Introduction to Data Science Version 3

Here’s one last point to cap off this thinking. Even with a sample of 100,000 there is some variation in samples. That’s why the 0.58638 from the sum() function above does not exactly match the theoretical 0.58303 from the ppois() function above. We can ask R to tell us how much variation there is around one of these probabilities using the poisson.test() function like this:

poisson.test(58638, 100000)

95 percent confidence interval:
0.5816434 0.5911456

We’ve truncated a little of the output in the interests of space: What you have left is the upper and lower bounds on a 95% confidence interval. Here’s what a confidence interval is: For 95% of the samples that we could generate using rpois(), using a sample size of 100,000, and a desired mean of 10, we will get a result that lies between 0.5816434 and 0.5911456 (remember that this resulting proportion is calculated as the total number of events whose delay time is 10 or less). So we know what would happen for 95% of the rpois() samples, but the assumption that statisticians also make is that if a natural phenomenon, like the arrival time of tweets, also fits the Poisson distribution, that this same confidence interval would be operative. So while we know that we got 0.58638 in one sample on the previous page, it is likely that future samples will vary by a little bit (about 1%). Just to get a feel for what happens to the confidence interval with smaller samples, look at these:

poisson.test(5863, 10000)

95 percent confidence interval:
0.5713874 0.6015033

poisson.test(586, 1000)

95 percent confidence interval:
0.5395084 0.6354261

poisson.test(58, 100)

95 percent confidence interval:
0.4404183 0.7497845

We’ve bolded the parameters that changed in each of the three commands above, just to emphasize that in each case we’ve reduced the sample size by a factor of 10. By the time we get to the bottom look how wide the confidence interval gets. With a sample of 100 events, of which 58 had delays of 10 seconds or less, the confidence interval around the proportion of 0.58 ranges from a low of 0.44 to a high of 0.75! That’s huge! The confidence interval gets wider and wider as we get less and less confident about the accuracy of our estimate. In the case of a small sample of 100 events, the confidence interval is very wide, showing that we have a lot of uncertainty about our estimate that 58 events out of 100 will have arrival delays of 10 or less. Note that you can filter out the rest of the stuff that poisson.test() generates by asking specifically for the “conf.int” in the output that is returned:

poisson.test(58, 100)$conf.int

[1] 0.4404183 0.7497845
attr(,"conf.level")
[1] 0.95

The bolded part of the command line above shows how we used the $ notation to get a report of just the bit of output that we wanted from poisson.test(). This output reports the exact same confidence interval that we saw on the previous page, along with a reminder in the final two lines that we are looking at a 95% confidence interval.