Question

The following is a set of data from a sample of n=11 items. Complete parts​ (a) through (c):

X Y

34 17

16 8

6 3

40 20

14 7

30 15

8 4

38 19

36 18

36 18

8 4

a. Compute the sample covariance.

​(Round to three decimal places as​ needed.)

b. Compute the coefficient of correlation.r=

​(Round to three decimal places as​ needed.)

c. How strong is the relationship between X and​ Y? Explain.

A. The variables X and Y have a strong positive correlation because as X​ increases, Y tends to increase also.

B.The variables X and Y have a perfect negative correlation because all points fall on a straight line with a negative slope.

C.The variables X and Y have a perfect positive correlation because all points fall on a straight line with a positive slope.

D.The variables X and Y have no correlation.

Answer 1

a) The sample covariance is

b) The coefficient of correlation is

Here

Let

and ,  
and so on

Using the bove formula ,

.

Verify using R command below.

X <- c(34, 16 , 6 , 40 , 14, 30 , 8 ,38 ,36 , 36, 8 )
Y <- c( 17 , 8 , 3 , 20 , 7 , 15 , 4 , 19 , 18, 18 , 4)
cov(X,Y)
cor(X,Y)

> cov(X,Y)
[1] 93.78182
> cor(X,Y)
[1] 1

c) Since , The variables X and Y have a perfect positive correlation because all points fall on a straight line with a positive slope.

Correct choice is (C).