At the 0.05 level of significance, determine whether Land makes a significant co

Question

At the 0.05 level of significance, determine whether Land makes a significant contribution to the regression model.

Select one:

a. There is evidence that the variable Land contributes to a model already containing Age.

b. There is no evidence that the variable Land contributes to a model already containing Age.

Explain your determination using the p-value of the slope coefficient.

Explain your determination using the 95% interval explanation (hint: your answer should mention the number 0.)

At the 0.05 level of significance, determine whether Age makes a significant contribution to the regression model.

Which variables should be included in the final model?

Select one:

a. Land

b. Age

c. Both Land and Age

d. Neither Land nor Age

APPraise TruPerty Sae 9 Sycamore Road 4660 02297 46 21 Jefferson St 3640 51 0.2192 29 38 Hiding Post Lane 429.0 0.1630 4 Poppy Lane 5484 4608 18 5Damid Drive 405.9 02549 46 15 Frands Terrace 341 0.2290 88 48 23 Gufoy Street 315.0 01808 7 05015 17 Carlyle Drive 749.7 8 Craft Avenue 2177 02229 52

Explanation / Answer

I am using R software to solve this.

At first we need to load the data into R as below:

Appraised <- c(466,364,429,548.4,405.9,374.1,315,749.7,217.7)
Land <- c(0.2297,0.2192,0.1630,0.4608,0.2549,0.2290,0.1808,0.5015,0.2229)
Age <- c(46,51,29,18,46,88,48,7,52)
Data <- data.frame(Appraised,Land,Age)

First, lets fit the model only with Age variable.

model_Age <- lm(Appraised~Age,Data)
summary(model_Age)

Call:
lm(formula = Appraised ~ Age, data = Data)

Residuals:
Min 1Q Median 3Q Max
-169.786 -64.460 -9.231 50.869 154.873

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 627.080 83.742 7.488 0.000139 ***
Age -4.608 1.742 -2.645 0.033181 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 114.6 on 7 degrees of freedom
Multiple R-squared: 0.4999,   Adjusted R-squared: 0.4284
F-statistic: 6.996 on 1 and 7 DF, p-value: 0.03318

Now lets add Land variable to the model

model_Age_Land <- lm(Appraised~Age+Land,Data)
summary(model_Age_Land)

Call:
lm(formula = Appraised ~ Age + Land, data = Data)

Residuals:
Min 1Q Median 3Q Max
-154.708 -31.338 -3.212 64.084 76.966

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 291.956 162.875 1.793 0.1232
Age -1.868 1.842 -1.014 0.3497
Land 796.741 353.423 2.254 0.0651 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 91.1 on 6 degrees of freedom
Multiple R-squared: 0.7292,   Adjusted R-squared: 0.639
F-statistic: 8.079 on 2 and 6 DF, p-value: 0.01985

After including Land variable, the Adjusted r squared has increased from 0.4284 to 0.639, though p-value of land variable is 0.0651 which is greater than 0.05 significance level. So there is no evidence that the variable Land contributes to a model already containing Age based on the p-value of slope coefficient.

95% confidence interval for coefficients can be determined using confint function.

confint(model_Age_Land)

2.5 % 97.5 %
(Intercept) -106.583719 690.496481
Age -6.375626 2.639394
Land -68.054488 1661.536646

The confidence interval includes 0. This indicates this model is no good.

Yes, at 0.05 level of significance, age does make a significant contribution, as its o value is 0.033 as shown above.

Lets fit the model only with Land variable:

model_Land <- lm(Appraised~Land,Data)
summary(model_Land)

Call:
lm(formula = Appraised ~ Land, data = Data)

Residuals:
Min 1Q Median 3Q Max
-159.965 -19.168 -9.842 81.310 113.222

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 147.37 78.91 1.868 0.10403   
Land 1033.17 266.16 3.882 0.00604 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 91.28 on 7 degrees of freedom
Multiple R-squared: 0.6828,   Adjusted R-squared: 0.6375
F-statistic: 15.07 on 1 and 7 DF, p-value: 0.006039

This model gives better results than Age only model.

So only Land variable should be included in the final model

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