In: Statistics and Probability
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 | 8 |
Good | 35 | 35 | 10 |
Fair | 25 | 10 | 12 |
(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 independent of
the reputation of the company.
Ha: The proportion of companies with excellent
management is equal across companies with differing
reputations.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. H0: Quality of
management is not independent of the reputation of the
company.
Ha: The proportion of companies with excellent
management is not equal across companies with differing
reputations.H0: Quality of management is not
independent of the reputation of the company.
Ha: 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 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 not independent of the rating for the reputation of the company. 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 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 ---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 ---Select--- associated independent .
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.
b) applying chi square test:
Expected | Ei=row total*column total/grand total | excellent | good | fair | Total |
excellent | 36.5000 | 25.5500 | 10.9500 | 73 | |
good | 40.0000 | 28.0000 | 12.0000 | 80 | |
fair | 23.5000 | 16.4500 | 7.0500 | 47 | |
total | 100 | 70 | 30 | 200 | |
chi square χ2 | =(Oi-Ei)2/Ei | excellent | good | fair | Total |
excellent | 0.3356 | 0.0118 | 0.7947 | 1.1422 | |
good | 0.6250 | 1.7500 | 0.3333 | 2.7083 | |
fair | 0.0957 | 2.5290 | 3.4755 | 6.1003 | |
total | 1.0564 | 4.2909 | 4.6036 | 9.9508 |
test statistic X2 =9.951
p value =0.0413
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 excellent
For companies with an good reputation, the largest column probability corresponds to good
For companies with a fair reputation, the largest column probability corresponds to fair
the two ratings are associated