Question

The questions involve the data set for asking prices of Richmond townhouses obtained on 2014.11.03.
For your subset, the response variable is:
asking price divided by 10000:
askpr=c(53.9, 50.8, 46.8, 48.8, 62.9, 68.8, 62.8888, 58.68, 54.8, 79.99, 55.8, 60.8, 73.8, 56.88, 25.9, 47.9, 65.8, 45.99, 59.8, 50.8, 57.8, 55.2, 54.8, 52.4, 56.8, 81.9, 40.8, 48.5, 51.68, 58.8, 40.8, 33.7, 68.5, 53.8, 41.99, 58.39, 68.5, 73.9, 79.8, 26.99, 68.8, 47.8, 78.8, 71.99, 57.5, 54.98, 77.8, 57.8, 53.8, 74.8)
The explanatory variables are:
(i) finished floor area divided by 100
ffarea=c(11.84, 16.6, 16.2, 14.8, 14, 16.9, 15.77, 13.96, 11.26, 22, 13.06, 13.2, 17.54, 15.78, 6.1, 12.1, 13.45, 16.01, 17.63, 12.27, 13.84, 15.3, 15.46, 16.22, 15.5, 20.95, 14, 14.8, 15.1, 17.37, 12.26, 12, 13.59, 10.95, 12.9, 15.09, 15.76, 15.15, 15.25, 10.5, 15.95, 13.34, 19.48, 15.05, 13.46, 13.06, 16.5, 12.01, 12.22, 17.48)
(ii) age
age=c(15, 23, 30, 50, 5, 8, 6, 9, 0, 20, 0, 3, 9, 17, 11, 7, 1, 25, 26, 17, 10, 9, 41, 25, 23, 19, 38, 24, 20, 26, 29, 28, 2, 18, 44, 8, 4, 0, 3, 37, 18, 32, 11, 8, 10, 1, 3, 0, 9, 5)
(iii) monthly maintenance fee divided by 10
mfee=c(21, 19.9, 16, 25, 19.6, 19.4, 35.7, 22, 24.8, 26.7, 18.6, 18.9, 18.2, 17.3, 17.1, 18, 18.2, 33.7, 32, 25.2, 16, 16.9, 31, 36.4, 17.4, 34.8, 23, 16.1, 24.5, 31, 19.8, 25.9, 17, 24.7, 23.2, 20.3, 22.1, 22.2, 35, 28, 23.6, 24.5, 20.4, 22.3, 22.1, 19.6, 25.4, 14.2, 18.5, 29.7)
(iv) number of bedrooms
beds=c(2, 4, 4, 3, 3, 4, 3, 3, 2, 3, 3, 3, 4, 4, 1, 3, 3, 3, 5, 2, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 2, 3, 2, 3, 4, 4, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4)
You are to make a prediction of the response variable when ffarea=16, age=6, mfee=26, beds=4.

You are to fit three multiple regression models with the response variable askpr:
(i) 2 explanatory variables ffarea, age
(ii) 3 explanatory variables ffarea, age, mfee
(iii) 4 explanatory variables ffarea, age, mfee, beds
After you have copied the above R vectors into your R session, you can get a dataframe with
richmondtownh=data.frame(cbind(askpr,ffarea,age,mfee,beds))

Please use 3 decimal places for the answers below which are not integer-valued

Part a) The values of adjusted R2 for the above models with 2, 3 and 4 explanatory variables are respectively: 2 explanatory: 3 explanatory: 4 explanatory: Part b) For the best of these 3 models based on adjusted R2, the number of explanatory variables is: Part c) For the best of these 3 models based on adjusted R2, the least squares coefficient for ffarea is and a 95% confidence interval for Brarea is to Part d) For the best of these 3 models based on adjusted R2 , get the prediction, SE and 95% prediction interval when the future values of the explanatory variables are: ffarea-16, age-6, mfee-26, beds-4. prediction and the upper endpoint of the 95% prediction interval is and its SE

Answer 1

Answer:

Part a.

Adjusted R square for

2 explanatory 0.805

3 explanatory 0.805

4 explanatory 0.805

Part b.

Number of explanatory variables: 2

Part c.

Coefficient for ffarea = 3.519

95% CI = (2.907, 4.131)

Part d:

Prediction: 68.104      SE= 5.727

Upper endpoint of 95% prediction: 79.859

Rcode:

askpr=c(53.9, 50.8, 46.8, 48.8, 62.9, 68.8, 62.8888, 58.68, 54.8, 79.99, 55.8, 60.8, 73.8, 56.88, 25.9, 47.9, 65.8, 45.99, 59.8, 50.8, 57.8, 55.2, 54.8, 52.4, 56.8, 81.9, 40.8, 48.5, 51.68, 58.8, 40.8, 33.7, 68.5, 53.8, 41.99, 58.39, 68.5, 73.9, 79.8, 26.99, 68.8, 47.8, 78.8, 71.99, 57.5, 54.98, 77.8, 57.8, 53.8, 74.8)

