Question

Each person in a representative sample of 446 college students age 18 to 24 was classified according to age and to the response to the following question: "How often have you used a credit card to buy items knowing you wouldn't have money to pay the bill when it arrived?" Possible responses were never, rarely, sometimes, or frequently. The responses are summarized in the table.

	Age 18 to 20	Age 21 to 22	Age 23 to 24
Never	73	62	29
Rarely	35	34	32
Sometimes	31	42	40
Frequently	12	24	32

Do these data provide evidence that there is an association between age group and the response to the question? Test the relevant hypotheses using α = 0.01.

State the appropriate null and alternative hypotheses.

H₀: There is no association between age group and the response to the question.
H_a: There is an association between age group and the response to the question.

H₀: The proportions falling into each of the three age groups are the same for all four responses to the question.
H_a: The proportions falling into each of the three age groups are not the same for all four responses to the question.

H₀: There is an association between age group and the response to the question.
H_a: There is no association between age group and the response to the question.

H₀: The proportions falling into each of the three age groups are not the same for all four responses to the question.
H_a: The proportions falling into each of the three age groups are the same for all four responses to the question.

Find the test statistic and P-value. (Use technology. Round your test statistic to three decimal places and your P-value to four decimal places.)

X²=

P-value=

State the conclusion in the problem context.

Reject H₀. There is convincing evidence that there is an association between age group and the response to the question.

Reject H₀. There is not convincing evidence that there is an association between age group and the response to the question.

Fail to reject H₀. There is not convincing evidence that there is an association between age group and the response to the question.

Fail to reject H₀. There is convincing evidence that there is an association between age group and the response to the question.

Answer 1

Solution:

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

Null hypothesis: H₀: There is no association between age group and the response to the question.

Alternative hypothesis: H_a: There is an association between age group and the response to the question.

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 = 4

Number of columns = c = 3

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

α = 0.01

Critical value = 16.81189

(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	Total
R1	73	62	29	164
R2	35	34	32	101
R3	31	42	40	113
R4	12	24	32	68
Total	151	162	133	446

Expected Frequencies
	Column variable
Row variable	C1	C2	C3	Total
R1	55.52466	59.56951	48.90583	164
R2	34.19507	36.6861	30.11883	101
R3	38.25785	41.04484	33.69731	113
R4	23.02242	24.69955	20.27803	68
Total	151	162	133	446

Calculations
(O - E)
17.47534	2.430493	-19.9058
0.804933	-2.6861	1.881166
-7.25785	0.955157	6.302691
-11.0224	-0.69955	11.72197

(O - E)^2/E
5.500031	0.099166	8.102144
0.018948	0.196672	0.117494
1.376877	0.022228	1.178845
5.277194	0.019813	6.776037

Chi square = ∑[(O – E)^2/E] = 28.68545

Chi square = 28.685

P-value = 0.0001

(By using Chi square table or excel)

P-value < α = 0.01

So, we reject the null hypothesis

Reject H₀. There is convincing evidence that there is an association between age group and the response to the question.

There is sufficient evidence to conclude that there is a association between age group and the response to the question.