Question

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	9.5	13.8	9.0	3.1	5.4	4.8	13.0	14.9	10.2	4.9	3.9	4.7	13.5
Firefighters	2.0	4.0	1.7	1.2	2.2	1.3	3.3	5.2	3.5	2.7	1.4	1.9	3.6

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 =

(b) What is the level of significance?

State the null and alternate hypotheses.

H_o: ρ_s < 0;  H₁: ρ_s = 0H_o: ρ_s = 0; H₁: ρ_s > 0     H_o: ρ_s = 0;  H₁: ρ_s ≠ 0H_o: ρ_s = 0; H₁: ρ_s < 0

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


(d) Find the P-value interval of the sample test statistic.
P-value <  ---Select--- 0.001 0.002 0.01 0.05

(e) Conclude the test.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.     At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(f) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that a monotonic relationship exists between number of fire fighters and police.Fail to reject the null hypothesis, there is insufficient evidence that a monotonic relationship exists between number of fire fighters and police.     Reject the null hypothesis, there is sufficient evidence that a monotonic relationship exists between number of fire fighters and police.Reject the null hypothesis, there is insufficient evidence that a monotonic relationship exists between number of fire fighters and police.

Answer 1

Let be the true value of Spearman's correlation between number of fire fighters and police. We want to test the hypothesis that a monotonic relationship exists between number of fire fighters and police. In other words we want to test if

The following are the hypotheses

First we sort the values of Police in descending order and assign ranks as below

Area	Police	Firefighters	Police Rank x
8	14.9	5.2	1
2	13.8	4	2
13	13.5	3.6	3
7	13	3.3	4
9	10.2	3.5	5
1	9.5	2	6
3	9	1.7	7
5	5.4	2.2	8
10	4.9	2.7	9
6	4.8	1.3	10
12	4.7	1.9	11
11	3.9	1.4	12
4	3.1	1.2	13

Next we sort the fire fighters in descending order and assign the ranks

Area	Police	Firefighters	Police Rank x	Firefighter Rank y
8	14.9	5.2	1	1
2	13.8	4	2	2
13	13.5	3.6	3	3
9	10.2	3.5	5	4
7	13	3.3	4	5
10	4.9	2.7	9	6
5	5.4	2.2	8	7
1	9.5	2	6	8
12	4.7	1.9	11	9
3	9	1.7	7	10
11	3.9	1.4	12	11
6	4.8	1.3	10	12
4	3.1	1.2	13	13

a) sort the above table of area and fill the rest of the columns

Area	Police	Firefighters	Police Rank x	Firefighter Rank y	d=x-y	d^2
1	9.5	2	6	8	-2	4
2	13.8	4	2	2	0	0
3	9	1.7	7	10	-3	9
4	3.1	1.2	13	13	0	0
5	5.4	2.2	8	7	1	1
6	4.8	1.3	10	12	-2	4
7	13	3.3	4	5	-1	1
8	14.9	5.2	1	1	0	0
9	10.2	3.5	5	4	1	1
10	4.9	2.7	9	6	3	9
11	3.9	1.4	12	11	1	1
12	4.7	1.9	11	9	2	4
13	13.5	3.6	3	3	0	0
					sum of d^2=	34

b) the level of significance alpha is 0.05

The hypotheses are

c)

The sample size n=13. Since this is less than 30 we will not use the normal approximation for test statistics.

the rank correlation coefficient is

the value of the sample test statistics is 0.907

d) This is a 2 tailed test. The p-value is the sum of area under both the tails.

The sample size n=13. we use the table for Spearman rank correlation and against the row for n=13 we go right and find the the value closest to 0.907 is 0.7912 , corresponding to the area 0.002.

Since this is the total area under both the tails, we use the p-value as 0.002

e) The p-value is less than the significance level alpha = 0.05. Hence we reject the null hypothesiss

ans: At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant

f) Since we reject the null hypothesis we can conclude that there is sufficient evidence that a monotonic relationship exists between number of fire fighters and police

ans: Reject the null hypothesis, there is sufficient evidence that a monotonic relationship exists between number of fire fighters and police