Question

3. 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?

1. State the Null and Alternative hypothesis.

2. What is the statistic you would use to analyze this?

3. State your decision rule:

4. Show your calculation:

5. What is your conclusion?

Quality Rating and Education level are independent

OR

Quality Rating and Education level are NOT independent

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: Two categorical variables the customer’s Education level and Quality Rating are independent.

Alternative hypothesis: H_a: Two categorical variables the customer’s Education level and Quality Rating are dependent.

We are given level of significance = α = 0.01

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 3

Number of columns = c = 4

Degrees of freedom = df = (r – 1)*(c – 1) = 2*3 = 6

α = 0.05

Critical value = 16.81189

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Education
Quality Rating	Some HS	HS Grad	Some College	College Grad	Total
Average	55	80	50	35	220
Outstanding	60	105	65	70	300
Exceptional	35	65	35	45	180
Total	150	250	150	150	700

Expected Frequencies
	Education
Quality Rating	Some HS	HS Grad	Some College	College Grad	Total
Average	47.14286	78.57143	47.14286	47.14286	220
Outstanding	64.28571	107.1429	64.28571	64.28571	300
Exceptional	38.57143	64.28571	38.57143	38.57143	180
Total	150	250	150	150	700

Calculations
(O - E)
7.857143	1.428571	2.857143	-12.1429
-4.28571	-2.14286	0.714286	5.714286
-3.57143	0.714286	-3.57143	6.428571

(O - E)^2/E
1.309524	0.025974	0.17316	3.127706
0.285714	0.042857	0.007937	0.507937
0.330688	0.007937	0.330688	1.071429

Test Statistic = Chi square = ∑[(O – E)^2/E] = 7.221549

χ² statistic = 7.221549

P-value = 0.300844

(By using Chi square table or excel)

P-value > α = 0.01

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that two categorical variables the customer’s Education level and Quality Rating are independent.

There is NO relationship between Education and Quality Rating.

Quality Rating and Education level are independent.