Question

A study showed that 14 of 180 publicly traded business services companies failed a test for compliance with Sarbanes-Oxley requirements for financial records and fraud protection. Assuming that these are a random sample of all publicly traded companies, construct a 95% confidence interval for the overall noncompliance proportion. (Round your answers to 4 decimal places.)

Answer 1

The 95% confidence interval for the overall noncompliance proportion is (0.0387, 0.1169).

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

For this problem, we have that:

`n = 180, \pi = (14)/(180) = 0.0778`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.0778 - 1.96\sqrt{(0.0778*0.9222)/(180)} = 0.0387`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.0778 + 1.96\sqrt{(0.0778*0.9222)/(180)} = 0.1169`

The 95% confidence interval for the overall noncompliance proportion is (0.0387, 0.1169).