Question

During a research, the amount of Internet users was measured. Each time three random groups of 10,000 people of the average age of 20, 40, and 60 were considered. The data are as follows (Xı - the number of years since the beginning of the research, X2 - age, y - number of users): y 1300 1050 140 2900 2400 490 4350 4010 710 6090 5530 1120 X1 1 1 3 3 3 5 5 5 7 7 7 X2 20 40 60 20 40 60 20 40 60 20 40 60 1 (a) Fit a multiple linear regression model using Xi and X2 as the regressors. Round your answers to one decimal place (e.g. 98.7). V = Intl (b) Estimate 02. Round your answer to the nearest integer (e.g. 9876). (c) Find the standard errors of the regression coefficients. Round your answers to one decimal place (e.g. 98.7). se(B) = se(B) = se (B2) = (d) Use the model to predict the number of users in random group of 40 years after 4 years of research. Round your answer to the nearest integer (e.g. 9876). Number of users =

Answer 1

Solution :-

y = c(1300,1050,140,2900,2400,490,4350,4010,710,6090,5530,1120)

x1 = c(1,1,1,3,3,3,5,5,5,7,7,7)

x2 = c(20,40,60,20,40,60,20,40,60,20,40,60)

mlr = lm(y~x1+x2)

summary(mlr)


## Output 

> summary(mlr)

Call:
lm(formula = y ~ x1 + x2)

Residuals:
    Min      1Q  Median      3Q     Max 
-1566.5  -632.7   158.1   558.9  1321.0 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  3283.83     888.40   3.696  0.00495 **
x1            567.17     124.40   4.559  0.00137 **
x2            -76.13      17.03  -4.469  0.00156 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 963.6 on 9 degrees of freedom
Multiple R-squared:  0.8191,    Adjusted R-squared:  0.7789 
F-statistic: 20.38 on 2 and 9 DF,  p-value: 0.0004552

(a) y hat =  3283.83 + (567.17) * x₁ + ( -76.13 ) * x₂

(b)  Residual standard error : 963.6 on 9 degrees of freedom

SSRes = 963.6

=> sigma² hat = SSRes / df = 963.6 / 9 = 107.066666667

So, sigma² hat = 107

(c) se(intercept) = 888.40

se( beta₁ hat ) = 124.40

se( beta₂ hat ) = 17.03

(d) Given that : x₁ = 4 ; x₂ = 40

=> Number of users = 3283.83 + (567.17) * 4  + ( -76.13 ) * 40 = 2507.31