Note:

In the first question, “coefficients” simply refer to the parameter estimates under the “Est” column heading. No other value in the table from
summary()
is considered a coefficient.
Car Horsepower and Gas Mileage I
 The results of
summary()
are in Table 1. Interpretations of the coefficients are below. The yintercept (46.93) is the estimated mean mpg for cars with no horsepower.
 The slope (i.e., the “hp” “Estimate”) says that for every increase in one horsepower the mpg will decrease by 0.22, on average.
 Interpretations of the pvalues in Table 1 are below.
 The “Intercept” pvalue (p<0.00005) shows that the mean mpg for cars with no horsepower is different than zero (a nonsensical significance test).
 The slope (in “hp” row) pvalue (p<0.00005) shows that the slope is significantly different from zero, which indicates that there is a significant relationship between the horsepower and mpg of a car.
 The coefficient of determination (“multiple rsquared”; 0.78) is the proportion of the total variability in mpg (ignoring horsepower) that is explained away by knowing a horsepower value.
 Confidence intervals for the model coefficients are in Table 2. Interpretations for each confidence interval are below.
 The “intercept” CI says that the mean mpg for cars with no horsepower is between 43.0 and 50.8.
 The slope (i.e., “hp” row) CI says that the mpg will decrease between 0.18 and 0.25 for a one unit increase in horespower.
 The ANOVA table is shown in Table 3. How each degreesoffreedom is calculated is below.
 The regression df (in the “hp” row) df is one less than the number of parameters estimated (2 – intercept and slope).
 The residual df is the number of observations (42) minus the number of parameters estimated (2).
 The total df is not shown on the table but is equal to the number of observations minus 1.
 The meanings of each MS in Table 3 are below.
 The MS regression (in the “hp” row) is the variance in mpg that can be explained by knowing the value of horespower.
 The MS residual is the variance in mpg after considering horsepower or the variability of individuals around the bestfit line (i.e., the full model).
 The total MS is not shown in the table but is the variance in mpg or the variability of individuals around the grand mean (i.e., the simple model).
 The F test statistic is the ratio of variability in mpg explained by knowing the value of horespower to the variability unexplained even after knowing the value of the horsespower. The F (141.5) and corresponding pvalue (p<0.00005) show that the full model including the slope is significantly “better” than the simple model with no slope. Thus, a slope “is needed” and it can be concluded that there is a significant relationship between a car’s horespower and its gas mileage.
 There is a significant relationship as indicated by the very small slope and Ftest pvalues (p<0.00005).
 The relationship between a car’s mpg and horsepower is shown in Figure 1.
Table 1: Summary of the linear regression of mpg on horsepower.
Estimate Std. Error t value Pr(>t)
(Intercept) 46.92659 1.92184 24.42 < 2e16
hp 0.21762 0.01829 11.90 1.03e14

Residual standard error: 3.096 on 40 degrees of freedom
Multiple Rsquared: 0.7796, Adjusted Rsquared: 0.7741
Fstatistic: 141.5 on 1 and 40 DF, pvalue: 1.027e14
Table 2: Confidence intervals for coefficients of the linear regression of mpg on horsepower.
2.5 % 97.5 %
(Intercept) 43.0424051 50.810780
hp 0.2545932 0.180651
Table 3: ANOVA table for simple linear regression results of mpg on horsepower.
Df Sum Sq Mean Sq F value Pr(>F)
hp 1 1356.83 1356.83 141.53 1.027e14
Residuals 40 383.48 9.59
Figure 1: Scatterplot of mpg on horsepower for cars with the bestfit line.
R Appendix.
car < read.csv("CarMPG.csv")
lm1 < lm(mpg~hp,data=car)
summary(lm1)
confint(lm1)
anova(lm1)
fitPlot(lm1,xlab="Horsepower",ylab="MPG")