Questions

Data

library(car)
pres<-Prestige
head(pres)
##                     education income women prestige census type
## gov.administrators      13.11  12351 11.16     68.8   1113 prof
## general.managers        12.26  25879  4.02     69.1   1130 prof
## accountants             12.77   9271 15.70     63.4   1171 prof
## purchasing.officers     11.42   8865  9.11     56.8   1175 prof
## chemists                14.62   8403 11.68     73.5   2111 prof
## physicists              15.64  11030  5.13     77.6   2113 prof

1. Is at least one of the predictors X1 , X2 , . . . , Xp useful in predicting the response?

Yes. The census, education, income and type were good explanaitors of the prestige.

2. Do all the predictors help to explain Y, or is only a subset of the predictors useful?

Only a subset mentioned above.

3. How well does the model fit the data?

It does well. \(R^2 = 0.8407\)

pres.lm6<-lm(pres$prestige~pres$education+pres$income+pres$census+pres$type)
summary(pres.lm6)
## 
## Call:
## lm(formula = pres$prestige ~ pres$education + pres$income + pres$census + 
##     pres$type)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.0873  -4.9935   0.7435   4.9617  19.4891 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    -1.144e+01  7.823e+00  -1.462   0.1472    
## pres$education  3.947e+00  6.498e-01   6.075 2.76e-08 ***
## pres$income     9.365e-04  2.221e-04   4.217 5.79e-05 ***
## pres$census     1.125e-03  6.113e-04   1.840   0.0691 .  
## pres$typeprof   1.091e+01  4.645e+00   2.348   0.0210 *  
## pres$typewc     5.605e-01  3.062e+00   0.183   0.8551    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.006 on 92 degrees of freedom
##   (4 observations deleted due to missingness)
## Multiple R-squared:  0.8407, Adjusted R-squared:  0.8321 
## F-statistic: 97.12 on 5 and 92 DF,  p-value: < 2.2e-16

And the Akaike Information Critireon (AIC) of this model is the lowest. ###4. Given a set of predictor values, what response value should we predict, and how accurate is our prediction?###

\(prestige = -1.144e+01 + 3.947e+00\text{ education} + 9.365e-04\text{ income} + 1.125e-03\text{ census} + 1.091e+01\text{ typeprof} + 5.605e-01\text{ typewc}\)

plot(fitted(pres.lm6), residuals(pres.lm6))
abline(a=0, b=0)

The standard error will be 7.006.

R