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	3.47
Yes	3.34
Yes	3.93
Yes	3.67
No	2.92
No	3.88
No	3.47
No	2.71
No	3.83
No	2.79

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


(b) Using a two-tailed test at a 0.05 level of significance, state the decision to retain or reject the null hypothesis. Hint: You must first convert r to a t-statistic.

Retain the null hypothesis

or

Reject the null hypothesis.    

Answer 1

When Y is continuous and X is dichotomous. In this case, Pearson r, computed in the usual fashion, becomes the point-biserial correlation coefficient.

using ms-excel the correlation coefficient between x and y is 0.381 , here Yes=1 and No=0

( correlation coefficient between x and y is -10.381 , if we take Yes=0 and No=1)

Activity(x)	Freshman GPA (y)	Activity(x)
Yes	3.47	1
Yes	3.34	1
Yes	3.93	1
Yes	3.67	1
No	2.92	0
No	3.88	0
No	3.47	0
No	2.71	0
No	3.83	0
No	2.79	0

(b) Retain the null hypothesis ( that population correlation coefficient is zero)

null hypothesis H0: $\rho$ =0 and alternate hypothesis H0: $\rho\ne$ 0

we use t-test here and statistic t =r/sqrt[(1—r²)/(n—2)]=0.382/sqrt(1-0.381*0.381)/(10-1))=0.0459 with n-1=10-1=9 df

the two tailed critical t(0.05,9)=2.2622 is more than calculated t=0.0459, so we fail to reject ( or accept ) null hypothesis.