Question

Is there a relation between murder and arson? A random sample of 15 Midwest cities (over 10,000 population) gave the following information about annual number of murder and arson cases. (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
Murder	12	18	23	15	21	20	8	22	17	16	9	7	11	13	19
Arson	37	19	157	29	165	38	7	105	69	155	27	9	13	59	160

Use a 1% level of significance to test the claim that there is a monotone-increasing relationship between the ranks of murder and arson.

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

City	Murder Rank x	Arson 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 incidence of murder and arson.Fail to reject the null hypothesis, there is insufficient evidence that a monotonic increasing relationship exists between incidence of murder and arson.     Reject the null hypothesis, there is sufficient evidence that a monotonic increasing relationship exists between incidence of murder and arson.Reject the null hypothesis, there is insufficient evidence that a monotonic increasing relationship exists between incidence of murder and arson.

Answer 1

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

Answer:

Therefore, spearman rank correlation is:

Therefore, Spearman rank correlation is

(b) What is the level of significance?

Answer: The level of significance is

State the null and alternate hypotheses.

Answer:

(c) Compute the sample test statistic.

Answer: The test statisic is:

  

  

  

Therefore, the sample statistic is

(d) Find the P-value interval of the sample test statistic.

Answer:

(e) Conclude the test.

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

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

Answer: Reject the null hypothesis, there is sufficient evidence that a monotonic increasing relationship exists between incidence of murder and arson.