Question

5. Consider the following data (You are allowed to use built-in function in R, but the code must be attached) X 99.0 101.1 102.7 103.0 105.4 107.0 108.7 110.8 112.1 112.4 113.6 113.8 115.1 115.4 120.0 y 28.8 27.9 27.0 25.2 22.8 21.5 20.9 19.6 17.1 18.9 16.0 16.7 13.0 13.6 10.8 (a) Obtain sample statistics such as sample mean of y and x and swa = n. 2-1(;- T)?, Say = n-1 =1(xi – 7)(yi – ). (b) Obtain the equation of the regression line. (c) Create scatterplot of the data and graph the regression line. (d) Use regression line to predict the value y at x = 121. (e) What is the value of error sum of squares (SSE)? (f) Calculate residual standard error. (g) Calculate and interpret the coefficient of determination.

Answer 1

solution

Rcode:-

x=c(99,101.1,102.7,103.0,105.4,107,108.7,110.8,112.1,112.4,113.6,113.8,115.1,115.4,120)
y=c(28.8,27.9,27,25.2,22.8,21.5,20.9,19.6,17.1,18.9,16,16.7,13,13.6,10.8)
mean(x)
mean(y)
reg=lm(y~x);summary(reg)

plot(x,y)

output is

> mean(x)
[1] 109.34
> mean(y)
[1] 19.98667

Call:
lm(formula = y ~ x)

Residuals:
Min 1Q Median 3Q Max
-1.7754 -0.5727 -0.1325 0.6034 1.6818

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 118.90992 4.49912 26.43 1.10e-12 ***
x -0.90473 0.04109 -22.02 1.12e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.938 on 13 degrees of freedom
Multiple R-squared: 0.9739, Adjusted R-squared: 0.9719
F-statistic: 484.8 on 1 and 13 DF, p-value: 1.125e-11

(a)

sample mean of y is 19.98667

sample mean of x is   109.34

the Sxx is 6.101499 and Sxy is

b) the equation of regression line is  

$\widehat{Y}$=118.9099-0.9047 X

c)

O O O 20 100 105 110 115 120 X 15 25

d)the value of y at x=121 is 9.4369

e) SSE is 0.938

f)

#Residual Standard error (Like Standard Deviation)
k=length(reg$coefficients)-1;k #Subtract one to ignore intercept
SSE=sum(reg$residuals**2)
n=length(reg$residuals)
sqrt(SSE/(n-(1+k))) #Residual Standard Error

output is

> sqrt(SSE/(n-(1+k)))
[1] 0.9380352

g) coefficient of determination is 0.9739