Question

these are percentages of annual sales growth and net sales attributed to loyalty card usage at 74 noodels and company restaurant

Store	Growth%	Loyalty%	Store	Growth%	Loyalty%
1	-7.7	0.7	38	7.1	1.7
2	-7.5	2.0	39	7.3	1.2
3	-6.4	1.3	40	7.7	1.7
4	-4.4	1.4	41	7.8	2.4
5	-4.1	2.0	42	8.3	2.1
6	-2.0	2.5	43	8.3	1.5
7	-1.6	1.5	44	8.3	1.3
8	-1.3	1.2	45	8.5	1.4
9	-1.0	2.4	46	8.6	1.4
10	-0.6	2.4	47	8.8	0.8
11	0.2	2.4	48	8.8	2.1
12	0.7	0.7	49	9.2	2.1
13	0.8	1.6	50	9.2	1.5
14	1.3	2.3	51	9.4	1.9
15	1.5	1.2	52	10.6	1.2
16	1.8	2.4	53	10.8	2.7
17	1.9	2.6	54	10.9	1.9
18	1.9	2.6	55	10.9	0.9
19	4.2	0.9	56	11.0	1.2
20	4.4	2.2	57	11.0	1.7
21	4.5	1.8	58	11.1	0.5
22	4.7	1.5	59	11.5	2.3
23	5.0	1.5	60	11.6	1.7
24	5.1	1.4	61	13.5	1.4
25	5.1	1.4	62	14.3	2.4
26	5.3	2.1	63	15.0	2.1
27	5.6	2.3	64	15.6	0.6
28	6.1	1.5	65	16.9	2.9
29	6.1	1.4	66	17.1	1.6
30	6.3	1.4	67	18.0	0.9
31	6.3	1.4	68	18.0	0.9
32	6.6	1.3	69	18.9	2.0
33	6.6	1.2	70	19.0	2.1
34	6.8	0.5	71	19.1	2.4
35	6.8	1.5	72	22.6	2.0
36	6.9	1.7	73	25.5	0.9
37	7.0	2.1	74	28.9	2.0

(b) Find the correlation coefficient. (Round your answer to 3 decimal places. A negative value should be indicated by a minus sign.)    

(c-1) To test the correlation coefficient for significance at α = 0.01, fill in the following. (Use the rounded value of the correlation coefficient from part b in all calculations. For final answers, round t_calc to 3 decimal places and the p-value to 4 decimal places. Negative values should be indicated by a minus sign.)

tcalc:
p-value:

there is no significant co relation

1- yes

2- no

does it appear that loyalty card usage is associated with increase sales growth?

1- no

2- yes

Answer 1

Ʃx =	562
Ʃy =	123.7
Ʃxy =	933.74
Ʃx² =	8111.9
Ʃy² =	231.03
Sample size, n =	74
x̅ = Ʃx/n = 562/74 =	7.59459459
y̅ = Ʃy/n = 123.7/74 =	1.67162162
SSxx = Ʃx² - (Ʃx)²/n = 8111.9 - (562)²/74 =	3843.73784
SSyy = Ʃy² - (Ʃy)²/n = 231.03 - (123.7)²/74 =	24.2504054
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 933.74 - (562)(123.7)/74 =	-5.71135135

Correlation coefficient, r = SSxy/√(SSxx*SSyy)

= -5.71135/√(3843.73784*24.25041) = -0.0187 = -0.019

Null and alternative hypothesis:

Ho: ρ = 0 ; Ha: ρ ≠ 0

α = 0.01

Test statistic :  

t = r*√(n-2)/√(1-r²) = -0.0187 *√(74 - 2)/√(1 - (-0.0187)²) = -0.159

df = n-2 = 72

p-value = T.DIST.2T(ABS(-0.1588), 72) = 0.8743

Conclusion:

p-value > α , Fail to reject the null hypothesis.

No, there is no significant co-relation

No, it does not appear that loyalty card usage is associated with increase sales growth.