Question

Extracurricular Activit Yes Yes Yes Yes No No No No No No College Freshman GPA 3.56 3.34 3.95 3.72 2.93 3.80 3.40 2.78 3.84 2.78 (a) Code the dichotomous variable and then compute a point-biserial correlation coefficient. (Round your answer to three (b) Using a two-tailed test at a 0.05 level of significance, state the decision to retain or reject the null hypothesis. Hint: Yo O Retain the null hypothesis. O Reject the null hypothesis You may need to use the appropriate table in Appendix C to answer this question Additional Materials eBook 11. 1/2 points | Previous Answers PriviteraStats3 15.E.031 A therapist measures the relationship between a patient's expectations that therapy will be successful and the actual success of th Therapy Successful No Totals Yes 24 4 Yes Client expects No success Totals 7 15 28 22 31 19 (a) Compute the phi correlation coefficient. (Round your answer to three decimal places.)

Answer 1

First let us define point-biserial correlation and phi correlation:

1: Point-Biserial Correlation: The point-biserial correlation, r_pbi, is a special case of Pearson's correlation coefficient. It measures the relationship between two variables:

(a) One continuous variable (must be ratio or interval scale).

(b) One naturally binary variable.

Formula :

r_pb=(M₁-M₀)/S_n*$\sqrt{pq}$

• where M₁= mean of the group that received binary variable "1"
• M₀= mean of the group that received binary variable "0"
• S_n= Standard deviation for the entire test
• p= Proportion of the cases in the "0" group
• q= Proportion of the cases in the "1" group

2: Phi correlation: The Phi correlation is measure of association between two binary variables. It is also called Yule phi or mean contingency coefficient.

$\phi=(ad-bc)/\sqrt{[(a+b)(c+d)(a+c)(b+d)]}$

Solution:

1: Let us code our dichotomous variable as  Yes=1 and No=0.

M₁= mean of group that received yes= 3.6425

M0= mean of group that received No= 3.2483

S_n= Standard deviation for the entire test=$\sqrt{\sum (x_{i}-\bar{x})^{2}/(n-1)}$=0.440131

p= Proportion of the cases in the "0/No" group = 6/10=0.6

q= Proportion of the cases in the "1" group= 4/10=0.4

r_pb=(M₁-M₀)/S_n*$\sqrt{pq}$=(3.6425-3.2483)*$\sqrt{0.6*0.4}$/0.440131=0.43877339 = 0.4388

2: Let us look at our contingency table first:

Therapy Successful Yes No Totals 7 4 15 19 Yes 24 31 Client expects No success Totals 28 22

In our case, a=24 , b=7 , c=4, d=15

$\phi=(ad-bc)/\sqrt{[(a+b)(c+d)(a+c)(b+d)]}$

=(24*15-7*4)/$\sqrt{(24+7)*(4+15)*(24+4)*(7+15)}$

=(332)/$\sqrt{362824}$ = 0.551