Question

Is there a relation between incidents of child abuse and number of runaway children? A random sample of cities (over 10,000 population) gave the following information about the number of reported incidents of child abuse and the number of runaway children. (Reference: Federal Bureau of Investigation, U.S. Department of Justice.)

City	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Abuse cases	56	21	78	63	58	103	98	17	18	25	61	41	31	118	11
Runaways	410	249	757	629	518	706	604	263	404	291	514	416	335	674	436

Use a 1% level of significance to test the claim that there is a monotone-increasing relationship between the ranks of incidents of abuse and number of runaway children.

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

City	Abuse Cases Rank x	Runaways Rank y	d = x - y	d2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15				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 0.01 0.05 0.10

(e) Conclude the test.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.     At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 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 increasing relationship exists between incidents of child abuse and run-away children.
Fail to reject the null hypothesis, there is insufficient evidence that a monotonic increasing relationship exists between incidents of child abuse and run-away children.
Reject the null hypothesis, there is sufficient evidence that a monotonic increasing relationship exists between incidents of child abuse and run-away children.
Reject the null hypothesis, there is insufficient evidence that a monotonic increasing relationship exists between incidents of child abuse and run-away children.

Answer 1

part A)

Abuse cases	Runnway	Abuses cases rank	Runnways rank	d	d.square
56	410	8	10	-2	4
21	249	12	15	-3	9
78	757	4	1	3	9
63	629	5	4	1	1
58	518	7	6	1	1
103	706	2	2	0	0
98	604	3	5	-2	4
17	263	14	14	0	0
18	404	13	11	2	4
25	291	11	13	-2	4
61	514	6	7	-1	1
41	416	9	9	0	0
31	335	10	12	-2	4
118	674	1	3	-2	4
11	436	15	8	7	49
				total	94

part B)

Part C) sample test statistic rho =0.8321429

part D) p-value = 0.0001578

therefore p-value <0.001

part E)

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

part F)

Reject the null hypothesis, there is sufficient evidence that a monotonic increasing relationship exists between incidents of child abuse and run-away children.

Note : for part a) i used excel to compute the given table while part b and c I use r command as follows

Abuse_cases=c(56, 21, 78, 63, 58, 103, 98, 17, 18, 25, 61, 41, 31, 118, 11)

Runaways=c(410, 249, 757, 629, 518, 706, 604, 263, 404, 291, 514, 416, 335, 674, 436)

cor.test(Abuse_cases,Runaways,method = "spearman")

Spearman's rank correlation rho

data: Abuse_cases and Runaways
S = 94, p-value = 0.0001578
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
0.8321429