so the other good news is that when you're doing a problem everything works almost exactly the same so let's work through a problem let's see that it goes let's see how it goes almost exactly the same how many M M's in a bag they're supposed to be a thousand M M's in a bag but I claim it's more than that so you take a sample of 80 bags and from that you get X bar is 1036 and you don't have Sigma but from a sample you get S equals 97.3 so what's the first thing that we do write down hoh1 so what is it what's the claim okay ho is not a thousand ho is a statement what's the statement yes so for ho you get mu equals a thousand and what about for H1 mu is greater than a thousand all right so now we need to write down our things so first we write down mu X bar now what is Mu X bar that's a thousand now can we write down Sigma X bar there's no Sigma we can't have Sigma X bar so I guess instead of Sigma X bar what do we have s x bar how do you think we find S of X bar s divided by the square root of n so it works it's almost the same just a little bit different so in this case we get 97.3 divided by the square root of 80. let's work out what that equals 97.3 divided by the square root of 80. 10.8785 and what's the next thing we need to ask is the distribution normal so is this distribution normal is this distribution normal no why is it not normal we don't have Sigma we don't have Sigma all bets are off what distribution are we going to use we're going to use the T distribution just like we were like we were just talking about so is this normal no we don't have Sigma we have S so we're going to so use T distribution so now we're going to draw the picture but like I said the picture for T looks almost the same as the picture for s so you just pretty much draw the same picture so now how do we label this 0 goes in the middle but not for Z zero goes in the middle for what for t so label that with t uh let me put a box around that that's different okay so what else do we have a thousand goes in the middle for what X bar is still an X we're still talking about X bars it's still an X BAR value now what else did we mark off 1036 and then we shade to the right because it's greater than so now what do we have to find we don't find the z-score we find the t-score well guess what the formula for T is it's basically it's exactly the same so to find the T exactly so we use our value minus X bar minus mu X bar divided by before we had Sigma X bar now we're going to use SX bar of that so X bar was 10 36 mu X bar was a thousand and s x bar was 10.8785 so just have to work that out so this equals 36 divided by 10.8785 so that gives us a t score of 3.3093 so let's fill that in on the picture now what do we have to do now we have to find the area I'll use stack disk so what do we do analysis distribution and which distribution are we using the T distribution so what do I have to fill in okay so the T value was 3.3093 and now we need to find you have to change the degrees of freedom don't forget that that's important that is a difference uh compared to the z-score normal distribution so what's the degrees of freedom here well how do we figure out the degrees of freedom let's flip back to it so we remember that degrees of freedom is just n minus one exactly so let's act that this is important let's actually write this down in this problem so in addition to all these things we need to oops we need to write down degrees of freedom which I'm going to call DF so what is DF it's n minus 1 which is 80 minus one that's 79. so degrees of freedom 79 now we're ready click evaluate and it tells me what I need the area to the right is .0007 area to the left is 0.993 so now we're ready to make our conclusion what's our conclusion we are 0.993 confident that mu is greater than a thousand now say it in English that how do you say mu greater than a thousand in English in this problem there are more than a thousand M M's in a bag on average and what's your p-value 0.0007 and as Benny said we say we reject h0 except H1 with 0.993 confidence um so a couple of quick comments um in Excel if you use Excel and the T distribution it only takes positive T values try to put in a negative T value give you an error so just be aware of that for Excel stat disk is fine um next when you calculate your t or Z value that is called the test statistic that's just terminology okay um and really quickly what I have in the notes Here is a comparison between normal distribution and T distribution as we just saw everything is basically the same but a little bit different chart represents a summary of all those things um I'm sorry the T or Z value is called the test statistic and one other bit of notation uh that thing that's called the significance uh mathematicians like to put a little Greek letter for that it's called it's called Alpha so it's a funny looking X or somebody will say it looks like a fish whatever that's called Alpha um and one final comment the way that the textbook covers this subject so I mentioned already a couple times that there are several can write down your conclusion these all come from the fact that over the years there were several different approaches to how do you solve this problem I have taught you one approach I've taught you the approach that I think is the clearest and best approach the textbook does I think all the approaches so they sometimes they will sometimes ask questions in a slightly weird way that have to do with the approaches that they use if you see something like what is the anything anytime you see the phrase critical value skip it don't even ask me I don't understand this what does this mean because I'm not going to explain it to you I have tried it becomes very confusing anytime you see critical value skip it use this method and you'll be able to answer all the questions okay thank you get out of here so we've now done up through 10.3 so you can work on that