3 a) > model=lm(taste~Acetic+H2S+Lactic, data=cheddar)
> summary(model) lm(formula = taste ~ Acetic + H2S + Lactic, data = cheddar)
Residuals:
Min 1Q Median 3Q Max
-17.390 -6.612 -1.009 4.908 25.449
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -28.8768 19.7354 -1.463 0.15540
Acetic 0.3277 4.4598 0.073 0.94198
H2S 3.9118 1.2484 3.133 0.00425 **
Lactic 19.6705 8.6291 2.280 0.03108 *
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 10.13 on 26 degrees of freedom
Multiple R-squared: 0.6518, Adjusted R-squared: 0.6116
F-statistic: 16.22 on 3 and 26 DF, p-value: 3.81e-06
b) β0̂≈28.8768 β ̂Acetic≈0.3277 β ̂H2S≈3.9118 β ̂Lactic≈19.6705
c) P-value of H2S and Lactic is 0.00425 and 0.03108, less than 0.05, so both of them are statistically significant.
d) > cor(fitted(model),cheddar$taste)
[1] 0.8073256
> cor(fitted(model),cheddar$taste)^2
[1] 0.6517747
So the result approximately closes to the Multiple R-squared 0.6518.
e) Set another model like follow:
> model2=lm(taste~0+Acetic+H2S+Lactic, data=cheddar)
> summary(model2)

lm(formula = taste ~ 0 + Acetic + H2S + Lactic, data = cheddar)
Residuals:
Min 1Q Median 3Q Max
-15.4521 -6.5262 -0.6388 4.6811 28.4744
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Acetic -5.454 2.111 -2.583 0.01553 *
H2S 4.576 1.187 3.854 0.00065 ***
Lactic 19.127 8.801 2.173 0.03871 *
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 10.34 on 27 degrees of freedom
Multiple R-squared: 0.8877, Adjusted R-squared: 0.8752
F-statistic: 71.15 on 3 and 27 DF, p-value: 6.099e-13
> cor(fitted(model2),cheddar$taste)
[1]