Question

An electronics store has 4 branches in a large city. They are curious if sales in any particular department are different depending on location. They take a random sample of purchases throughout the 4 branches – the results are recorded below. Run an independence test for the data below at the 0.05 level of significance.

	Appliances	TV	Computers	Cameras	Cell Phones
Branch 1	54	28	61	24	81
Branch 2	44	21	55	23	92
Branch 3	49	18	49	30	72
Branch 4	51	29	65	29	102

What is the Test statistic (?2) and how is it done in excel?

Answer 1

The hypothesis being tested is:

H₀: Sales and location are independent

H_a: Sales and location are not independent

The output is:

		Col 1	Col 2	Col 3	Col 4	Col 5	Total
Row 1	Observed	54	28	61	24	81	248
	Expected	50.26	24.37	58.38	26.91	88.08	248.00
	O - E	3.74	3.63	2.62	-2.91	-7.08	0.00
	(O - E)² / E	0.28	0.54	0.12	0.31	0.57	1.82
Row 2	Observed	44	21	55	23	92	235
	Expected	47.63	23.09	55.32	25.50	83.46	235.00
	O - E	-3.63	-2.09	-0.32	-2.50	8.54	0.00
	(O - E)² / E	0.28	0.19	0.00	0.24	0.87	1.58
Row 3	Observed	49	18	49	30	72	218
	Expected	44.18	21.42	51.32	23.65	77.43	218.00
	O - E	4.82	-3.42	-2.32	6.35	-5.43	0.00
	(O - E)² / E	0.53	0.55	0.10	1.70	0.38	3.26
Row 4	Observed	51	29	65	29	102	276
	Expected	55.93	27.12	64.97	29.94	98.03	276.00
	O - E	-4.93	1.88	0.03	-0.94	3.97	0.00
	(O - E)² / E	0.44	0.13	0.00	0.03	0.16	0.76
Total	Observed	198	96	230	106	347	977
	Expected	198.00	96.00	230.00	106.00	347.00	977.00
	O - E	0.00	0.00	0.00	0.00	0.00	0.00
	(O - E)² / E	1.52	1.41	0.22	2.29	1.98	7.42
		7.42	chi-square
		12	df
		.8285	p-value

Expected Value = (Row Total*Column Total)/Marginal Total

For example, for 54,

Expected Value = (248*198)/977 = 50.26

For example, for 28,

Expected Value = (248*96)/977 = 24.37

Similarly, for all.

Find Observed - Expected(O - E) and (O - E)² / E.

Sum up from both sides to get the chi-square test statistic which is 7.42.

df = (row - 1)*(column - 1) = (4 - 1)* (5-1) = 12

p-value from the table is 0.8285.

Since the p-value (0.8285) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we can conclude that sales in any particular department are different depending on location.