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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.

User Erilem
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Answer:

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%.

User Lampbob
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