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	40	25	5
Good	35	35	10
Fair	25	10	15

(a)

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.

H₀: Quality of management is not independent of the reputation of the company.
H_a: The proportion of companies with excellent management is not equal across companies with differing reputations.

H₀: Quality of management is independent of the reputation of the company.
H_a: Quality of management is not independent of the reputation of the company.  

H₀: Quality of management is independent of the reputation of the company.
H_a: The proportion of companies with excellent management is equal across companies with differing reputations.

H₀: Quality of management is not independent of the reputation of the company.
H_a: Quality of management is independent of the reputation of the company.

Find the value of the test statistic. (Round your answer to three decimal places.)

Find the p-value. (Round your answer to four decimal places.)

p-value =

State your conclusion.

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.

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

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

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

(b)

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 (excellent, good, or fair) management quality. For companies with a good reputation, the largest column probability corresponds to  (excellent, good, or fair) management quality. For companies with a fair reputation, the largest column probability corresponds to (excellent, good, or fair) management quality. Since these highest probabilities correspond to (different, the same) ratings of quality of management and reputation, the two ratings are (independent or associated)

Answer 1

H0: Quality of management is independent of the reputation of the company.
Ha: Quality of management is not independent of the reputation of the company.  

2)

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	excellent	good	fair	Total
	excellent	35.0000	24.5000	10.5000	70.00
	good	40.0000	28.0000	12.0000	80.00
	fair	25.0000	17.5000	7.5000	50.00
	total	100.00	70.00	30.00	200.00
chi square    χ2	=(Oi-Ei)2/Ei	excellent	good	fair	Total
	excellent	0.714	0.010	2.881	3.6054
	good	0.625	1.750	0.333	2.7083
	fair	0.000	3.214	7.500	10.7143
	total	1.3393	4.9745	10.7143	17.028
test statistic X2 =	17.028

p value =

0.0019

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

b)

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