Question

The data below shows the age and favorite type of music of 779 randomly selected people. Age 15-21 21-30 30-40 40-50 Country Rock Pop Classical 21 45 90 33 68 55 42 48 65 31 57 60 39 25 53 Test the claim that preferred music type is dependent on age. Use a = 0.05.

Answer 1

Solution:

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

Null hypothesis: H₀: Two categorical variables music type and age are independent.

Alternative hypothesis: H_a: Two categorical variables music type and age are dependent.

We are given level of significance = α = 0.05

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 = 0.05

Critical value = 16.91898

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Music Type
Age	Country	Rock	Pop	Classical	Total
15 to 21	21	45	90	33	189
21 to 30	68	55	42	48	213
30 to 40	65	47	31	57	200
40 to 50	60	39	25	53	177
Total	214	186	188	191	779

Expected Frequencies
	Music Type
Age	Country	Rock	Pop	Classical	Total
15 to 21	51.92041	45.12709	45.61232	46.34018	189
21 to 30	58.51348	50.85751	51.40436	52.22465	213
30 to 40	54.94223	47.75353	48.26701	49.03723	200
40 to 50	48.62388	42.26187	42.7163	43.39795	177
Total	214	186	188	191	779

Calculations
(O - E)
-30.9204	-0.12709	44.38768	-13.3402
9.486521	4.14249	-9.40436	-4.22465
10.05777	-0.75353	-17.267	7.962773
11.37612	-3.26187	-17.7163	9.602054

(O - E)^2/E
18.41418	0.000358	43.19591	3.840304
1.538006	0.337418	1.720517	0.341747
1.841182	0.01189	6.177089	1.293013
2.661577	0.251759	7.347719	2.124512

Test Statistic = Chi square = ∑[(O – E)^2/E] = 91.09718

χ² statistic = 91.09718

P-value = 0.0000

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that two categorical variables music type and age are dependent.

There is sufficient evidence to conclude that music type is dependent on age.