Hi, my name is Brian Caffo and this this Mathematical Biostatistics Boot Camp 2, Lecture 2 on the subject of Power. So in this lecture, we're going to talk about power, calculating power. power, specifically for the T-test, and because the T-test involves maybe a little bit of math that we don't want to get into, we'll talk about how you can do it using Monte Carlo, which is a standard way to estimate power. So, as it's name would suggest, power is a good thing. You want more power. so power is the probability of rejecting the null hypothesis when, in fact, it's false. So the opposite of power, 1 minus power is the type two error rate. and that's the probability of failing to reject a null hypothesis when it's false. And we usually label that as beta. So, consider our previous example involving the Respiratory Disturbance Index. Here, we are testing whether or not mu was 30 versus the alternative, then mu is specifically greater than 30.Then the power the power is exactly the probability there are t statistics lies in the rejection region. Recall that we calculated the rejection region was if the normalized mean X bar minus 30,the value under the null hypothesis divided by the standard error of the mean, s over square root n. If that normalized mean was greater than a t quantile specifically, t1 minus alpha, n minus 1 quantile where alpha is the desired type one error rate. So if the statistic is larger than that t quantile, then we reject the null hypothesis. So then the power is the probability that we reject the null hypothesis. In other words, the probability that our test statistic is larger than that quantile but, instead of calculated under the null hypothesis that mu was 30, calculated under the assumption of the alternative hypothesis that mu is specifically greater than 30. So, in the even that mu a approaches 30, then this probability approaches alpha, the type one error rate. but on the other hand, if it is not specifically 30, if it's actually a power calculation, so the mu is not 30 then this calculation depends on the specific value of mu a that we plug in. So, to do a power calculation, you actually have to know the value under the null alternative that you want to plug in and then you can perform the calculation.