Question

1. A researcher would like to predict the dependent variable YY from the two independent variables X1X1 and X2X2 for a sample of N=15N=15 subjects. Use multiple linear regression to calculate the coefficient of multiple determination and test the significance of the overall regression model. Use a significance level α=0.05α=0.05.

X1X1	X2X2	YY
66.4	76.4	58
34.6	39	65.5
32.7	23.1	65.8
44.4	71.2	73.3
57.3	50.8	57.9
32.7	48	74.6
53.3	51.4	64.4
48.3	51.1	59.2
66.9	81.4	59.4
31.2	40.1	59.1
44.6	33.1	54.8
45.5	49.7	65.7
30.5	9.9	44.4
62.8	75.3	66.7
56.1	61	51.2

SSreg=
SSres=
R2=
F=
P-value =

2.

Complete the missing information for this regression model. Note:  N=23N=23.

ˆYY^	=	29.692	+	2.311X12.311X1	−-	1.374X21.374X2	+	X3X3
		2.195		1.005		1.016		1.415	Standard Errors
		13.527		2.3					t-ratios
								0.059	P-values

(Except for P-values, report all values accurate to 3 decimal places. For P-values, report accurate to 4 decimal places.)

Answer 1

In order to solve this question I used R software.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')
> head(d)
    X1   X2    Y
1 66.4 76.4 58.0
2 34.6 39.0 65.5
3 32.7 23.1 65.8
4 44.4 71.2 73.3
5 57.3 50.8 57.9
6 32.7 48.0 74.6
> attach(d)
The following objects are masked from d (pos = 3):

    X1, X2, Y

> fit=lm(Y~X1+X2)
> summary(fit)

Call:
lm(formula = Y ~ X1 + X2)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.9976 -3.0325  0.4088  3.9166  8.1187 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  68.8802     6.0662  11.355 8.93e-08 ***
X1           -0.6906     0.2119  -3.259  0.00684 ** 
X2            0.4928     0.1358   3.630  0.00345 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.89 on 12 degrees of freedom
Multiple R-squared:  0.528,     Adjusted R-squared:  0.4493 
F-statistic: 6.711 on 2 and 12 DF,  p-value: 0.01106

> anova(fit)
Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value   Pr(>F)   
X1         1   8.57    8.57  0.2469 0.628242   
X2         1 457.15  457.15 13.1757 0.003452 **
Residuals 12 416.36   34.70                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 

SSReg =SS(X1) +SS(X2) = 8.57 + 457.15 = 465.72

SSres = 416.36

R2 = 0.528

F = 6.711 [ It is in the output of summary(fit) command ]

P-value = 0.01106