Question

#32

Is there a relation between police protection and fire protection? A random sample of large population areas gave the following information about the number of local police and the number of local fire-fighters (units in thousands). (Reference: Statistical Abstract of the United States.)

Area	1	2	3	4	5	6	7	8	9	10	11	12	13
Police	10.8	13.3	17.0	16.3	7.6	3.2	5.3	15.4	2.7	4.7	4.1	6.1	5.6
Firefighters	3.1	2.5	4.8	2.9	3.3	1.2	2.4	3.4	1.0	1.5	2.3	2.7	0.9

Use a 5% level of significance to test the claim that there is a monotone relationship (either way) between the ranks of number of police and number of firefighters.

(a) Rank-order police using 1 as the largest data value. Also rank-order firefighters using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test.

Area	Police Rank x	Firefighters Rank y	d = x - y	d2
1 2 3 4 5 6 7 8 9 10 11 12 13				Σd2 =

(c) Compute the sample test statistic. (Use 3 decimal places.)

Answer 1

a) H0: there is no monotone relationship (either way) between the ranks of number of police and number of firefighters.

H1: there is a monotone relationship (either way) between the ranks of number of police and number of firefighters.

Let the los be alpha = 5%

X	Y	Ranks of X	Ranks of Y	di = R(xi)-R(yi)	di^2
10.8	3.1	5	4	1	1
13.3	2.5	4	7	-3	9
17	4.8	1	1	0	0
16.3	2.9	2	5	-3	9
7.6	3.3	6	3	3	9
3.2	1.2	12	11	1	1
5.3	2.4	9	8	1	1
15.4	3.4	3	2	1	1
2.7	1	13	12	1	1
4.7	1.5	10	10	0	0
4.1	2.3	11	9	2	4
6.1	2.7	7	6	1	1
5.6	0.9	8	13	-5	25
				Total =	62

c) Test for correlation coefficient:

Test Statistic t = 4.930