Question

A record of violent crimes for a random sample of cities from the list of U.S. cities with the fewest crimes is shown below. At significance level 0.05, is there sufficient evidence to indicate a relationship between the city and the type of crime committed? (Contingency table )

	Forcible Rape	Robbery	Aggravated assault
Cary, NC	14	35	70
Amherst, NY	10	33	76
Simi Valley, CA	14	37	77
Norman, OK	47	36	53

Answer 1

Claim: There is a relationship between the city and the type of crime committed.

The null and alternative hypothesis is

H0: There is no relationship between the city and the type of crime committed.

H1: There is a relationship between the city and the type of crime committed.

Level of significance = 0.05

Test statistic is

O: Observed frequency
E: Expected frequency.
E = ( Row total*Column total) / Grand total

	Robbery	Aggravated assault
Forcible Rape				Total
Cary, NC	14	35	70	119
Amherst, NY	10	33	76	119
Simi Valley, CA	14	37	77	128
Norman, OK	47	36	53	136
Total	85	141	276	502

O	E	(O-E)	(O-E)^2	(O-E)^2/E
14	20.1494	-6.1494	37.81515	1.876738
35	33.4243	1.575697	2.482822	0.074282
70	65.42629	4.573705	20.91878	0.31973
10	20.1494	-10.1494	103.0104	5.112329
33	33.4243	-0.4243	0.180033	0.005386
76	65.42629	10.57371	111.8032	1.708843
14	21.67331	-7.67331	58.87964	2.716689
37	35.95219	1.047809	1.097903	0.030538
77	70.3745	6.625498	43.89722	0.623766
47	23.02789	23.97211	574.6621	24.95505
36	38.1992	-2.1992	4.836495	0.126612
53	74.77291	-21.7729	474.0595	6.339991
			Total	43.89

Degrees of freedom = ( Number of rows - 1 ) * ( Number of column - 1) = ( 4 - 1) * (3 - 1) = 3 * 2 = 6

Critical value = 12.592

( From chi-square table)

Test statistic > critical value we reject null hypothesis.

Conclusion:

There is a relationship between the city and the type of crime committed.