Question

Please label and show each step

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		50		35	55	80
Outstanding		65		70	60	105
Exceptional		35		45	35	65

Using a level of significance of 0.05, 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

(a)

H₀: Quality Rating and Education are independent

H₁: Quality Rating and Education are not independent

(b)

$\chi^2$ statistic will be analysed here.

(c)

If,

$\chi^2_{calculated} > \chi^2_{\alpha, df}$,

Then we reject the null hypothesis

(d)

Original Table with marginal totals:

	Some HS	HS Grad	Some College	College Grad	Total (f.j)
Average	50	35	55	80	220
Outstanding	65	70	60	105	300
Exceptional	35	45	35	65	180
Total (fi.)	150	150	150	250	700 = f..

Table for,

$\frac{f_{ij}^2}{f_{i.} * f_{.j}}$

	Some HS	HS Grad	Some College	College Grad	Total
Average	0.076	0.037	0.092	0.116	0.321
Outstanding	0.094	0.109	0.08	0.147	0.43
Exceptional	0.045	0.075	0.045	0.094	0.259
Total	0.215	0.221	0.217	0.357	1.01

So, the value of chi-squared statistic is 700*(1.01-1) = 7 and the corresponding degrees of freedom is (3-1)*(4-1) = 6.

Now, critical value at 5% level of significance is . Since the calculated value is less that the critical value $\chi^2_{0.05, 6} = 12.592$ , we fail to reject the null hypothesis. Hence, there is insufficient evidence to conclude that Quality Rating and Education are related.

(e)

Quality Rating and Education are independent.

** If the answer does not match please comment.