We have found that the dataset is insufficient to make a strong conclusion about whether automatic cars or manual cars have lower fuel consumption. We have tried to fit 12 models, using one or several of the given variables as regressors. However,
Models that favor automatic cars do not show suffient level of statistical significance.
Models that favor manual cars are confounded and cannot be trusted either.
The current document is the appendix showing details of the model comparison.
Settting the options and loading the data
library(knitr)
data(mtcars)The data set mtcars has 32 observations of 11 variables. We are interested in the relation between am and mpg. Obviously, other variables affect fuel consumption too, most notably the displacement disp and the weight wt.
We’ll do the following operations with the dataset:
Exclude variables cyl, gearm and carb from further analysis (since they are almost entirely dependent on am and disp).
Exclude drat and vs from analysis too because we don’t know what they are.
Change miles per gallon to litres per 100 km for better interpretability (bigger engine - higher fuel consumption) and for convenience of working in the metric system.
C <- mtcars[,c(3,4,6,7,9)]
C$lp100 <- 235.214583/mtcars$mpg
names(C) ## [1] "disp" "hp" "wt" "qsec" "am" "lp100"Here is table of correlation coefficients of the new dataset C:
## disp hp wt qsec am lp100
## disp 1.00 0.79 0.89 -0.43 -0.59 0.88
## hp 0.79 1.00 0.66 -0.71 -0.24 0.76
## wt 0.89 0.66 1.00 -0.17 -0.69 0.89
## qsec -0.43 -0.71 -0.17 1.00 -0.23 -0.39
## am -0.59 -0.24 -0.69 -0.23 1.00 -0.54
## lp100 0.88 0.76 0.89 -0.39 -0.54 1.00Here is table of correlation coefficients of the original dataset mtcars:
## mpg cyl disp hp drat wt qsec vs am gear carb
## mpg 1.00 -0.85 -0.85 -0.78 0.68 -0.87 0.42 0.66 0.60 0.48 -0.55
## cyl -0.85 1.00 0.90 0.83 -0.70 0.78 -0.59 -0.81 -0.52 -0.49 0.53
## disp -0.85 0.90 1.00 0.79 -0.71 0.89 -0.43 -0.71 -0.59 -0.56 0.39
## hp -0.78 0.83 0.79 1.00 -0.45 0.66 -0.71 -0.72 -0.24 -0.13 0.75
## drat 0.68 -0.70 -0.71 -0.45 1.00 -0.71 0.09 0.44 0.71 0.70 -0.09
## wt -0.87 0.78 0.89 0.66 -0.71 1.00 -0.17 -0.55 -0.69 -0.58 0.43
## qsec 0.42 -0.59 -0.43 -0.71 0.09 -0.17 1.00 0.74 -0.23 -0.21 -0.66
## vs 0.66 -0.81 -0.71 -0.72 0.44 -0.55 0.74 1.00 0.17 0.21 -0.57
## am 0.60 -0.52 -0.59 -0.24 0.71 -0.69 -0.23 0.17 1.00 0.79 0.06
## gear 0.48 -0.49 -0.56 -0.13 0.70 -0.58 -0.21 0.21 0.79 1.00 0.27
## carb -0.55 0.53 0.39 0.75 -0.09 0.43 -0.66 -0.57 0.06 0.27 1.00They show some light on why the best models are those including disp and wt.
