Question

To test whether extracurricular activity is a good predictor of college success, a college administrator records whether students participated in extracurricular activities during high school and their subsequent college freshman GPA.

Extracurricular Activity	College Freshman GPA
Yes	3.57
Yes	3.32
Yes	3.86
Yes	3.72
No	2.93
No	3.88
No	3.46
No	2.71
No	3.86
No	2.84

(a) Code the dichotomous variable and then compute a point-biserial correlation coefficient. (Round your answer to three decimal places.)

Answer 1

Y : extracurricular activities

X: College Freshman GPA

Code the dichotomous variable Y: extracurricular activities as '1' for Yes and '0' for No

By coding dichotomous variable Y: extracurricular activities we get

Y: Extracurricular	Coded : Y: Extracurricular	X: CollegeFreshman GPA
Yes	1	3.57
Yes	1	3.32
Yes	1	3.86
Yes	1	3.72
No	0	2.93
No	0	3.88
No	0	3.46
No	0	2.71
No	0	3.86
No	0	2.84

Now , dichotomous variable Y has the two values 0 and 1. If we divide the data set into two groups, group 1 which received the value "1" on Y and group 2 which received the value "0" on Y

point-biserial correlation coefficient

where s_n is the standard deviation used when data are available for every member of the population:

M₁ being the mean value on the continuous variable X for all data points in group 1, and M₀ the mean value on the continuous variable X for all data points in group 2. Further, n₁ is the number of data points in group 1, n₀ is the number of data points in group 2 and n is the total sample size.

n = 10 ; total sample size

Y: Extracurricular	Coded : Y: Extracurricular	X: CollegeFreshman GPA
Yes	1	3.57
Yes	1	3.32
Yes	1	3.86
Yes	1	3.72
No	0	2.93
No	0	3.88
No	0	3.46
No	0	2.71
No	0	3.86
No	0	2.84
	Total	34.15

	X:	(x-)	(x-)2
	3.57	0.155	0.024025
	3.32	-0.095	0.009025
	3.86	0.445	0.198025
	3.72	0.305	0.093025
	2.93	-0.485	0.235225
	3.88	0.465	0.216225
	3.46	0.045	0.002025
	2.71	-0.705	0.497025
	3.86	0.445	0.198025
	2.84	-0.575	0.330625
Total	34.15		1.80325

Group 1 for Y=1

Y: Extracurricular	Coded : Y: Extracurricular	X:
Yes	1	3.57
Yes	1	3.32
Yes	1	3.86
Yes	1	3.72
Total		14.47

n₁ is the number of data points in group 1 = 4

M₁ mean value on the continuous variable X for all data points in group 1

group 2 for value "0" on Y

Coded Y :Extracurricular	X
0	2.93
0	3.88
0	3.46
0	2.71
0	3.86
0	2.84
Total	19.68

n₀ is the number of data points in group 2 = 5

M₀ mean value on the continuous variable X for all data points in group 2

point-biserial correlation coefficient

point-biserial correlation coefficient = 0.3894