all right in this video we are talking about calculating our type two error and we also introduced the idea of power of test so power of test is really simple it is just one minus beta once you have your type two eror error you just take one minus that value and the power of the test represents the probability of rejecting H subo given that a specific alternative is true so essentially meaning that you rejected when you should have and that's going to be this down here so your AG soot is false and we rejected it so we made this decision here that's our power of the test all right so now we're going to jump into calculating this so consider the example of Weights of males at a college from the type one and type two error video so that what we just talked about and we want to find the type two error when Mew is equal to 70 so if the actual population mean is 70 and we are still running this test of 67 and 69 as our critical values what is our type two error going to be if the actual population mean is 70 so the way that we do these problems and I'll do it more fully in an example in class um I cheated a little bit here and copied some diagrams from the textbook um but I'll show you a little bit better how to do this in class so we're going to start off with our original curve so that's this H subo curve curve on this side so if you just look at this portion this should look familiar so this is the curve from last time we have our 68 and then we have our reject regions of 69 67 and 69 and then we overlay our new curve of H1 and H1 is going to be um our new curve or hypothesis curve um and so we have this new curve that's saying okay this is our true mean as it relates to this new curve what are our odds of rejecting so if this is our curve what are our chances that we do not reject so I said that a little bit backwards at first we want to know what is the probability that we we're working here so we are given a false H subo so we are now saying we originally worked with 68 of our H subo and we found that to be false because it's actually equal to 70 so we know our H subo is false so as it relates to our 70 curve where are we do not rejecting our H subo and so our do not reject region is in here from our previous problem because our reject region is this section we need to find the do not reject region so we want to find the area under the new curve where H sub o is false so H sub o is false everywhere under this curve so everything under here is false but also where we do not reject H subo so that's going to be all of this area under the curve and remember that theoretically this curve goes on infinite so the standard normal curve is asymptotic um so theoretically this goes all the way down here and there could be a tiny little sliver here too so we end up subtracting that out and you'll see when we do our example here in a second so we want to find what is this shaded region this is the Shaded region where we um did not reject given that our mu is actually equal to 70 and so that's what this says up here given that our mean is actually 70 so using our mean 70 curve what is the probability that we did not reject or that it fell between 67 and 69 and so that's the definition of our type two error so to tackle that we need to find what these Z values represent for this curve so to do that we're going to have Z1 and Z2 Z1 is going to be equal to so this is the xar minus mu over Sigma over root n equation again all same numbers from last time I lied now we are using a sample size of 64 so our n now is 64 and our Sigma is still 3.6 so when we do that our xar is going to be that point on the curve so that's going to be 67 and now we want it in relation to our new mean so we're going to use the 70 ID 3.6 / the square < TK of 64 which gives us 6.67 and Z2 is going to be 69 - 70 / 3.6 over < TK 64 which is equal to - 2.22 so to find our probability we need to find the we want to find this section under this curve so to do that we need to take the probability of Z less than 2.22 minus the probability of Z less than 6.67 if that doesn't come straight to your mind now is a great time to pause the video and make sure that you understand where those numbers come from this relates back to some of the original stuff we were doing with standard normal curve so make sure that you understand how I got this equation set up so then when I pull those numbers from my Z table I get 0.0132 minus 0 is equal to 0.0132 and this is my Beta value so then my power is going to equal 1 minus beta which is going to be somewhere around 0.87 no that was terrible math 0. 987 Yes somewhere around 0987 for my power of test that's a really great number 98% strength in our test is great so now let's do another example now let's find the type two error when the mean is actually 68.5 and so now we have this curve this is our when H subo is false so h o is false anywhere under this curve and this is when our mu is equal to 68.5 and we know H is false because 68 does not equal 68.5 um and then we want to find where under our false curve are we in our do not reject region so now our do not reject region is defined by these two lines so this is our do not reject region but we specifically want the area under this curve so same method as before we're going to take our Z1 and z two values so we have Z1 is equal to 67 minus the MU of the curve that we want to relate it to we want to relate it to our new or false curve which is 68.5 divided by 3.6 over theare < TK of 64 and that is equal to - 3.33 and then we have Z2 which is 69us 68.5 / 3.6 / < TK of 64 and that is equal to 1.11 now we need to find our probabilities again so it's going to be the probability of Z less than 1.11 minus probability of Z less than - 3.33 so we're taking this entire area and then we are subtracting out this section making that go away so that we are just left with that Center section and so then from that we get 0.866 5 minus 0.00004 which is equal to 0.866 1 this is my Beta so from that I can get my power of the test equal to 1 minus beta equal to about 0.349 I think off the top of my head and that's really bad we only have 13% power with this um and we have 87% type two error and so this relates back to what I was saying at the end of the last video where the further away our true mean is the more likely we are to be able to detect that if we're only off by 0.5 we're almost never going to pick that up there's very little chance that we're really going to start to see that until we have really large samples um whereas when we are really far off we only have a small chance of not picking that up of seeing that hey this is not right um so that is type two error and power of the test and like I said we'll do a handful of these examples um but this is probably one of the most confusing topics because there's so many different overlapping curves and things to translate between so definitely dedicate a little bit of extra time to understanding this topic in order to be successful with type two error and power of the test