ffarea=c(11.84, 16.6, 16.2, 14.8, 14, 16.9, 15.77, 13.96, 11.26, 22, 13.06, 13.2, 17.54, 15.78, 6.1, 12.1, 13.45, 16.01, 17.63, 12.27, 13.84, 15.3, 15.46, 16.22, 15.5, 20.95, 14, 14.8, 15.1, 17.37, 12.26, 12, 13.59, 10.95, 12.9, 15.09, 15.76, 15.15, 15.25, 10.5, 15.95, 13.34, 19.48, 15.05, 13.46, 13.06, 16.5, 12.01, 12.22, 17.48)

age=c(15, 23, 30, 50, 5, 8, 6, 9, 0, 20, 0, 3, 9, 17, 11, 7, 1, 25, 26, 17, 10, 9, 41, 25, 23, 19, 38, 24, 20, 26, 29, 28, 2, 18, 44, 8, 4, 0, 3, 37, 18, 32, 11, 8, 10, 1, 3, 0, 9, 5)

mfee=c(21, 19.9, 16, 25, 19.6, 19.4, 35.7, 22, 24.8, 26.7, 18.6, 18.9, 18.2, 17.3, 17.1, 18, 18.2, 33.7, 32, 25.2, 16, 16.9, 31, 36.4, 17.4, 34.8, 23, 16.1, 24.5, 31, 19.8, 25.9, 17, 24.7, 23.2, 20.3, 22.1, 22.2, 35, 28, 23.6, 24.5, 20.4, 22.3, 22.1, 19.6, 25.4, 14.2, 18.5, 29.7)

beds=c(2, 4, 4, 3, 3, 4, 3, 3, 2, 3, 3, 3, 4, 4, 1, 3, 3, 3, 5, 2, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 2, 3, 2, 3, 4, 4, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4)

richmondtownh=data.frame(cbind(askpr,ffarea,age,mfee,beds))

Model1 <- lm(askpr ~ ffarea+age, data= richmondtownh)

summary(Model1)

Model2 <- lm(askpr ~ ffarea+age+mfee, data= richmondtownh)

summary(Model2)

Model3 <- lm(askpr ~ ffarea+age+mfee+beds, data= richmondtownh)

summary(Model3)

confint(Model1, 'ffarea', level=0.95)

newdata = data.frame(ffarea=16,age=6)

predict(Model1, newdata, interval="prediction",conf.level=.95)

R output:

Call:

lm(formula = askpr ~ ffarea + age, data = richmondtownh)

Residuals:

     Min       1Q   Median       3Q      Max

-11.1668 -4.4598 -0.0047   3.8634 12.6005

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 15.2711     4.5777   3.336 0.00167 **

ffarea        3.5189     0.3044 11.560 2.43e-15 ***

age          -0.5780     0.0639 -9.046 7.35e-12 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.735 on 47 degrees of freedom

Multiple R-squared: 0.8128,    Adjusted R-squared: 0.8048

F-statistic:   102 on 2 and 47 DF, p-value: < 2.2e-16

>

> Model2 <- lm(askpr ~ ffarea+age+mfee, data= richmondtownh)

> summary(Model2)

Call:

lm(formula = askpr ~ ffarea + age + mfee, data = richmondtownh)

Residuals:

     Min       1Q   Median       3Q      Max

-12.5179 -4.6707   0.0602   3.7629 10.7233

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 13.59960    4.86405   2.796 0.00753 **

ffarea       3.40351    0.32486 10.477 9.06e-14 ***

age         -0.59727    0.06664 -8.963 1.17e-11 ***

mfee         0.15875    0.15654   1.014 0.31585   

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.733 on 46 degrees of freedom

Multiple R-squared: 0.8169,    Adjusted R-squared: 0.8049

F-statistic: 68.39 on 3 and 46 DF, p-value: < 2.2e-16

> Model3 <- lm(askpr ~ ffarea+age+mfee+beds, data= richmondtownh)

> summary(Model3)

Call:

lm(formula = askpr ~ ffarea + age + mfee + beds, data = richmondtownh)

Residuals:

    Min      1Q Median      3Q     Max

-12.150 -4.676   0.081   3.869 10.782

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 15.94889   5.35275   2.980 0.00464 **

ffarea       3.62634    0.38817   9.342 4.19e-12 ***

age         -0.59718    0.06657 -8.971 1.39e-11 ***

mfee         0.09248    0.16872   0.548 0.58633   

beds        -1.35115    1.29136 -1.046 0.30101   

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.727 on 45 degrees of freedom

Multiple R-squared: 0.8212,    Adjusted R-squared: 0.8053

F-statistic: 51.67 on 4 and 45 DF, p-value: 2.933e-16

confint(Model1, 'ffarea', level=0.95)

          2.5 %   97.5 %

ffarea 2.906502 4.131211

>

> newdata = data.frame(ffarea=16,age=6)

> predict(Model1, newdata, interval="prediction",conf.level=.95)

       fit      lwr      upr

1 68.10448 56.34961 79.85935

>