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		-8.0				0.5			38			7.3				2.4
2		-7.5				2.5			39			7.5				1.6
3		-6.4				2.4			40			7.8				1.9
4		-5.2				1.8			41			8.0				2.1
5		-5.0				2.1			42			8.1				1.6
6		-1.7				1.6			43			8.1				1.3
7		-0.7				2.1			44			8.5				2.5
8		-0.5				2.2			45			8.5				2.3
9		-0.5				1.2			46			8.6				1.4
10		-0.5				2.6			47			8.7				0.8
11		0.3				2.6			48			8.8				2.7
12		0.9				0.8			49			9.0				2.1
13		0.9				1.9			50			9.1				1.4
14		1.2				1.3			51			9.2				2.1
15		1.7				2.2			52			10.5				2.0
16		1.8				2.4			53			10.8				1.7
17		1.9				2.5			54			10.8				1.4
18		2.0				1.9			55			11.0				0.9
19		4.0				0.8			56			11.3				1.8
20		4.3				2.1			57			11.4				1.2
21		4.5				1.4			58			11.6				0.7
22		4.7				2.2			59			11.8				1.5
23		4.9				1.5			60			11.8				2.1
24		5.2				1.8			61			13.5				1.2
25		5.2				2.4			62			14.1				1.5
26		5.3				1.6			63			14.3				1.9
27		5.4				1.2			64			15.1				0.7
28		5.5				2.0			65			15.5				2.1
29		5.6				2.6			66			15.9				1.6
30		5.7				2.0			67			16.0				0.9
31		5.9				1.5			68			16.2				2.6
32		6.0				1.9			69			16.2				1.4
33		6.4				2.3			70			17.8				2.2
34		6.4				0.6			71			18.8				2.1
35		6.6				1.9			72			18.9				1.3
36		6.6				2.3			73			19.8				0.6
37		6.7				1.2			74			24.0				1.7

(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 = 0.01, 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

Answer 1

Answer(b):

The coefficient of correlation between two variables can be obtained by

Answer(c-1)

We have to test the below null hypothesis at α=0.01

H0: ρ=0

HA: ρ≠0

The test statistic to test this hypothesis is

tcalc = -1.723

The p-value for above test is 0.0892

p-value=0.0892

since the p-value is greater than 0.05, we do not have enough evidence against H0 to reject it, so we fail to reject the null hypothesis.