Question

The Wall Street Journal Corporate Perceptions Study 2011 surveyed readers and asked how each rated the quality of management and the reputation of the company for over 250 worldwide corporations. Both the quality of management and the reputation of the company were rated on an excellent, good, and fair categorical scale. Assume the sample data for 200 respondents below applies to this study.

Quality of Management	Reputation of Company
	Excellent	Good	Fair
Excellent	41	25	5
Good	34	35	10
Fair	25	10	15

1)Use a 0.05 level of significance and test for independence of the quality of management and the reputation of the company.

-State the null and alternative hypotheses.

- Find the value of the test statistic

-Find the p-value

-State your conclusion.;

a)Reject H₀. We conclude that the rating for the quality of management is independent of the rating for the reputation of the company.

b)Reject H₀. We conclude that the rating for the quality of management is not independent of the rating for the reputation of the company.     

c)Do not reject H₀. We cannot conclude that the ratings for the quality of management and the reputation of the company are not independent.

d)Do not reject H₀. We cannot conclude that the rating for the quality of management is independent of the rating of the reputation of the company.

2)If there is a dependence or association between the two ratings, discuss and use probabilities to justify your answer.

For companies with an excellent reputation, the largest column probability corresponds to  ---Select--- excellent good fair management quality. For companies with a good reputation, the largest column probability corresponds to  ---Select--- excellent good fair management quality. For companies with a fair reputation, the largest column probability corresponds to  ---Select--- excellent good fair management quality. Since these highest probabilities correspond to  ---Select--- the same different ratings of quality of management and reputation, the two ratings are

Answer 1

Solution :

Expected Table :

Quality of Management	Reputation of company
	Excellent	Good	Fair
Excellent	(71 * 100) / 200 = 35.5	(71 * 70) / 200 = 24.85	(71 * 30) / 200 = 10.65	71
Good	(79 * 100) / 200 = 39.5	(79 * 70) / 200 = 27.65	(79 * 30) / 200 = 11.85	79
Fair	(50 * 100) / 200 = 25	(50 * 70) / 200 = 17.5	(50 * 30) / 200 = 7.5	50
Total	100	70	30	200

Percentage	Oi	Ei	( Oi - Ei )	( Oi - Ei )2	( Oi - Ei )2 / Ei
	41	35.5	5.5	30.25	0.8521
	25	24.85	0.15	0.0225	0.0009
	5	10.65	-5.65	31.9225	2.9974
	34	39.5	-5.5	30.25	0.7658
	35	27.65	7.35	54.0225	1.9538
	10	11.85	-1.85	3.4225	0.2888
	25	25	0	0	0
	10	17.5	-7.5	56.25	3.2143
	15	7.5	7.5	56.25	7.5
Total	200	200	0	262.39	17.5731

To Test :-

Hypothesis :-

H0 :- Quality of management and reputation of company are independent

H1 :- Quality of management and reputation of company are dependent

Test Statistic :-
X2 = Σ (Oi - Ei )2 / Ei
X2 = 17.573

Test Criteria :-
Reject null hypothesis if X2 > X2(α, (r-1)(c-1))
Critical value X2(0.05, (3-1) (3-1)) = X2(0.05,4) = 9.488 ( From chi square table )
Since, 17.5731 > 9.488
Conclusion = Reject null hypothesis

Decision based on P value
P (X2 > 17.5731) = 0.0015
Reject null hypothesis if P value < α = 0.05
P value = 0.0015 < 0.05, hence we reject null hypothesis
Conclusion = Reject null hypothesis

Option B is correct.

There are more than 1 questions, as per the Q&A guidelines we are allowed to do only one question. If you want to get the answers for the rest of the parts, please post the question in a new post.

Please give me a thumbs-up if this helps you out. Thank you!