# How to calculate confidence intervals for the percentage of people who will download a song on a website?

So let's say that I am trying to find confidence intervals for the percentage of people who will download a song on a website. We've had 1000 people visit the site in one year and 100 people downloaded the song. Now, I can generate confidence intervals to estimate the expected download rate of visitors to our site.

– whuber
Commented Sep 30, 2011 at 20:48

In the past year you have had an average of $\frac{1000}{365}\approx 2.7397$ a day visiting the site, of which $10\%$ have downloaded the song, so about $0.27397$ downloads a day.

Let's assume that each arrival is independent of others and each download decision given an arrival is independent of others, taking those to be the mean of a Poisson distribution for daily visitors and the probability of downloads respectively, going forward; this suggests a Poisson distribution for downloads as well. Then you get the following

Number  Prob visits Prob downloads
0   0.064588040 0.760352907
1   0.176953534 0.208315865
2   0.242402101 0.028536420
3   0.221371782 0.002606066
4   0.151624508 0.000178498
5   0.083081922 9.78069E-06
6   0.037936951 4.46607E-07
7   0.014848122 1.74797E-08
8   0.005084973 5.98621E-10
9   0.001547937 1.82229E-11
10  0.000424092 4.99256E-13


Depending on how you frame it, your confidence interval for the number of downloads a day might be $[0,2]$ or $[0,1]$.

A $0\%$ daily download rate is by far the most likely outcome. A $2\%$ download rate in a day is rather unlikely, as you might need from $40$ to $66$ visitors that day (and some rounding), and a single download, and that is rather unlikely.