Question

A quality survey asked recent customers of their experience at a local department store. One question asked for the customers rating on their service using categorical responses of average, outstanding, and exceptional. Another question asked for the applicant’s education level with categorical responses of Some HS, HS Grad, Some College, and College Grad. The sample data below are for 700 customers who recently visited the department store. Education Quality Rating Some HS HS Grad Some College College Grad Average 55 80 50 35 Outstanding 60 105 65 70 Exceptional 35 65 35 45 Using a level of significance of 0.01, is there evidence to suggest that the customer’s Education level and Quality Rating are independent? In other words, is there a relationship or is there NO relationship between Education and Quality Rating? a. State the Null and Alternative hypothesis. b. What is the statistic you would use to analyze this? c. State your decision rule: d. Show your calculation: e. What is your conclusion? Quality Rating and Education level are independent OR Quality Rating and Education level are NOT independent

Answer 1

a.

Null hypothesis : H_o : Quality Rating and Education level are independent

Alternate hypothesis : H_a : Quality Rating and Education level are not independent

b.

The Chi-Square test of independence is being used to determine if there is a significant relationship between two categorical variables Quality Rating and Education level.

Test Statistic

O : Observed Count

E: Expected Count

c. Decision Rule

= 0.01

Degrees of freedom : df = (Number of rows - 1)*(Number of columns - 1) = (3-1)*(4-1) = 2*3=6

Critical value of for : 0.01 and 6 degrees of freedom = 16.812

If Value of the test Statistic > Critical value of : 16.812 ; Reject the null hypothesis ; Else Fail to reject the null hypothesis

d.

O : Observed Count

	Education
Quality Rating	Some Hs	HS Grad	Some College	College Grad
Average	55	80	50	35
Outstanding	60	105	65	70
Exceptional	35	65	35	45

E : Expected Count

	Education
Quality Rating	Some Hs	HS Grad	Some College	College Grad	Total
Average	(220*150)/700	(220*250)/700	(220*150)/700	(220*150)/700	220
Outstanding	(300*150)/700	(300*250)/700	(300*150)/700	(300*150)/700	300
Exceptional	(180*150)/700	(180*250)/700	(180*150)/700	(180*150)/700	180
Total	150	250	150	150	700

E:

	Education
Quality Rating	Some Hs	HS Grad	Some College	College Grad	Total
Average	47.1429	78.5714	47.1429	47.1429	220
Outstanding	64.2857	107.1429	64.2857	64.2857	300
Exceptional	38.5714	64.2857	38.5714	38.5714	180
Total	150	250	150	150	700

O	E	O-E	(O-E)2	(O-E)2/E
55	47.1429	7.8571	61.7347	1.3095
60	64.2857	-4.2857	18.3673	0.2857
35	38.5714	-3.5714	12.7551	0.3307
80	78.5714	1.4286	2.0408	0.0260
105	107.1429	-2.1429	4.5918	0.0429
65	64.2857	0.7143	0.5102	0.0079
50	47.1429	2.8571	8.1633	0.1732
65	64.2857	0.7143	0.5102	0.0079
35	38.5714	-3.5714	12.7551	0.3307
35	47.1429	-12.1429	147.4490	3.1277
70	64.2857	5.7143	32.6531	0.5079
45	38.5714	6.4286	41.3265	1.0714
			Total	7.2215

As Value of the test Statistic : 7,2215 < Critical value of : 16.812 ; Fail to reject the null hypothesis.

Not enough evidence to conclude that the Quality Rating and Education level are not independent.

e.

Conclusion :

Quality Rating and Education level are independent.