Question

clearly state answers a,b and c please.

The accompanying data on x = head circumference z score (a comparison score with peers of the same age - a positive score suggests a larger size than for peers) at age 6 to 14 months and y- volume of cerebral grey matter (in ml) at age 2 to 5 years were read from a graph in an article Cerebral Grey Head Circumfer- Matter (ml) 2-5 yr 6-14 Months ence x Scores at 680 -0.75 690 1.2 700 -0.30 720 0.25 740 0.30 740 1.5 750 1.1 750 2.0 760 780 1.1 790 2.0 810 2.1 815 820 825 2.8 2.2 0.90 2.35 2.3 2.2 835 840 845 (a) What is the value of the correlation coefficient? (Give the answer to two decimal places.) (b) Find the equation of the least squares line. (Give the answer to two decimal places.) 9 - (c) Predict the volume of cerebral grey matter for a child whose head circumference 2 score at age 12 months was 1.8. (Give the answer to two decimal places.)

Answer 1

Solution-:

Let, X= Head cirumference Z-score at 6 -14 months and Y= Cerebral grey Matter (ml) 2-5 yr.

> X=c(-0.75,1.2,-0.30,0.25,0.30,1.5,1.1,2.0,1.1,1.1,2.0,2.1,2.8,0.90,2.35,2.3,2.2);X
[1] -0.75 1.20 -0.30 0.25 0.30 1.50 1.10 2.00 1.10 1.10 2.00 2.10
[13] 2.80 0.90 2.35 2.30 2.20
> Y=c(680,690,700,720,740,750,750,760,780,790,810,815,820,825,835,840,845);Y
[1] 680 690 700 720 740 750 750 760 780 790 810 815 820 825 835 840 845
> #(a) Correlation
> r=cor(X,Y);round(r,2)
[1] 0.8
> #Comment:Here,r=0.80 , it shows highly positive correlation.
> #(b)For regression equation
> fit=lm(Y~X);fit

Call:
lm(formula = Y ~ X)

Coefficients:
(Intercept) X
716.7 43.6

> cr=coef(fit);cr
(Intercept) X
716.72436 43.59756
> mcr=matrix(cr);mcr
[,1]
[1,] 716.72436
[2,] 43.59756
> a=mcr[1,1];a
[1] 716.7244
> round(a,2)
[1] 716.72
> b=mcr[2,1];b
[1] 43.59756
> round(b,2)
[1] 43.6
> #The equation of the line of regression of Y on X is: Y= m*X+b or y=a+b*X
> #The equation of the line of regression of Y on X is:Y=716.72+43.60*X   
> #(c) #When X=1.8 then, Y=716.72+43.60*X=716.72+43.60*1.8=795.20

R-Code:

X=c(-0.75,1.2,-0.30,0.25,0.30,1.5,1.1,2.0,1.1,1.1,2.0,2.1,2.8,0.90,2.35,2.3,2.2);X
Y=c(680,690,700,720,740,750,750,760,780,790,810,815,820,825,835,840,845);Y
#(a) Correlation
r=cor(X,Y);round(r,2)
#Comment:Here,r=0.80 , it shows highly positive correlation.
#(b)For regression equation
fit=lm(Y~X);fit
cr=coef(fit);cr
mcr=matrix(cr);mcr
a=mcr[1,1];a
round(a,2)
b=mcr[2,1];b
round(b,2)  
#The equation of the line of regression of Y on X is: Y= m*X+b or y=a+b*X
#The equation of the line of regression of Y on X is:Y=716.72+43.60*X   
#(c) #When X=1.8 then, Y=716.72+43.60*X=716.72+43.60*1.8=795.20