Lab 7
Scott Ogden
November 12, 2016
Exercise 1
download.file("http://www.openintro.org/stat/data/mlb11.RData", destfile = "mlb11.RData")
load("mlb11.RData")
summary(mlb11)## team runs at_bats hits
## Arizona Diamondbacks: 1 Min. :556.0 Min. :5417 Min. :1263
## Atlanta Braves : 1 1st Qu.:629.0 1st Qu.:5448 1st Qu.:1348
## Baltimore Orioles : 1 Median :705.5 Median :5516 Median :1394
## Boston Red Sox : 1 Mean :693.6 Mean :5524 Mean :1409
## Chicago Cubs : 1 3rd Qu.:734.0 3rd Qu.:5575 3rd Qu.:1441
## Chicago White Sox : 1 Max. :875.0 Max. :5710 Max. :1600
## (Other) :24
## homeruns bat_avg strikeouts stolen_bases
## Min. : 91.0 Min. :0.2330 Min. : 930 Min. : 49.00
## 1st Qu.:118.0 1st Qu.:0.2447 1st Qu.:1085 1st Qu.: 89.75
## Median :154.0 Median :0.2530 Median :1140 Median :107.00
## Mean :151.7 Mean :0.2549 Mean :1150 Mean :109.30
## 3rd Qu.:172.8 3rd Qu.:0.2602 3rd Qu.:1248 3rd Qu.:130.75
## Max. :222.0 Max. :0.2830 Max. :1323 Max. :170.00
##
## wins new_onbase new_slug new_obs
## Min. : 56.00 Min. :0.2920 Min. :0.3480 Min. :0.6400
## 1st Qu.: 72.00 1st Qu.:0.3110 1st Qu.:0.3770 1st Qu.:0.6920
## Median : 80.00 Median :0.3185 Median :0.3985 Median :0.7160
## Mean : 80.97 Mean :0.3205 Mean :0.3988 Mean :0.7191
## 3rd Qu.: 90.00 3rd Qu.:0.3282 3rd Qu.:0.4130 3rd Qu.:0.7382
## Max. :102.00 Max. :0.3490 Max. :0.4610 Max. :0.8100
## I’d use a scatter plot
library(ggplot2)
ggplot(mlb11,aes(x=mlb11$at_bats,y=mlb11$runs))+geom_point(stat="identity",fill="red",colour="dark red")
These data look linear but with a decent but of noise.
cor(x=mlb11$at_bats,y=mlb11$runs)## [1] 0.610627Exercise 2
It has a relatively high correlation coefficient for this number of data. It seems to be a positive relationship.
Exercise 3
The smallest sum of squares I got was 149683.9
m1 <- lm(runs ~ at_bats, data = mlb11)Exercise 4
m2<-lm(runs~homeruns,data=mlb11)
summary(m2)##
## Call:
## lm(formula = runs ~ homeruns, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -91.615 -33.410 3.231 24.292 104.631
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 415.2389 41.6779 9.963 1.04e-10 ***
## homeruns 1.8345 0.2677 6.854 1.90e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 51.29 on 28 degrees of freedom
## Multiple R-squared: 0.6266, Adjusted R-squared: 0.6132
## F-statistic: 46.98 on 1 and 28 DF, p-value: 1.9e-07Y= 415.2 + 1.8345*homeruns
Positive relationship
Exercise 5
plot(mlb11$runs ~ mlb11$at_bats)
abline(m1)
Predicted runs ~ -2789.2+.6305*5578 = 727.7 runs
plot(m1$residuals ~ mlb11$at_bats)
abline(h = 0, lty = 3)
abline(v=5578)
Residual~ -20 so it is an overestimate
Exercise 6
Doesn’t appear to be a pattern It indicated possible linearity with noise.
hist(m1$residuals)
qqnorm(m1$residuals)
qqline(m1$residuals) 
Exercise 7
Nearly normal with some stepping maybe. The nearly normal residuals does appear to be met.
Exercise 8
Constant variability condition on the QQ plot appears to be met as there are relatively similar pockets of variability. Nothing extreme or noteworthy.
Problem 1
Batting average
plot(x=mlb11$bat_avg,y=mlb11$runs)
At first glace this appears very linear.
