Question

Suppose we have the following values on the number of customers (X) and the average profits (Y) for fifteen stores:

Store                   Customers (X)          Average Profits (Y)

A                               161                              157

B                               99                              93

C                               135                              136

D                               120                              123

E                                164                              153

F                                221                              241

G                               179                              201

H                               204                              206

I                                 214                              229

J                                 101                              135

K                               231                              224

L                                206                              195

M                               248                              242

N                               107                              115

O                               205                              197

Use the Spearman’s Rank Correlation test at the 0.05 level to see if X and Y are significantly related.

Answer 1

a.

X	Y
161	157
99	93
135	136
120	123
164	153
221	241
179	201
204	206
214	229
101	135
231	224
206	195
248	242
107	115
205	197

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

	X	Y	X*Y	X2	Y2
	161	157	25277	25921	24649
	99	93	9207	9801	8649
	135	136	18360	18225	18496
	120	123	14760	14400	15129
	164	153	25092	26896	23409
	221	241	53261	48841	58081
	179	201	35979	32041	40401
	204	206	42024	41616	42436
	214	229	49006	45796	52441
	101	135	13635	10201	18225
	231	224	51744	53361	50176
	206	195	40170	42436	38025
	248	242	60016	61504	58564
	107	115	12305	11449	13225
	205	197	40385	42025	38809
Sum =	2595	2647	491221	484513	500715

The correlation coefficient rr 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

b.

where ρ corresponds to the population correlation.

The sample size is n = 15, so then the number of degrees of freedom is df = n-2 = 15 - 2 = 13

The corresponding t-statistic to test for the significance of the correlation is:

The p-value is computed as follows:

Since we have that p-value = 4.648477e-09 < .0001, it is concluded that the null hypothesis H0​ is rejected.

Therefore, it is concluded that there is enough evidence to claim that the population correlation ρ is different than 0, at the 0.05 significance level.

Therefore, it is concluded that there is enough evidence to claim that X and Y are significantly related.