Question

1.) Researchers wondered whether the size of a​ person's brain was related to the​ individual's mental capacity. They selected a sample of 5 females and 5 males and measured their MRI image pixel counts and IQ scores. The data is reported to the right. Complete parts ​(a) through ​(d) below.

Females		Males
MRI	IQ	MRI	IQ
833, 868	132	965,353	133
951,545	137	1,079,550	141
790,619	135	949,395	140
866,662	130	1,038,438	139
852,244	132	935,494	141

​(a) Draw a scatter diagram treating MRI count as the explanatory variable and IQ as the response variable. Choose the correct diagram below.

​(b) Compute the linear correlation coefficient between MRI count and IQ. Are MRI count and IQ linearly​ related? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice

​(c) Draw a scatter​ diagram, but use a different plotting symbol for each gender. Choose the correct diagram below.

​(d) Compute the linear correlation coefficient between MRI count and IQ for females. Compute the linear correlation coefficient between MRI count and IQ for males.

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)
Gender MRI IQ X
1 Female 833868 132 NA
2 Female 951545 137 NA
3 Female 790619 135 NA
4 Female 866662 130 NA
5 Female 852244 132 NA
6 Male 965353 133 NA
> attach(d)
The following objects are masked from d (pos = 3):

Gender, IQ, MRI, X

> plot(MRI,IQ)
> cor.test(MRI,IQ)

Pearson's product-moment correlation

data: MRI and IQ
t = 2.6234, df = 8, p-value = 0.03049
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.08814668 0.91702092
sample estimates:
cor
0.6800316

> plot(MRI[Gender=='Female'],IQ[Gender=='Female'], xlab='MRI', ylab='IQ',
+ xlim=c(800000,1080000), ylim=c(130,145),col='blue', pch=8)
> par(new=TRUE)
> plot(MRI[Gender=='Male'],IQ[Gender=='Male'], xlab='MRI', ylab='IQ',
+ xlim=c(800000,1080000), ylim=c(130,145), col='red', pch=16)
> cor(MRI[Gender=='Female'],IQ[Gender=='Female'])
[1] 0.3693489
> cor(MRI[Gender=='Male'],IQ[Gender=='Male'])
[1] 0.2170374
>

Que.a

Que.b

The linear correlation coefficient between MRI count and IQ is 0.6800316 .

Since p-value for correlation test is 0.03049, whixh is less than 0.05, hence we conclude that there is significant relationship between MRI count and IQ.

Que.c

Que.d

The linear correlation coefficient between MRI count and IQ for females is 0.3693489

The linear correlation coefficient between MRI count and IQ for males is 0.2170374