Here are all the statistics values justifying our conclusions:
fit.disp <- lm(lp100 ~ disp , data=C)
summary(fit.disp)##
## Call:
## lm(formula = lp100 ~ disp, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.15 -0.90 0.25 1.22 3.57
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.4276 0.7059 9.1 3.9e-10 ***
## disp 0.0274 0.0027 10.1 3.3e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.87 on 30 degrees of freedom
## Multiple R-squared: 0.774, Adjusted R-squared: 0.767
## F-statistic: 103 on 1 and 30 DF, p-value: 3.32e-11fit.disp.wt <- lm(lp100 ~ disp + wt , data=C)
anova(fit.disp,fit.disp.wt)## Analysis of Variance Table
##
## Model 1: lp100 ~ disp
## Model 2: lp100 ~ disp + wt
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 104.5
## 2 29 78.7 1 25.8 9.51 0.0045 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(fit.disp.wt)##
## Call:
## lm(formula = lp100 ~ disp + wt, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.013 -0.822 0.372 1.044 2.553
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.18321 1.22266 2.60 0.0144 *
## disp 0.01321 0.00519 2.54 0.0165 *
## wt 2.02797 0.65757 3.08 0.0045 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.65 on 29 degrees of freedom
## Multiple R-squared: 0.83, Adjusted R-squared: 0.818
## F-statistic: 70.7 on 2 and 29 DF, p-value: 7.01e-12fit.disp.am <- lm(lp100 ~ disp + am , data=C)
anova(fit.disp,fit.disp.am)## Analysis of Variance Table
##
## Model 1: lp100 ~ disp
## Model 2: lp100 ~ disp + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 104
## 2 29 104 1 0.276 0.08 0.78summary(fit.disp.am)##
## Call:
## lm(formula = lp100 ~ disp + am, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.147 -0.955 0.203 1.116 3.607
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.65172 1.08064 6.16 1.0e-06 ***
## disp 0.02687 0.00341 7.89 1.1e-08 ***
## am -0.23458 0.84615 -0.28 0.78
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.9 on 29 degrees of freedom
## Multiple R-squared: 0.775, Adjusted R-squared: 0.759
## F-statistic: 49.9 on 2 and 29 DF, p-value: 4.13e-10fit.disp.wt.am <- lm(lp100 ~ disp + wt +am , data=C)
summary(fit.disp.wt.am)##
## Call:
## lm(formula = lp100 ~ disp + wt + am, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.189 -0.673 0.186 1.012 2.299
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.59494 1.78465 0.89 0.3791
## disp 0.01276 0.00516 2.47 0.0198 *
## wt 2.42850 0.73108 3.32 0.0025 **
## am 0.99168 0.81744 1.21 0.2352
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.63 on 28 degrees of freedom
## Multiple R-squared: 0.838, Adjusted R-squared: 0.821
## F-statistic: 48.4 on 3 and 28 DF, p-value: 3.31e-11fit.wt <- lm(lp100 ~ wt, data=C)
summary(fit.wt)##
## Call:
## lm(formula = lp100 ~ wt, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.230 -1.042 0.221 1.067 2.742
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.451 1.104 1.31 0.2
## wt 3.514 0.329 10.68 9.6e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.79 on 30 degrees of freedom
## Multiple R-squared: 0.792, Adjusted R-squared: 0.785
## F-statistic: 114 on 1 and 30 DF, p-value: 9.57e-12fit.wt.am <- lm(lp100 ~ wt + am, data=C)
anova(fit.wt,fit.wt.am)## Analysis of Variance Table
##
## Model 1: lp100 ~ wt
## Model 2: lp100 ~ wt + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 96.3
## 2 29 91.1 1 5.18 1.65 0.21summary(fit.wt.am)##
## Call:
## lm(formula = lp100 ~ wt + am, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.622 -0.956 0.079 1.133 2.954
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.301 1.748 -0.17 0.86
## wt 3.915 0.451 8.68 1.5e-09 ***
## am 1.136 0.884 1.28 0.21
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.77 on 29 degrees of freedom
## Multiple R-squared: 0.803, Adjusted R-squared: 0.79
## F-statistic: 59.1 on 2 and 29 DF, p-value: 5.84e-11fit.hp <- lm(lp100 ~ hp, data=C)
summary(fit.hp)##
## Call:
## lm(formula = lp100 ~ hp, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.169 -1.334 -0.165 0.570 7.355
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.44908 1.07380 6.01 1.4e-06 ***
## hp 0.04299 0.00665 6.46 3.8e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.54 on 30 degrees of freedom
## Multiple R-squared: 0.582, Adjusted R-squared: 0.568
## F-statistic: 41.8 on 1 and 30 DF, p-value: 3.84e-07fit.hp.am <- lm(lp100 ~ hp + am, data=C)
anova(fit.hp,fit.hp.am)## Analysis of Variance Table
