Question

The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan areas. The data are shown in the following table. Do these data provide evidence to reject the null hypothesis that ρ = 0 in favor of ρ ≠ 0 at α = .05? (Give your answers correct to three decimal places.)

Population	9.7	0.2	3.2	7.2	0.5	3.7	4.8	3.3	2.2	3.5
Crime Rate	13	8.7	8.8	9.2	6.8	8.4	9.3	7	7.1	6.6

(a) Calculate r.


(ii) Calculate the critical region.
(smaller value)
(larger value)
(b) State the appropriate conclusion.

Reject the null hypothesis, there is not significant evidence that ρ ≠ 0. Reject the null hypothesis, there is significant evidence that ρ ≠ 0.     Fail to reject the null hypothesis, there is not significant evidence that ρ ≠ 0. Fail to reject the null hypothesis, there is significant evidence that ρ ≠ 0.

Answer 1

The provided data are shown in the table below

X	Y
9.7	13
0.2	8.7
3.2	8.8
7.2	9.2
0.5	6.8
3.7	8.4
4.8	9.3
3.3	7
2.2	7.1
3.5	6.6

Also, the following calculations are needed to compute the correlation coefficient:

	X	Y	X*Y	X2	Y2
	9.7	13	126.1	94.09	169
	0.2	8.7	1.74	0.04	75.69
	3.2	8.8	28.16	10.24	77.44
	7.2	9.2	66.24	51.84	84.64
	0.5	6.8	3.4	0.25	46.24
	3.7	8.4	31.08	13.69	70.56
	4.8	9.3	44.64	23.04	86.49
	3.3	7	23.1	10.89	49
	2.2	7.1	15.62	4.84	50.41
	3.5	6.6	23.1	12.25	43.56
Sum =	38.3	84.9	363.18	221.17	753.03

The correlation coefficient r is computed using the following expression:

where

In this case, based on the data provided, we get that

Therefore, based on this information, the sample correlation coefficient is computed as follows

which completes the calculation.

The following needs to be tested:

where ρ corresponds to the population correlation.

The sample size is n = 10, so then the number of degrees of freedom is df = n-2 = 10 - 2 = 8

The corresponding critical correlation value r_c​ for a significance level of α=0.05, for a two-tailed test is:

r_c = 0.632

Observe that in this case, the null hypothesis H0​:ρ=0 is rejected if |r| > r_c = 0.632

Based on the sample correlation provided, we have that |r| = 0.776 > r_c = 0.632 , from which is concluded that the null hypothesis is rejected.