Question

In a survey of delinquent peers, a small sample of adolescents is asked to rate the overall delinquency of their closest friends on a scale of 1 to 10. Later, the same survey asks the respondents to self-report their delinquency rating on the same scale. The data for twelve subjects are as follows (high scores signify high levels of delinquency):                                               

Respondent	Delinquent Peer Rating	Self-Report Rating
A	10	8
B	10	9
C	9	8
D	9	6
E	8	9
F	8	3
G	7	7
H	7	2
I	6	3
J	6	4
K	5	2
L	5	3

Compute the Pearson's r and test the significance of the obtained correlation (use α .05) using Table H of Appendix C.

Answer 1

Answer: The Pearson's r is given by r = .

It comes out to be r =  0.7715 (rounded to 4 decimal places)

TO test its significance, we construct our null and alternative hypothesis to be:

H0: rho = 0 vs Ha: rho not equal to 0.

Where rho is the unknown population correlation coefficient between x and y.

The test statistic for the given test is given by T = (r * sqrt(n-2))/sqrt(1-(r*r)), where n = sample size, r = sample correlation coefficient. Sqrt refers to square root function.

We reject H0 if |T(observed)| > t(alpha/2, (n-2)). Where |T(observed)| is the absolute value of the observed test statistic, t(alpha/2,(n-2)) is the upper alpha/2 point of the t-distribution with (n-2) degrees of freedom. Alpha is the level of significance.

n=12. r is calculated as above.

Here |Tobserved| =  3.835123 and t(alpha/2, (n-2)) = 2.228139. (Obtained from the inverse t-table).

Thus we see that |T(observed)| > t(alpha/2, (n-2)). So reject H0 and conclude on the basis of the given sample at a 5% level of significance that there is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.