Question

 

Below are a set of heights (in inches) and GPA scores for a sample of 6 students.
Height, GPA
60, 4.0
55, 3.2
62, 3.7
58, 3.9
49, 2.4
61, 2.8

a) Find the correlation coefficient (r) for these two variables by hand

b) Find the mean and standard deviation for each variable.

c) Find the equation of the regression line to predict GPA from height by hand

d) Based on this data, what is the GPA prediction for a student who is 56 inches tall?

e) How good is this prediction?

Answer 1

For this problem a set of Heights (in inches) and GPA scores for a sample of 6 students is given.below,

Height(in inches)	GPA
60	4.0
55	3.2
62	3.7
58	3.9
49	2.4
61	2.8
Total = 345	Total = 20

a) Now we find the correlation coefficient (r) for these two variable Height (X=independent variable) and GPA (Y=dependent variable). Now correlation between Height (X=independent variable) and GPA (Y=dependent variable) is defined as,

   

TABLE FOR CALCULATION:-

Xi(Height)	Yi(GPA)	Xi-	Yi-	(Xi-)(Yi-)	(Xi-)2	(Yi-)2
60	4.0	2.5	0.6667	1.66675	6.25	0.444489
55	3.2	-2.5	-0.1333	0.33325	6.25	0.017769
62	3.7	4.5	0.3667	1.65015	20.25	0.134469
58	3.9	0.5	0.5667	0.28335	0.25	0.321149
49	2.4	-8.5	-0.9333	7.93305	72.25	0.871049
61	2.8	3.5	-0.5333	-1.86655	12.25	0.284409
Total = 345	Total = 20	Total = 0	Total = 0.0002	Total = 10	Total = 117.5	Total = 2.073333

c) Now we want to find the regression line to predict GPA i.e. we want to fine the regression line of GPA on Height. Which is linear regression line and given by in the form ,

=> GPA = a + b Height

where a and b are estimated by,

d) Based on this data GPA score of a student who is 56 inches tall is given by the regression line of GPA on Height,

GPA = -1.55995 + 0.0851 Height

GPA = -1.55995 + 0.0851 56 = 3.20565

So, GPA score of a student who is 56 inches tall is 3.20565

e) In general from common sense we can say that Height is not a good predictor of GPA. But based on this data, Correlation coefficient between Height and GPA is 0.6407 which is moderately high so according to this correlation coefficient height is good predictor of GPA.