all right in this video we are continuing with our discussion on confidence intervals for chapter nine in this video specifically we are moving into difference in in means or difference between two means and in the previous video we were talking about just one sample we we're looking at using one xar to estimate one mu in this example we are looking at two samples so we have one xar and well we have two x bars and then we also have two M's and we're looking at estimating what the difference between those two samples are so in this video we're going to kind of break down what our different scenarios are and the different equations we might have when we're talking difference of means we have two main questions we need to ask similar to the previous discussion question one is is this an independent sample or are the samples independent of one another and you can either answer that with yes or no and then you have a second question which is do you know population variance keep in mind if you know population standard deviation you also know your population variance and so then again you can answer this yes or no and so we have four different situations that might come out of this so we're going to pair this up with our equations above so in the first one this is for two independent samples with known variances so if we answer yes to both questions yes it's independent yes we know the population variances then this is scenario one in scenario one we have xar minus xar 2 z alpha over two plug in our sigmas and N values calculated so this is fairly similar to what we've been working on it's just expanded to add in that second sample variance in the next situation we have two independent samples with variances unknown but assumed equal so that's going to be here yes yes we have independent samples no we don't know the population variance so after we ask ourselves do you know population variance we actually have a third question Q3 are they assumed equal and so to understand if these are assumed equal if the r itio of our sample variances S12 over S2 squar is between one2 and two if that is true then yes we assume Sigma 1^ 2 is equal to Sigma 2^ s that our variances are equal they are still unknown but we assume they are equal if no if our ratio does not fall in this range then no assume Sigma 1^ 2 does not equal Sigma 2^ 2 so situation two is independent samples unknown population variance assumed equal so this is situation two down here so then again we would just plug in all of our numbers until we get what we need and then situation three we have independent samples variances unknown and assume different so now we have independent samples yes unknown population variance assumed different so this is situation three so this is situation three and then in the situation where we have dependent samples so not independent samples when we have dependent samples this is equation four or situation four and I forgot to copy this equation down but it's on the top of the next page of your formula sheet and this is called paired observations so now we have to use them as paired observations because they are dependent on each other we can't use them as two separate samples because they depend on one another and so for that our equation for paired observations is D Bar minus t Alpha over 2 SD over squ < TK of n less than mu D less than D Bar plus dot dot dot so this same thing goes on that side and like I said that's on your formula sheet on the next page for paired observations so that's when we have dependent samples so those are the different situations that we have when we are looking at difference between means with two samples so the main difference is here um when we have unknown variances just like when we were working with our confidence intervals for mean when our variances are unknown we are using T so we have t in both of these and when our variances are known we're using Z and so that's true between what we talked about last time and this time unknown variance we use our T table variance known population variance known we're using our Z table and then whenever we are using our T distribution we have to calculate our degrees of freedom so you can see down here we have degrees of freedom here and here our degrees of freedom are really messy it is truly that whole equation um so keep an eye out for that and then when we have equal variances we have to calculate our pulled variance or pulled standard deviation so that's that formula but if they are assumed different we just use our sample standard deviation sample variances separately so those are some of the differences between these and so then if we do a problem which we're not going to in this video because there's so many different examples we'd have to work through we'll just tackle that in class if we were to do a problem when we get our solution if we have a negative lower limit and a positive upper limit that tells us that zero is a possible value and so it could be possible that mu1 minus mu2 equals zero and so mu1 minus mu2 equals 0 is a possibility which means that mu1 could equal mu2 and so because zero is a possible outcome we cannot conclude a difference cannot conclude a difference so that is one conclusion that we could have another conclusion we might find that we have a negative lower limit and a negative upper limit so I'm going to backtrack for a second here so if we have -2 less than mu1 minus mu2 and positive2 now you can really see what I mean by negative lower limit positive upper limit zero is a potential value in between there so down here now if we have six less than and -4 it's going to be a negative number the difference between means is going to be negative and so if mu1 minus mu2 equals a negative let's say it equals -1 then we would have mu1 is equal to mu2 minus one so if we let mu1 equal 100 then mu2 for these to equal out mu2 has to equal 100 one so what that means is yes there is a difference and mu2 is larger than mu1 so we just worked a super quick sample here to show that is true so I'm going to erase that if we have both negatives then U2 is larger and then conversely if we have a positive lower limit and a positive upper limit we're going to say yes there's a difference and now mu1 is larger than mu2 so to work a quick example mu1 minus mu2 if it's a positive number let's say the true difference is positive 5 then we get mu1 is equal to mu2 + 5 so if we let mu1 equal 100 again then we get that mu2 is equal to 95 so that gives us 100 is equal to 95 + 5 equal 100 so then our equation is true which means that these numbers are correct and now we can see that mu1 is larger than mu2 all right we will work some examples in class please come ready with any questions that you have and we'll talk about them there thanks