugar 0.0090945 ## Plain:Regular-Frosted:Low Sugar 0.0047177 ## Powdered:Regular-Frosted:Low Sugar 0.2109797 ## Glazed:Low Sugar-Frosted n Sprinkles:Low Sugar 0.0000000 ## Plain:Low Sugar-Frosted n Sprinkles:Low Sugar 0.0000000 ## Powdered:Low Sugar-Frosted n Sprinkles:Low Sugar 0.0000000 ## Frosted:Regular-Frosted n Sprinkles:Low Sugar 0.0000000 ## Frosted n Sprinkles:Regular-Frosted n Sprinkles:Low Sugar 0.9991271 ## Glazed:Regular-Frosted n Sprinkles:Low Sugar 0.0000000 ## Plain:Regular-Frosted n Sprinkles:Low Sugar 0.0000000 ## Powdered:Regular-Frosted n Sprinkles:Low Sugar 0.0000000 ## Plain:Low Sugar-Glazed:Low Sugar 1.0000000 ## Powdered:Low Sugar-Glazed:Low Sugar 0.9708515 ## Frosted:Regular-Glazed:Low Sugar 0.0000000 ## Frosted n Sprinkles:Regular-Glazed:Low Sugar 0.0000000 ## Glazed:Regular-Glazed:Low Sugar 0.9999374 ## Plain:Regular-Glazed:Low Sugar 0.9999993 ## Powdered:Regular-Glazed:Low Sugar 0.7436526 ## Powdered:Low Sugar-Plain:Low Sugar 0.9775604 ## Frosted:Regular-Plain:Low Sugar 0.0000000 ## Frosted n Sprinkles:Regular-Plain:Low Sugar 0.0000000 ## Glazed:Regular-Plain:Low Sugar 0.9999713 ## Plain:Regular-Plain:Low Sugar 0.9999998 ## Powdered:Regular-Plain:Low Sugar 0.7710767 ## Frosted:Regular-Powdered:Low Sugar 0.0000000 ## Frosted n Sprinkles:Regular-Powdered:Low Sugar 0.0000000 11 ## Glazed:Regular-Powdered:Low Sugar 0.9996415 ## Plain:Regular-Powdered:Low Sugar 0.9966047 ## Powdered:Regular-Powdered:Low Sugar 0.9998461 ## Frosted n Sprinkles:Regular-Frosted:Regular 0.0000000 ## Glazed:Regular-Frosted:Regular 0.0000000 ## Plain:Regular-Frosted:Regular 0.0000000 ## Powdered:Regular-Frosted:Regular 0.0000000 ## Glazed:Regular-Frosted n Sprinkles:Regular 0.0000000 ## Plain:Regular-Frosted n Sprinkles:Regular 0.0000000 ## Powdered:Regular-Frosted n Sprinkles:Regular 0.0000000 ## Plain:Regular-Glazed:Regular 1.0000000 ## Powdered:Regular-Glazed:Regular 0.9601107 ## Powdered:Regular-Plain:Regular 0.8985937 12 Question 3 Based on the interaction plot and Tukey’s test, any of the plain, powered or glazed toppings are low in sugar content regardless of whether the donut was made with regular or low sugar. The low sugar version of the donut has less sugar than the regular version, but for these three types of topping they are not statistically different. Therefore, all six combinations of- plain:regular powered:regular glazed:regular plain:low sugar powered:low sugar glazed:low sugar are statistically lower than the remaining donuts and are good choices. Donuts made with frosted and frosted n sprinkles are the worst topppings with regard to the amount of sugar they contain regardless of whether the donut used regular or low sugar. 13 Questions 4 - 6 Asparagus Problem #Asparagus A asparagus <- read.csv("~/Documents/STAT 5303 Spring 2020/Data/asparagus.csv") attach(asparagus) ## The following object is masked from donuts: ## ## X str(asparagus) ## 'data.frame': 24 obs. of 5 variables: ## $ X : int 1 2 3 4 5 6 7 8 9 10 ... ## $ K : int 65 54 41 33 56 52 54 48 56 50 ... ## $ cooking.method: Factor w/ 2 levels "Grilled","Steamed": 2 2 2 2 2 2 2 2 2 2 ... ## $ cooking.time : int 4 8 12 16 4 8 12 16 4 8 ... ## $ age : Factor w/ 3 levels "mature","old",..: 3 3 3 3 1 1 1 1 2 2 ... asparagus$cooking.time.z<-cooking.time