Question

In a random sample of 92 automobile engine crankshaft bearings, 24 have a surface finish that is rougher than the specifications allow. Construct a 99% two-sided confidence interval for p, the proportion of bearings with surface finish rougher than allowed specification. What is the upper bound of this 2-sided confidence interval

Answer 1

The 99% two-sided confidence interval for p, the proportion of bearings with surface finish rougher than allowed specification is (0.1430, 0.3788). The upper bound of this 2-sided confidence interval is 0.3788.

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)`.

In a random sample of 92 automobile engine crankshaft bearings, 24 have a surface finish that is rougher than the specifications allow.

This means that
`n = 92, \pi = (24)/(92) = 0.2609`

99% confidence level

So
`\alpha = 0.01`, z is the value of Z that has a pvalue of
`1 - (0.01)/(2) = 0.995`, so
`Z = 2.575`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.2609 - 2.575\sqrt{(0.2609*0.7391)/(92)} = 0.1430`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.2609 + 2.575\sqrt{(0.2609*0.7391)/(92)} = 0.3788`

The 99% two-sided confidence interval for p, the proportion of bearings with surface finish rougher than allowed specification is (0.1430, 0.3788). The upper bound of this 2-sided confidence interval is 0.3788.