##
## Model 1: lp100 ~ hp
## Model 2: lp100 ~ hp + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 193
## 2 29 132 1 61.7 13.6 0.00093 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(fit.hp.am)##
## Call:
## lm(formula = lp100 ~ hp + am, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.760 -1.142 -0.365 0.761 6.471
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.39063 1.04357 8.04 7.2e-09 ***
## hp 0.03783 0.00575 6.57 3.3e-07 ***
## am -2.91573 0.79053 -3.69 0.00093 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.13 on 29 degrees of freedom
## Multiple R-squared: 0.716, Adjusted R-squared: 0.696
## F-statistic: 36.5 on 2 and 29 DF, p-value: 1.21e-08fit.qsec <- lm(lp100 ~ qsec, data=C)
summary(fit.qsec)##
## Call:
## lm(formula = lp100 ~ qsec, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.809 -2.362 -0.282 1.836 9.971
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.644 6.531 4.23 0.0002 ***
## qsec -0.834 0.364 -2.29 0.0292 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.62 on 30 degrees of freedom
## Multiple R-squared: 0.149, Adjusted R-squared: 0.121
## F-statistic: 5.25 on 1 and 30 DF, p-value: 0.0292fit.qsec.am <- lm(lp100 ~ qsec + am, data=C)
anova(fit.qsec,fit.qsec.am)## Analysis of Variance Table
##
## Model 1: lp100 ~ qsec
## Model 2: lp100 ~ qsec + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 394
## 2 29 201 1 193 27.9 1.2e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(fit.qsec.am)##
## Call:
## lm(formula = lp100 ~ qsec + am, data = C)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.521 -1.424 -0.391 0.835 7.927
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 35.618 4.978 7.16 7.1e-08 ***
## qsec -1.164 0.272 -4.28 0.00018 ***
## am -5.138 0.973 -5.28 1.2e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.63 on 29 degrees of freedom
## Multiple R-squared: 0.566, Adjusted R-squared: 0.536
## F-statistic: 18.9 on 2 and 29 DF, p-value: 5.54e-06If we used the original miles per gallon, we would basically have the same picture, maybe a little worse because of slightly smaller values of the variance proportion explained by the model:
fit.disp <- lm(mpg ~ disp , data=mtcars)
summary(fit.disp)##
## Call:
## lm(formula = mpg ~ disp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.892 -2.202 -0.963 1.627 7.231
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.59985 1.22972 24.07 < 2e-16 ***
## disp -0.04122 0.00471 -8.75 9.4e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.25 on 30 degrees of freedom
## Multiple R-squared: 0.718, Adjusted R-squared: 0.709
## F-statistic: 76.5 on 1 and 30 DF, p-value: 9.38e-10fit.disp.wt <- lm(mpg ~ disp + wt , data=mtcars)
anova(fit.disp, fit.disp.wt)## Analysis of Variance Table
##
## Model 1: mpg ~ disp
## Model 2: mpg ~ disp + wt
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 317
## 2 29 247 1 70.5 8.29 0.0074 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(fit.disp.wt)##
## Call:
## lm(formula = mpg ~ disp + wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.409 -2.324 -0.768 1.772 6.348
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.96055 2.16454 16.15 4.9e-16 ***
## disp -0.01772 0.00919 -1.93 0.0636 .
## wt -3.35083 1.16413 -2.88 0.0074 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.92 on 29 degrees of freedom
## Multiple R-squared: 0.781, Adjusted R-squared: 0.766
## F-statistic: 51.7 on 2 and 29 DF, p-value: 2.74e-10fit.disp.am <- lm(mpg ~ disp + am , data=mtcars)
anova(fit.disp,fit.disp.am)## Analysis of Variance Table
##
## Model 1: mpg ~ disp
## Model 2: mpg ~ disp + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 317
## 2 29 300 1 16.9 1.63 0.21summary(fit.disp.am)##
## Call:
## lm(formula = mpg ~ disp + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.638 -2.475 -0.563 2.233 6.839
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.84808 1.83407 15.18 2.5e-15 ***
## disp -0.03685 0.00578 -6.37 5.7e-07 ***
## am 1.83346 1.43610 1.28 0.21
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.22 on 29 degrees of freedom
## Multiple R-squared: 0.733, Adjusted R-squared: 0.715
## F-statistic: 39.9 on 2 and 29 DF, p-value: 4.75e-09fit.disp.wt.am <- lm(mpg ~ disp + wt +am , data=mtcars)
summary(fit.disp.wt.am)##
## Call:
## lm(formula = mpg ~ disp + wt + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.489 -2.411 -0.723 1.750 6.329
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.67591 3.24061 10.70 2.1e-11 ***
