A recent report suggests that Chief Information Officers (CIO's) who report directly to Chief Financial Officers (CFO's) rather than Chief Executive Officers (CEO's) are more likely to have IT agendas that deal with cost cutting and compliance. In a random sample of 858 companies, it was found that CIO's reported directly to CFO's in 170 out of 528 service firms and in 98 out of 330 manufacturing companies.
a) Determine the χ2 statistic.
b) Determine the p-value.
c) Give 95% confidence interval for the difference in proportions of companies in which the CIO reports directly to CFO between service and manufacturing firms.
Solution:
From given information, we have following summarized data:
CFO |
CEO |
Total |
|
Service Firm |
170 |
358 |
528 |
Manufacturing Firm |
98 |
232 |
330 |
Total |
268 |
590 |
858 |
Here, we have to use chi square test for independence of two categorical variables.
Null hypothesis: H0: Reporting department and type of firm are independent.
Alternative hypothesis: Ha: Reporting department and type of firm are not independent.
We assume level of significance = α = 0.05
a) Determine the χ2 statistic
Test statistic formula is given as below:
Chi square = ∑[(O – E)^2/E]
Where, O is observed frequencies and E is expected frequencies.
E = row total * column total / Grand total
We are given
Number of rows = r = 2
Number of columns = c = 2
Degrees of freedom = df = (r – 1)*(c – 1) = 1*1 = 1
α = 0.05
Critical value = 3.841459
(by using Chi square table or excel)
Calculation tables for test statistic are given as below:
Observed Frequencies |
|||
Column variable |
|||
Row variable |
CFO |
CEO |
Total |
Service |
170 |
358 |
528 |
Manufacturing |
98 |
232 |
330 |
Total |
268 |
590 |
858 |
Expected Frequencies |
|||
Column variable |
|||
Row variable |
CFO |
CEO |
Total |
Service |
164.9231 |
363.0769 |
528 |
Manufacturing |
103.0769 |
226.9231 |
330 |
Total |
268 |
590 |
858 |
Calculations |
|
(O - E) |
|
5.076923 |
-5.07692 |
-5.07692 |
5.076923 |
(O - E)^2/E |
|
0.156286 |
0.070991 |
0.250057 |
0.113585 |
Chi square = ∑[(O – E)^2/E] = 0.59092
b) Determine the p-value.
P-value = 0.442064
(By using Chi square table or excel)
P-value > α = 0.05
So, we do not reject the null hypothesis
There is sufficient evidence to conclude that Reporting department and type of firm are independent.
c) Give 95% confidence interval for the difference in proportions of companies in which the CIO reports directly to CFO between service and manufacturing firms.
Confidence interval for difference between two population proportions:
Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]
Where, P1 and P2 are sample proportions for first and second groups respectively.
WE are given
Confidence level = 95%
Critical Z value = 1.96
(by using z-table)
N1 = 528, N2 = 330, X1 = 170, X2 = 98
P1 = X1/N1 = 170/528 = 0.321969697
P2 = X2/N2 = 98/330 = 0.296969697
P1 – P2 = 0.321969697 - 0.296969697 = 0.025
SE = sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]
SE = sqrt[(0.321969697*(1 - 0.321969697)/528) + (0.296969697*(1 - 0.296969697)/330)]
SE = 0.0323
Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]
Confidence interval = 0.025 ± 1.96*0.0323
Confidence interval = 0.025 ± 0.0634
Lower limit = 0.025 - 0.0634 = -0.0384
Upper limit = 0.025 + 0.0634 = 0.0884
Confidence interval = (-0.0384, 0.0884)
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