Question

The billing department of a national cable service company is conducting a study of how customers pay their monthly cable bills. The cable company accepts payment in person at a local​ office, by​ mail, by credit​ card, or by electronic funds transfer from a bank account. The cable company randomly sampled 400 customers to determine if there is a relationship between the​ customer's age and the payment method used. The accompanying sample results were obtained. Based on the sample​ data, can the cable company conclude that there is a relationship between the age of the customer and the payment method​ used? Conduct the appropriate test at the α=0.01 level of significance.

Payment Method	20-30	31-40	41-50	Over 50
In person	9	12	11	12
By mail	29	66	71	56
By credit card	25	19	5	6
By Funds Transfer	23	34	16	6

A) Calculate the test statistic

B) Calculate the p-value

C) Conclusion

Answer 1

Solution:

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

Null hypothesis: H₀: There is no relationship between the age of the customer and the payment method ​used.

Alternative hypothesis: H_a: There is a relationship between the age of the customer and the payment method​ used.

We are given level of significance = α = 0.01

A) Calculate the test statistic

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 = 4

Number of columns = c = 4

Degrees of freedom = df = (r – 1)*(c – 1) = 3*3 = 9

α = 0.01

Critical value = 21.66599

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	C1	C2	C3	C4	Total
R1	9	12	11	12	44
R2	29	66	71	56	222
R3	25	19	5	6	55
R4	23	34	16	6	79
Total	86	131	103	80	400

Expected Frequencies
	Column variable
Row variable	C1	C2	C3	C4	Total
R1	9.46	14.41	11.33	8.8	44
R2	47.73	72.705	57.165	44.4	222
R3	11.825	18.0125	14.1625	11	55
R4	16.985	25.8725	20.3425	15.8	79
Total	86	131	103	80	400

Calculations
(O - E)
-0.46	-2.41	-0.33	3.2
-18.73	-6.705	13.835	11.6
13.175	0.9875	-9.1625	-5
6.015	8.1275	-4.3425	-9.8

(O - E)^2/E
0.022368	0.40306	0.009612	1.163636
7.349946	0.618348	3.348329	3.030631
14.67912	0.054138	5.927725	2.272727
2.130128	2.553145	0.926991	6.078481

Chi square = ∑[(O – E)^2/E] = 50.56839

B) Calculate the p-value

P-value = 0.00

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

C) Conclusion

There is sufficient evidence to conclude that there is a relationship between the age of the customer and the payment method​ used.