Question

A philanthropic organisation sent free mailing labels and greeting cards to a random sample of 100 comma 000 potential donors on their mailing list and received 5066 donations. What is the 99% confidence interval?

Answer 1

Answer:

The 99% confidence interval for the proportion of donors who donated is (0.0547, 0.0585).

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 $1 - \frac{\alpha}{2}$.

For this problem, we have that:

$n = 100000, \pi = \frac{5066}{100000} = 0.0566$

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 lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.0566 - 2.575\sqrt{\frac{0.0566*0.9434}{100000}} = 0.0547$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.0566 + 2.575\sqrt{\frac{0.0566*0.9434}{100000}} = 0.0585$

The 99% confidence interval for the proportion of donors who donated is (0.0547, 0.0585).