Let's look at the earlier example. Suppose we poll 1,000 likely voters and find that 58% approve of the way the president handles his job. So the formula for the standard error of the percentage is sigma over square root n. And then we can convert into a percentage simply by multiplying with 100%. And sigma is given by square root p(1-p) where p is the proportion of all voters who approve. That's simply using a formula which we looked at earlier. So, in this case, p is unknown to us, because we don't know the proportion of all 140 million voters who approve. The bootstrap principle then says to replace sigma by s, which is the standard deviation of the 0/1 labels in our sample. And if we compute that standard deviation, we get square root 0.58(1- 0.58), which is 0.49. And by the way, that's exactly the same answer we would have gotten if we had plugged in. Instead of p here, our estimate from the sample. So now we can write down our 95% confidence interval for p. The formula says we take the estimate in our sample, which is 58%. Then we do plus minus z, and since we have a 95% confidence level, we take z = 2. And then we have the standard error which has the formula given above, so we get 0.49 divided by square root sample size. And we can write this as an interval, 54.9% to 61.1%. Now let's look at our second example, where we estimate the speed of light using the average of 30 measurements. So now, in this case, what is the population or the probability histogram that we are sampling from? We have to think through the question first before we can apply the bootstrap principle. Now, the reason why we get 30 different measurements in the first place is because each measurement is off by a chance error. So we have a model where each measurement = the true speed of light + a measurement error. And we can think of the measurement error as a random error. This chance error follows some kind of probability histogram that we don't know. But we can use the bootstrap principle to estimate the standard deviation of this probability histogram. Using the standard deviation of the sample. By the way, the reason why this works is that we are looking at the standard deviation, s, of the measurements on the left hand side. And we want to use that standard deviation as an estimate. Of the standard deviation sigma of the measurement errors on the right hand side. So notice there is an added term, which is the speed of light. But remember that adding a fixed number to the measurements will not change the standard deviation. That's the reason why the bootstrap principle works in this case. Later on, we will see how we can use the bootstrap principle in even more complicated situations than these.