Question

Below are percentages for annual sales growth and net sales attributed to loyalty card usage at 74 Noodles & Company restaurants.

Annual Sales Growth (px;) and Loyalty Card Usage (px; of Net Sales) (n = 74 restaurants)
Store	Growth%				Loyalty%				Store		Growth%				Loyalty%
1		-6.6				1.9			38			7.4				1.2
2		-6.2				2.3			39			7.5				1.7
3		-6.1				2.2			40			7.5				2.0
4		-5.9				2.0			41			7.5				1.4
5		-3.8				2.3			42			7.6				1.2
6		-1.9				1.7			43			7.6				1.9
7		-1.9				1.9			44			7.9				1.9
8		-1.8				1.3			45			8.1				1.7
9		-0.9				1.6			46			8.3				2.7
10		-0.5				2.3			47			8.7				0.7
11		0.4				1.9			48			8.9				2.0
12		0.4				1.4			49			8.9				1.3
13		0.6				2.3			50			8.9				2.2
14		0.8				2.2			51			9.2				1.7
15		0.8				2.0			52			10.8				1.8
16		1.0				1.6			53			10.9				1.5
17		1.8				1.3			54			10.9				1.6
18		2.0				2.4			55			11.2				2.5
19		4.0				0.7			56			11.2				2.4
20		4.2				1.4			57			11.4				1.3
21		4.3				2.4			58			11.6				0.6
22		4.3				1.2			59			11.9				2.4
23		4.4				2.5			60			11.9				1.3
24		4.6				2.7			61			13.0				1.5
25		5.0				1.5			62			14.7				1.8
26		5.2				1.7			63			14.8				2.0
27		5.3				1.5			64			15.4				2.4
28		5.7				1.9			65			15.8				1.3
29		5.8				2.5			66			17.6				2.2
30		6.0				1.8			67			17.7				0.5
31		6.4				2.2			68			18.1				1.9
32		6.6				1.5			69			18.3				1.6
33		6.6				2.0			70			18.4				2.2
34		6.7				1.4			71			21.4				1.7
35		6.8				2.0			72			23.8				1.7
36		7.2				2.0			73			24.2				0.6
37		7.3				2.3			74			25.6				2.0

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

(c-1) To test the correlation coefficient for significance at α = 0.1, 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

(c-2) There is no significant correlation.
  

• False

• True

(d) Does it appear that increased loyalty card usage is associated with decreased sales growth?
  

• Yes

• No

Next

Answer 1

b) the correlation coefficient r = -0.201

c)

alpha = 0.1

Hypothesis:

H0​:ρ=0

HA​:ρ̸​=0

t cal = r*SQRT((n-2)/(1-r^2)) = -0.201*SQRT((74-2)/(1-(-0.201)^2)) = -1.741

P value = 0.0859(Use t table)

d)

P value < 0.1, Reject H0

False, There is significant correlation.

e)

Week and negative correlation between Annual Sales Growth and Loyalty Card Usage

Yes, it does appear that increased loyalty card usage is associated with decreased sales growth

(But actually we can not say Annual Sales Growth decreasing with Loyalty Card Usage, it has a week relation)

(Because of above test we are saying it is decreasing)