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	12.3	28.4	24.5	14.3	7.5	27.7	16.2	20.1
y	22	77	39	23	15	30	15	25

(a) Use a calculator to verify that Σx = 151.0, Σx² = 3252.58, Σy = 246, Σy² = 10438 and Σxy = 5430.8.

Compute r. (Round to 3 decimal places.)

(b) Compute the sample correlation coefficient r for each of the following data sets and show that r is the same for both. (Use 3 decimal places.)

(i)	x	3	7	9
	y	1	2	5

(ii)	x	1	2	5
	y	3	7	9

	r
(i)
(ii)

Answer 1

a)

S.No	X	Y	X12	Y12	XY
1	12.3	22	151.29	484.0000	270.60
2	28.4	77	806.56	5929.0000	2186.80
3	24.5	39	600.25	1521.0000	955.50
4	14.3	23	204.49	529.0000	328.90
5	7.5	15	56.25	225.0000	112.50
6	27.7	30	767.29	900.0000	831.00
7	16.2	15	262.44	225.0000	243.00
8	20.1	25	404.01	625.0000	502.50
	ΣX	207.106	ΣX2	ΣY2	ΣXY
total	151.0	246.0000	3252.6	10438.00	5430.8

ΣX =	151.000
ΣY=	246.000
ΣX2 =	3252.580
ΣY2 =	10438.000
ΣXY =	5430.800

	sx=(√(Σx2-(Σx)2/n)/(n-1))=		7.5825
	sy=(√(Σy2-(Σy)2/n)/(n-1))=		20.2608
	Cov=sxy=(ΣXY-(ΣXΣY)/n)/(n-1)=		112.5071
	r=Cov/(Sx*Sy)=		0.732

b)

i)

S.No	X	Y	(x-x̅)2	(y-y̅)2	(x-x̅)(y-y̅)
1	3	1	11.1111	2.78	5.5556
2	7	2	0.4444	0.44	-0.4444
3	9	5	7.1111	5.44	6.2222
Total	19	8	18.6667	8.67	11.3333
Mean	6.333	2.67	SSX	SSY	SXY

correlation coefficient r=

Sxy/(√Sxx*Syy) =

0.891

ii)

S.No	X	Y	(x-x̅)2	(y-y̅)2	(x-x̅)(y-y̅)
1	1	3	2.7778	11.11	5.5556
2	2	7	0.4444	0.44	-0.4444
3	5	9	5.4444	7.11	6.2222
Total	8	19	8.6667	18.67	11.3333
Mean	2.667	6.33	SSX	SSY	SXY

correlation coefficient r=

Sxy/(√Sxx*Syy) =

0.891