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

Answer:

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

Step-by-step 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.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

Sample of 252 customers, 208 are satisfied:

This means that $n = 252, \pi = \frac{208}{252} = 0.8254$

99% confidence level

So $\alpha = 0.01$, z is the value of Z that has a pvalue of $1 - \frac{0.01}{2} = 0.995$, so $Z = 2.575$.

The upper limit of this interval is:

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

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%.