Question

A market researcher believes that brand perception of one of the company's products may vary between different groups. After interviewing 415 persons, the following data was compiled. Can we conclude that brand perception is dependent on age?

Age	Favorable	Unfavorable	Neutral	Total
18-30	89	18	19	126
30-45	77	59	59	195
Over 45	68	12	14	94
Total	234	89	92	415

Step 1 of 8: State the null and alternative hypothesis.

Step 2 of 8: Find the expected value for the number of particpants who are 18-30 years old and have a favorable perception of the brand. Round your answer to one decimal place.

Step 3 of 8: Find the expected value for the number of particpants who are 30-45 years old and have a unfavorable perception of the brand. Round your answer to one decimal place.

Step 4 of 8: Find the value of the test statistic. Round your answer to three decimal places.

Step 5 of 8: Find the degrees of freedom associated with the test statistic for this problem.

Step 6 of 8: Find the critical value of the test at the 0.1 level of significance. Round your answer to three decimal places.

Step 7 of 8: Make the decision to reject or fail to reject the null hypothesis at the 0.1 level of significance.

Step 8 of 8: State the conclusion of the hypothesis test at the 0.1 level of significance.

Answer 1

Step 1:

H0: The  brand perception is independent on age

H1: The  brand perception is not independent on age

Step 2: The expected value for the number of particpants who are 18-30 years old and have a favorable perception of the brand is (126*234)/415 = 71.1

Step 3:

The expected value for the number of particpants who are 30-45 years old and have a unfavorable perception of the brand is 195*89/415 = 41.8

Step 4:

The Expected frequencies are
	Favorable	Unfavorable	Neutral	Total
18-30	71.046	27.022	27.933	126
30-45	109.952	41.819	43.229	195
Over 45	53.002	20.159	20.839	94.00
Total	234	89	92	415

The Chi-square contribution values are

Oi	Ei	(Oi-Ei)^2 /Ei
89	71.046	4.5371
18	27.022	3.0122
19	27.933	2.8568
77	109.952	9.8755
59	41.819	7.0587
59	43.229	5.7536
68	53.002	4.244
12	20.159	3.3022
14	20.839	2.2444
	Total:	42.8845

Test Statistic, X^2: 42.884

Step 5: Degrees of freedom: 4
Step 6:Critical X^2: 7.779
Step 7: Since Chi-square value > Chi-square critical value so we reject H0

Step 8: Thus we conclude that the  brand perception is not independent on age