Question

A series of questions on sports and world events was asked of 14 randomly selected young adult naturalized citizens. The results were translated into sports and world ecents "knowledge" scores. The scores were:

Citizen	Sports	World Events
J.C. McCarthy	47	49
A.N. Baker	12	10
B.B. Beebe	62	76
L.D. Gaucet	81	92
C.A. Jones	90	86
J.N. Lopez	35	42
A.F. Nissen	61	61
L.M. Zaugg	87	75
J.B. Simon	59	86
J. Goulden	40	61
A.A. Fernandez	87	18
A.M. Carbo	16	75
A.O. Smithy	50	51
J.J. Pascal	60	61

A: Determine the degree of association between how the citizens ranked with respect to knowledge of sports and how they ranked on world events.

B: At the .05 significance level, is the rank correlation between the sports and world events "knowledge" scores greater than zero?

Answer 1

CORRELATION

( X)	( Y)	X^2	Y^2	X*Y
47	49	2209	2401	2303
12	10	144	100	120
62	76	3844	5776	4712
81	92	6561	8464	7452
90	86	8100	7396	7740
35	42	1225	1764	1470
61	61	3721	3721	3721
87	75	7569	5625	6525
59	86	3481	7396	5074
40	61	1600	3721	2440
87	18	7569	324	1566
16	75	256	5625	1200

calculation procedure for correlation
sum of (x) = 787
sum of (y) = 843
sum of (x^2) = 52379
sum of (y^2) = 58635
sum of (x*y) = 50533
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ sum (x*y - N *(sum (x/N) * (sum (y/N) ]/n-1
= 50533 - [ 14 * (787/14) * (843/14) ]/14- 1
= 224.597
and now to calculate r( x,y) = 224.597/ (SQRT(1/14*50533-(1/14*787)^2) ) * ( SQRT(1/14*50533-(1/14*843)^2)
=224.597 / (24.11*23.716)
=0.393
value of correlation is =0.393

& with above we conclude that correlation ( r ) is = 0.3928> 0 ,positive correlation

HYPOTHESIS TEST

Given that,
value of r =0.393
number (n)=14
null, Ho: row(ρ) =0
alternate, H1: row(ρ)>0
level of significance, α = 0.05
from standard normal table,right tailed t α/2 =1.782
since our test is right-tailed
reject Ho, if to > 1.782
we use test statistic (t) = r / sqrt(1-r^2/(n-2))
to=0.393/(sqrt( ( 1-0.393^2 )/(14-2) )
to =1.481
|to | =1.481
critical value
the value of |t α| at los 0.05% is 1.782
we got |to| =1.481 & | t α | =1.782
make decision
hence value of |to | < | t α | and here we do not reject Ho

no evidence that rank correlation between the sports and world events "knowledge" scores greater than zero