Question

A recent survey of 252 customers, selected at random from a database with 12,861 customers, found that 208 are satisfied with the service they are receiving. Find the upper bound of a 99% confidence interval for the percentage satisfied for all customers in the database.

Answer 1

The upper bound of a 99% confidence interval for the percentage satisfied for all customers in the database is 88.70%.

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)`.

Sample of 252 customers, 208 are satisfied:

This means that
`n = 252, \pi = (208)/(252) = 0.8254`

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 upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.8254 + 2.575\sqrt{(0.8254*0.1746)/(252)} = 0.8870`

As a percentage:

100%*0.8870 = 88.70%

The upper bound of a 99% confidence interval for the percentage satisfied for all customers in the database is 88.70%.