asparagus$cooking.time<-as.factor(cooking.time) attach(asparagus) ## The following objects are masked from asparagus (pos = 3): ## ## age, cooking.method, cooking.time, K, X ## ## The following object is masked from donuts: ## ## X # Since there is no replication in the study, # the three-way interaction is used as the error. model.1<-aov(K~(cooking.method+age+cooking.time)^2,data=asparagus) summary(model.1) ## Df Sum Sq Mean Sq F value Pr(>F) ## cooking.method 1 126.0 126.04 10.894 0.0164 * ## age 2 170.1 85.04 7.351 0.0243 * ## cooking.time 3 794.5 264.82 22.890 0.0011 ** ## cooking.method:age 2 3.6 1.79 0.155 0.8598 ## cooking.method:cooking.time 3 57.5 19.15 1.655 0.2741 ## age:cooking.time 6 326.9 54.49 4.709 0.0406 * ## Residuals 6 69.4 11.57 ## — ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1 # The residual df = 6, so we pool the non significant # two-way interactions, cooking.method:age and # cooking.method:cooking:time. model.1a<-with(asparagus,aov(K~cooking.method+age+ cooking.time + age:cooking.time)) summary(model.1a) ## Df Sum Sq Mean Sq F value Pr(>F) 14 ## cooking.method 1 126.0 126.04 10.628 0.0076 ** ## age 2 170.1 85.04 7.171 0.0102 * ## cooking.time 3 794.5 264.82 22.329 5.55e-05 *** ## age:cooking.time 6 326.9 54.49 4.594 0.0142 * ## Residuals 11 130.5 11.86 ## — ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1 plot(model.1a,which=1) 35 40 45 50 55 60 65 6 4 2 0 2 4 6 Fitted values R es id ua ls aov(K ~ cooking.method + age + cooking.time + age:cooking.time) Residuals vs Fitted 9 21 15 plot(model.1a,which=2) 15 2 1 0 1 2 2 1 0 1 2 Theoretical Quantiles St an da rd ize d re sid ua ls aov(K ~ cooking.method + age + cooking.time + age:cooking.time) Normal Q Q 9 21 15 outlierTest(model.1a) ## No Studentized residuals with Bonferroni p < 0.05 ## Largest |rstudent|: ## rstudent unadjusted p-value Bonferroni p ## 9 2.967808 0.014099 0.33837 boxCox(model.1a,lambda=seq(-2,5)) 2 1 0 1 2 3 4 5 66 64 62 60 58 λ lo g lik e lih oo d 95% Question 4. 16 The residuals are not as stabled as dersired, but a pattern isn’t obvious. The normality plot looks fine. A BoxCox transformation will be investigated. Since 1 is included in the 95% CI of the Box-Cox transformation, a log transformation will not help. The potassium content will be analyzed on the original scale. Question 5 Now let’s examine cooking.time and the cooking.time by age interaction on the quantitative scale. model.1b<-with(asparagus,aov(K~cooking.method+age+ cooking.time.z + I(cooking.time.z^2)+I(cooking.time.z^3) + age:cooking.time.z + age:I(cooking.time.z^2)+ age:I(cooking.time.z^3))) summary(model.1b) ## Df Sum Sq Mean Sq F value Pr(>F) ## cooking.method 1 126.0 126.0 10.628 0.00760 ** ## age 2 170.1 85.0 7.171 0.01015 * ## cooking.time.z 1 785.4 785.4 66.224 5.55e-06 *** ## I(cooking.time.z^2) 1 2.0 2.0 0.172 0.68619 ## I(cooking.time.z^3) 1 7.0 7.0 0.591 0.45825 ## age:cooking.time.z 2 315.2 157.6 13.289 0.00116 ** ## age:I(cooking.time.z^2) 2 1.1 0.5 0.046 0.95554 ## age:I(cooking.time.z^3) 2 10.6 5.3 0.448 0.65031 ## Residuals 11 130.5 11.9 ## — ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1 model.1c<-with(asparagus,lm(K~cooking.method+age+ cooking.time.z + I(cooking.time.z^2)+I(cooking.time.z^3) + age:cooking.time.z + age:I(cooking.time.z^2)+ age:I(cooking.time.z^3))) summary(model.1c) ## ## Call: ## lm(formula = K ~ cooking.method + age + cooking.time.z + I(cooking.time.z^2) + ## I(cooking.time.z^3) + age:cooking.time.z + age:I(cooking.time.z^2) + ## age:I(cooking.time.z^3)) ## ## Residuals: ## Min 1Q Median 3Q Max ## -5.292 -1.354 0.000 1.354 5.292 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 71.33333 11.67854 6.108 7.65e-05 *** ## cooking.method1 2.29167 0.70297 3.260 0.0076 ** ## age1 10.66667 16.51595 0.646 0.5316 ## age2 -18.83333 16.51595 -1.140 0.2784 ## cooking.time.z -4.27778 4.46520 -0.958 0.3586 ## I(cooking.time.z^2) 0.35937 0.49317 0.729 0.4814 ## I(cooking.time.z^3) -0.01259 0.01637 -0.769 0.4582 17 ## age1:cooking.time.z -4.63889 6.31474 -0.735 0.4779 ## age2:cooking.time.z 4.94444 6.31474 0.783 0.4502 ## age1:I(cooking.time.z^2) 0.59375 0.69745 0.851 0.4127 ## age2:I(cooking.time.z^2) -0.51563 0.69745 -0.739 0.4752 ## age1:I(cooking.time.z^3) -0.01997 0.02316 -0.862 0.4070 ## age2:I(cooking.time.z^3) 0.01780 0.02316 0.768 0.4584 ## — ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1 ## ## Residual standard error: 3.444 on 11 degrees of freedom ## Multiple R-squared: 0.9157, Adjusted R-squared: 0.8238 ## F-statistic: 9.96 on 12 and 11 DF, p-value: 0.0002941 interaction.plot(cooking.time.z,age,K) 35 40 45 50 55 60 65 cooking.time.z m e a n o f K 4 8 12 16 age mature old young The interaction plot shows that the potassium content of young asparagus declines more rapidly than mature and old asparagus with increasing cooking time. Mature asparagus decline the least with increasing cooking time. Old asparagus decline is in between the rapid decline of young asparagus and slow decline of mature asparagus. It isn’t necessary to discuss cooking.method since the question asked about cooking.time and age. However in a consulting situation, you would report your findings regarding the cooking.method since it is statistically significant too. Question 6 This question is thrown out. There is a statistically significant age by cooking.time interaction which makes the answer to the debate vague. 18 Question 7 # Type I error My thumb is fine (H0 is true), but I think it is broken (reject H0). I will go to the doctor. Type II error My thumb is broken (Ha is true), but I think it will be fine (fail to reject H0). I will not go to the doctor. Power My thumb is broken (Ha is true), and I think it is broken (reject H0). I will go to the doctor. Type II error is more risky healthwise, since I think my thumb is fine (fail to reject H0) when in reality it is broken (Ha is true). I have a broken thumb, but will not go to the doctor. 19 Question 8 # This is an observational study. There does not seem to be a way to construct an experiment to investiage the relationship between car cost and whether they stop or not. You cannot randomly assign a cost to a car. It comes with a cost already determined by the manufacturer. You cannot create homogeneous experimental units in this study. It seems the easiest way to conduct this study is to observe the car make and model and whether they stop or not. Collecting the car make and model will allow the investigator to determine its cost and relate it to whether they stop or not. This type of study is an observational study. 20