Question

The sample data below are the index of exposure (x) to radioactive waste for nine different Oregon counties and cancer mortality rate (y) (deaths per 100,000). X 2.49 2.57 3.41 1.25 1.62 3.83 11.64 6.41 8.34 y 147.1 130.1 129.9 113.5 137.5 162.3 207.5 177.9 210.3 a) Find the linear correlation coefficient r. (2 points) b) Do the data provide sufficient evidence to conclude that index of exposure and cancer mortality rate are linearly correlated? (3 points) c) Find the linear regression equation and find the mortality rate for a county in Oregon with an index exposure of 5.6. (4 points)

Answer 1

	x	y	x - x	y-ý	(y-7) (x - 2)	(x - x)2	2 (y-
	2.49	147.1	-2.128	-10.244	21.798	4.527	104.949
	2.57	130.1	-2.048	-27.244	55.791	4.193	742.260
	3.41	129.9	-1.208	-27.444	33.147	1.459	753.198
	1.25	113.5	-3.368	-43.844	147.658	11.342	1922.335
	1.62	137.5	-2.998	-19.844	59.489	8.987	393.802
	3.83	162.3	-0.788	4.956	-3.904	0.621	24.558
	11.64	207.5	7.022	50.156	352.203	49.312	2515.580
	6.41	177.9	1.792	20.556	36.840	3.212	422.531
	8.34	210.3	3.722	52.956	197.112	13.855	2804.291
Total	41.56	1416.1	0.000	0.000	900.135	97.507	9683.502
Mean	4.618	157.344

Answer(a):

The correlation coefficient between two variables is given by following formula

r E-)*(x - ) E(y-7)2(x - 2)2

r T = 900.135 197.507 * 9683.502

r= T 900.135 971.7061

r= 0.926

The coefficient of correlation between index of exposure and cancer mortality rate is 0.926

We have got r=0.926 which is close to 1 and it indicates that there is strong positive correlation between the two variables.

Answer(b):

We have to test

H₀: ρ=0

H_A:ρ≠0

The test statistic to test this hypothesis is

* t = r In-2 V1-82

t = 0.926* 19-2 V1-0.9262

t = 0.926 * 2.646 0.3767

t 2.451 0.3767

t = 6.51

The p-value for this test is 0.0003

The obtained p-value is less than α=0.05 which suggests that we have enough evidence against H₀ to reject it and we can conclude that the correlation is highly significant.

Answer(c):

The regression equation between two variables can be given by following equation

û = b0 + b1* x

The estimate of b0 and b1 can be obtained by least square method.

The least square estimate of b1 can be given by following expression

b1 : Σy - y) * (x – x) Σ(x – x)2

b1 900.135 97.507

b1 = 9.23

The estimate of b0 can be given by

b0 = 7 – b1*

b0 = 157.344 - (9.23) * 4.618

b0 = 114.716

The final estimated linear regression equation of line can be given as

Û = 114.716 + 9.23x

Now we have to predict the mortality rate when index of exposure is 5.6, that means we have to find y when x=5.6 using above estimated equation.

Û = 114.716 + 9.23* 5.6

Û = 166.4

Hence the mortality rate for county in Oregon with an index exposure of 5.6 will be 166.4