## disp -0.01780 0.00937 -1.90 0.068 .
## wt -3.27904 1.32751 -2.47 0.020 *
## am 0.17772 1.48432 0.12 0.906
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.97 on 28 degrees of freedom
## Multiple R-squared: 0.781, Adjusted R-squared: 0.758
## F-statistic: 33.3 on 3 and 28 DF, p-value: 2.25e-09fit.wt <- lm(mpg ~ wt, data=mtcars)
summary(fit.wt)##
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.543 -2.365 -0.125 1.410 6.873
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.285 1.878 19.86 < 2e-16 ***
## wt -5.344 0.559 -9.56 1.3e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.05 on 30 degrees of freedom
## Multiple R-squared: 0.753, Adjusted R-squared: 0.745
## F-statistic: 91.4 on 1 and 30 DF, p-value: 1.29e-10fit.wt.am <- lm(mpg ~ wt + am, data=mtcars)
anova(fit.wt,fit.wt.am)## Analysis of Variance Table
##
## Model 1: mpg ~ wt
## Model 2: mpg ~ wt + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 278
## 2 29 278 1 0.00224 0 0.99summary(fit.wt.am)##
## Call:
## lm(formula = mpg ~ wt + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.530 -2.362 -0.132 1.403 6.878
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.3216 3.0546 12.22 5.8e-13 ***
## wt -5.3528 0.7882 -6.79 1.9e-07 ***
## am -0.0236 1.5456 -0.02 0.99
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.1 on 29 degrees of freedom
## Multiple R-squared: 0.753, Adjusted R-squared: 0.736
## F-statistic: 44.2 on 2 and 29 DF, p-value: 1.58e-09fit.hp <- lm(mpg ~ hp, data=mtcars)
summary(fit.hp)##
## Call:
## lm(formula = mpg ~ hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.712 -2.112 -0.885 1.582 8.236
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.0989 1.6339 18.42 < 2e-16 ***
## hp -0.0682 0.0101 -6.74 1.8e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.86 on 30 degrees of freedom
## Multiple R-squared: 0.602, Adjusted R-squared: 0.589
## F-statistic: 45.5 on 1 and 30 DF, p-value: 1.79e-07fit.hp.am <- lm(mpg ~ hp + am, data=mtcars)
anova(fit.hp,fit.hp.am)## Analysis of Variance Table
##
## Model 1: mpg ~ hp
## Model 2: mpg ~ hp + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 448
## 2 29 245 1 202 23.9 3.5e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(fit.hp.am)##
## Call:
## lm(formula = mpg ~ hp + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.384 -2.264 0.137 1.697 5.866
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.58491 1.42509 18.65 < 2e-16 ***
## hp -0.05889 0.00786 -7.50 2.9e-08 ***
## am 5.27709 1.07954 4.89 3.5e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.91 on 29 degrees of freedom
## Multiple R-squared: 0.782, Adjusted R-squared: 0.767
## F-statistic: 52 on 2 and 29 DF, p-value: 2.55e-10fit.qsec <- lm(mpg ~ qsec, data=mtcars)
summary(fit.qsec)##
## Call:
## lm(formula = mpg ~ qsec, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.88 -3.45 -0.72 2.28 11.65
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.114 10.030 -0.51 0.614
## qsec 1.412 0.559 2.53 0.017 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.56 on 30 degrees of freedom
## Multiple R-squared: 0.175, Adjusted R-squared: 0.148
## F-statistic: 6.38 on 1 and 30 DF, p-value: 0.0171fit.qsec.am <- lm(mpg ~ qsec + am, data=mtcars)
anova(fit.qsec,fit.qsec.am)## Analysis of Variance Table
##
## Model 1: mpg ~ qsec
## Model 2: mpg ~ qsec + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 929
## 2 29 353 1 576 47.4 1.5e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(fit.qsec.am)##
## Call:
## lm(formula = mpg ~ qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.345 -2.770 0.294 2.095 6.919
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -18.89 6.60 -2.86 0.0077 **
## qsec 1.98 0.36 5.50 6.3e-06 ***
## am 8.88 1.29 6.88 1.5e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.49 on 29 degrees of freedom
## Multiple R-squared: 0.687, Adjusted R-squared: 0.665
## F-statistic: 31.8 on 2 and 29 DF, p-value: 4.88e-08Our analysis has been originally created and run in RStudio v. 0.98.1080 under OS X 10.9.5.
Time and date the report has been generated: 2014-11-23 22:31:59 (Central European time).