Problem 2
m3<-lm(runs~bat_avg,data=mlb11)
summary(m3)##
## Call:
## lm(formula = runs ~ bat_avg, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -94.676 -26.303 -5.496 28.482 131.113
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -642.8 183.1 -3.511 0.00153 **
## bat_avg 5242.2 717.3 7.308 5.88e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 49.23 on 28 degrees of freedom
## Multiple R-squared: 0.6561, Adjusted R-squared: 0.6438
## F-statistic: 53.41 on 1 and 28 DF, p-value: 5.877e-08R^2 = 0.65 so this performs better than the at_bats model.
Problem 3
m4<-lm(runs~hits,data=mlb11)
summary(m4)##
## Call:
## lm(formula = runs ~ hits, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -103.718 -27.179 -5.233 19.322 140.693
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -375.5600 151.1806 -2.484 0.0192 *
## hits 0.7589 0.1071 7.085 1.04e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 50.23 on 28 degrees of freedom
## Multiple R-squared: 0.6419, Adjusted R-squared: 0.6292
## F-statistic: 50.2 on 1 and 28 DF, p-value: 1.043e-07m5<-lm(runs~strikeouts,data=mlb11)
summary(m5)##
## Call:
## lm(formula = runs ~ strikeouts, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -132.27 -46.95 -11.92 55.14 169.76
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1054.7342 151.7890 6.949 1.49e-07 ***
## strikeouts -0.3141 0.1315 -2.389 0.0239 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 76.5 on 28 degrees of freedom
## Multiple R-squared: 0.1694, Adjusted R-squared: 0.1397
## F-statistic: 5.709 on 1 and 28 DF, p-value: 0.02386m6<-lm(runs~wins,data=mlb11)
summary(m6)##
## Call:
## lm(formula = runs ~ wins, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -145.450 -47.506 -7.482 47.346 142.186
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 342.121 89.223 3.834 0.000654 ***
## wins 4.341 1.092 3.977 0.000447 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 67.1 on 28 degrees of freedom
## Multiple R-squared: 0.361, Adjusted R-squared: 0.3381
## F-statistic: 15.82 on 1 and 28 DF, p-value: 0.0004469m7<-lm(runs~stolen_bases,data=mlb11)
summary(m7)##
## Call:
## lm(formula = runs ~ stolen_bases, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -139.94 -62.87 10.01 38.54 182.49
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 677.3074 58.9751 11.485 4.17e-12 ***
## stolen_bases 0.1491 0.5211 0.286 0.777
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 83.82 on 28 degrees of freedom
## Multiple R-squared: 0.002914, Adjusted R-squared: -0.0327
## F-statistic: 0.08183 on 1 and 28 DF, p-value: 0.7769At bats: Rsq = .37 Bat avg: Rsq = .65 homeruns: Rsq =.63 hits: Rsq = .64 strikeouts: Rsq= .17 wins: Rsq = .36 Stolen: rsq = 0.003
At bats best predicts runs.
plot(mlb11$runs ~ mlb11$bat_avg)
abline(m3)
Problem 4
New variables
m8<-lm(runs~new_onbase,data=mlb11)
summary(m8)##
## Call:
## lm(formula = runs ~ new_onbase, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -58.270 -18.335 3.249 19.520 69.002
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1118.4 144.5 -7.741 1.97e-08 ***
## new_onbase 5654.3 450.5 12.552 5.12e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 32.61 on 28 degrees of freedom
## Multiple R-squared: 0.8491, Adjusted R-squared: 0.8437
## F-statistic: 157.6 on 1 and 28 DF, p-value: 5.116e-13plot(mlb11$runs ~ mlb11$new_onbase)
abline(m8)
m9<-lm(runs~new_slug,data=mlb11)
summary(m9)##
## Call:
## lm(formula = runs ~ new_slug, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -45.41 -18.66 -0.91 16.29 52.29
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -375.80 68.71 -5.47 7.70e-06 ***
## new_slug 2681.33 171.83 15.61 2.42e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 26.96 on 28 degrees of freedom
## Multiple R-squared: 0.8969, Adjusted R-squared: 0.8932
## F-statistic: 243.5 on 1 and 28 DF, p-value: 2.42e-15plot(mlb11$runs ~ mlb11$new_slug)
abline(m9)
m10<-lm(runs~new_obs,data=mlb11)
summary(m10)##
## Call:
## lm(formula = runs ~ new_obs, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43.456 -13.690 1.165 13.935 41.156
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -686.61 68.93 -9.962 1.05e-10 ***
## new_obs 1919.36 95.70 20.057 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 21.41 on 28 degrees of freedom
## Multiple R-squared: 0.9349, Adjusted R-squared: 0.9326
## F-statistic: 402.3 on 1 and 28 DF, p-value: < 2.2e-16plot(mlb11$runs ~ mlb11$new_obs)
abline(m10)
OnBase: rsq = .85 Slug: rsq=0.89 Obs: rsq=.934
These variables explain a lot more of the variance. They are better predictors based on the graphs and RSQ.
Problem5
Model diagnostics for OBS:
hist(m10$residuals)
qqnorm(m10$residuals)
qqline(m10$residuals)
Histogram nearly normal. Very little variance on the qq plot implying the variability condition is met.