So let's just reiterate when we're testing our alternative being that mu is greater than mu0. And then I'm going to specify power in terms of 1 minus beta, where beta is the type two error rate. Then we get this equation. Our power is equal to the probability that x bar is larger than mu0 plus the standard normal quintile times the standard error. Where this probability's calculated under the hypothesis that mu equals mu a. And then I simply reiterate that point here on the next line, where I'm explicitly stating that x bar is normal, with mean mu a and variance sigma squared over n. So the point I'd like to make, is that the unknowns in this equation are mu a, sigma, n, and beta, or equivalently to beta the power you want. The knowns are mu0 and usually you know alpha, you know exactly which type one error way that you would like. So, given these four unknowns and our two knowns, this is simply an equation. If you specify any three, you can solve for the fourth. So if you know the alternative mean that you'd like to detect. This a sigma that you're willing to assume, the n that you want then you can solve for power like we've done previously, otherwise what you could do, is you could say here's power that I want, here's a m, mean that I'd like to detect. And here's a sigma that I can tolerate, and then I'd like to solve for the n that would allow me to have that power, maybe at the process of planning a trial. And this is how power calculations usually work. You are particularly concerned most of the time with either n or beta. You usually either want to figure out for a particular power that I want, what n would I like to have, or should I have? Or, I can only conduct a study of this size, of size n, what is my power for doing that? Should I even waste my time? So, those are the two most common ways, but of course, you could also solve for mu a, or sigma. So I'm hoping at this point in the class, that you'll be able to take the work we've done for the greater than test, and do it for the less than test. H0, mu, less than mu0, just by extending the arguments from the previous set of slides. When you're testing not equal to just make sure you use the z1 minus alpha over 2, instead of, z1 minus alpha, that critical value. So for example, for alpha equal to 0.05, you want to make sure you use 1.96 and not 1.645. And then just do the one sided power for whatever direction mu a is, whether it's mu a greater than mu0, or Mu a less than Mu0. And this isn't, this is only approximately right, it omits a small component of power, but that, that component of power is only meaningful, if Mu a and Mu0 are close together. So let's go through some basic rules about power. As Alpha gets larger, our power gets larger. So the power of a 1 sided test is greater than the power of the associated 2 sided test, and we can see that by virtue of just dividing alpha by 2 in this, in this, bullet point above. The further mu a gets, gets from mu0, the higher the power is. As n goes up the sample mean has less variability, and so, we have higher power. As sigma goes down our sample mean has less variability, and so we're going to have higher power. Now here's an interesting fact, power usually doesn't depend on all of these separate parameters, but on some function of those parameters. And in this particular case, that function is one dimensional. So you really only need to know one number to, to know to, to calculate power, and it's basically the difference between the null and alternative means divided by the standard error. And power only depends on those things, or if you have power and this number then you can calculate, then you can calculate n. So this quantity the difference and the means divided by sigma, is called the effect size. And, what's nice about the effect size is that it's unit free, so when you subtract mu0 and minus mu a, mu, and divide it by sigma, you get a unit free quantity. And that's why the effect size is very useful because it, you know, being unit free it has some hope of being interpretable cross problems.