Question

Do larger universities tend to have more property crime? University crime statistics are affected by a variety of factors. The surrounding community, accessibility given to outside visitors, and many other factors influence crime rate. Let x be a variable that represents student enrollment (in thousands) on a university campus, and let y be a variable that represents the number of burglaries in a year on the university campus. A random sample of n = 8 universities in California gave the following information about enrollments and annual burglary incidents.

x	10.7	28.0	24.5	14.3	7.5	27.7	16.2	20.1
y	22	74	39	23	15	30	15	25

(b) Use a calculator to verify that Σx = 149.0, Σx² = 3193.22, Σy = 243, Σy² = 9985 and Σxy = 5280.8.

Compute r. (Round to 3 decimal places.)

Answer 1

b.

( X)	( Y)	X^2	Y^2	X*Y
10.7	22	114.49	484	235.4
28	74	784	5476	2072
24.5	39	600.25	1521	955.5
14.3	23	204.49	529	328.9
7.5	15	56.25	225	112.5
27.7	30	767.29	900	831
16.2	15	262.44	225	243
20.1	25	404.01	625	502.5

calculation procedure for correlation
sum of (x) = ∑x = 149
sum of (y) = ∑y = 243
sum of (x^2)= ∑x^2 = 3193.22
sum of (y^2)= ∑y^2 = 9985
sum of (x*y)= ∑x*y = 5280.8
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ ∑x*y - N *(∑x/N) * (∑y/N) ]/n-1
= 5280.8 - [ 8 * (149/8) * (243/8) ]/8- 1
= 94.366
and now to calculate r( x,y) = 94.366/ (SQRT(1/8*5280.8-(1/8*149)^2) ) * ( SQRT(1/8*5280.8-(1/8*243)^2)
=94.366 / (7.229*18.041)
=0.724
value of correlation is =0.724
coeffcient of determination = r^2 = 0.523
properties of correlation
1. If r = 1 Correlation is called Perfect Positive Correlation
2. If r = -1 Correlation is called Perfect Negative Correlation
3. If r = 0 Correlation is called Zero Correlation
& with above we conclude that correlation ( r ) is = 0.7236> 0 ,perfect positive correlation              

Answer:
r value = 0.724