Question

The Wall Street Journal Corporate Perceptions Study surveys readers and asks how each rats the quality of management and the reputation of the company for over 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 respondents below:
Reputation of Company
Quality of Management Excellent Good Fair
Excellent 40 25 5
Good 35 35 10
Fair 25 10 15
A. Use level of significance and test for independence of the quality of management and the reputation of the company. Compute the value of the test statistic (to 2 decimals).
B. If there is a dependence or association between the two ratings, discuss and use the probabilities to justify your answer.

Answer 1

Answer:

Step-by-step explanation:

$\text{The data given for the quality measurement and other criteria can be seen below:}$

Quality management      Excellent      Good      Fair      Total

Excellent                             40                   25         5             70

Good                                   35                    35        10            80

Fair                                       25                   10         15             50

Total                                    100                   70        30           200

The null and alternative hypothesis:

$\mathbf{H_o: \text{the quality management and reputation are independent} }$

$\mathbf{H_a: \text{the quality management and reputation are not independent} }$

The expected value is calculated by using the formula:

$E=\dfrac{row \ total \times column \ total }{table \ total}$

After using the formula to calculate the table above; we have:

The expected values of the data to be:

Quality management     Excellent         Good      Fair      Total

Excellent                            35                  24.5       10.5      70

Good                                  40                   28         1.2         80

Fair                                     25                  17.5        7.5         50

Total                                   100                  70        30           200

Degree of freedom = ( row - 1 ) × ( column -1 )

$= (3 - 1 ) \times (3 -1)$

$= 2\times 2$

$=4$

$\text{The p-value at level of significance of 0.05 and degree of freedom of 4 = }\mathbf{0.0019}$

Decision rule: To reject the $\mathbf{H_o}$ if the p-value is less than the ∝

Conclusion: We reject the $\mathbf{H_o}$ and conclude that quality management and reputation are not independent.

(B)$\text{The P-value here implies that in case that there is no dependence between the two ratings, }$$\text{the probability that they will seem to show at least this kind of strong dependence is 0.0019.}$

$\text{Thus, the probability that they will seem to show at least this kind of strong independence is =}$$\mathbf{0.0019}$