so the last thing I want to talk about is I want to summarize everything that we've done about confidence intervals and we have time to work on one maybe two problems um okay so here's the summary confidence intervals um so confidence interval summary if you're doing a confidence interval for a mu where you know Sigma it's the formula is xar plus or minus Z * Sigma xar if you don't know Sigma it's a t distribution and your form looks like xar plus orus T * sxar what about a confidence interval for proportion P the formula is p hat plus or minus Z * Sigma P hat okay now writing it out like this it's almost obv the the pattern should become very clear and obvious as to what's going on with the formula for a confidence interval um the only thing that remains to know is uh what's the formula for Sigma xar what's the formula for Sigma P hat um and that's that's a separate that's a separate thing that you guys know as well I like writing it like this the textbook likes to write the formula with everything all mushed in in there um so for example P hat plus or minus Z sub Alpha / 2 * square root of P Hat 1 minus P hat Over N the square root blah all that stuff so I don't like writing it like that because it obscures uh as far as I'm concerned what's really going on what's really going on is that the basic formula for confence Interval is always the same thing with these adjustments that you know Sigma xar vers versus Sigma P hat so that's one thing um okay something really important um when you're working on a confidence inter problem particularly if it's a if it's a word problem um start off your problem thinking about is this a problem about means or is it about proportions and it's not always immediately obvious uh for example in the question it might say um you might have to calculate the proportion for example it might say 67 out of 120 people said yes or we interviewed 120 people and 67 said yes so what is that problem is this problem about means or proportions it's about proportions and how do you find your P hat right it would be 67 divided 120 I have to say this is definitely one of the three most most common mistakes that I see is that people don't when they see 67 out of 120 they don't recognize that as a proportion and as part of a proportion problem and they simply say oh I guess my xar is 67 and of course once you do that I'll bottom the comment you made before well I don't have a sigma well the that becomes a problem when you try to calculate the formula Sigma ided theare root of n uh and then you and then as a someone grading the work I see all these kinds of problems from people not recognizing that this is really a proportion problem so that that's again that's a major thing to to keep in mind as you're doing this um and then finally we saw um sometimes we're we're told um we're given a constraint for what the margin of error E should be and then go ahead and find n and then we have formulas for that uh on the one side in case we're dealing with means on the other side in case you're dealing with proportions okay so that's that's the whole that's the that that's this whole topic boiled into half a page all right all right so let's let's look at a problem and do it from scratch okay are the people in support of stricter gun control laws out of 400 people surveyed 296 says so what is your 96% confidence interval uh what's your point estimate what's your margin of error Etc okay so how do we start this how do we start this question what do we say okay so what's the formula going to [Music] be there you go so in particular Elizabeth is saying that this problem is about a proportion okay so that's one important thing to recognize okay so here's our formula so the formula is going to guide our work so what do we need to figure out okay so even easier first I'm going to say let's find P hat then then yes we we should do that okay so a couple things so first of all what's P hat how do we find P hat no what is p hat mean in words well it's similar the xar but what is p hat meaning words P hat P hat represents the what it's the proportion of the sample whoever said that yes so p is the proportion from the sample so what is the proportion from the sample in numbers there you go so if you know what these things are it helps do the problem P hat is the proportion of the sample what's the proportion of the sample 296 out of 400 which is 74 next next thing next thing we can find albana Sigma P hat exactly so how do we find Sigma P hat so P hat so 74 Time 1 - 74 all over what divided by what divided by N400 so let's work that out 0219 next yes is it normal and how do we see if it's normal is that at least 10 so we got 400 74 cuz we're using p hat and then 1 - 74 is at at least 10 it's 76 something or other certainly bigger than 10 so it's normal now that we know it's normal what can we do draw picture and so what goes on the picture zero in the middle for Z and what else P hat values okay and what else point I'm sorry where does 74 go underneath the zero middle for p hat okay and what else and so where does 96 go the middle for it's the middle area right okay and so now we can start to figure stuff out so once we know that what else can we fill in point what 02 in each tail because it's 004 divid by two okay and then once we have that now what do we find zcore so area to the left is 02 knock it out there's our zcore 2.05 37 [Music] and now we have everything we need so the confidence interval is numbers 74 plus or minus z uh well zcore is 0 2.05 37 times Sigma P hat which is 0219 that all works out to 74 plus or minus .045 and that works out to 695 7850 um and we just St our conclusion so we're 96% confident that we're between and again these are proportions so you don't have to but I like to State it in common language so 695 is really 69.5% um so it's it's between 69.5% and 78.5% uh want stricter gun control laws um and also what is the point estimate yeah it's 74 that's our best guess and what is the margin of error point no yeah it's 0.0450 it's the whole plus minus part now again remember we're talking about proportions so that 74 is really the same as saying 74% and that margin of error is the same as saying 4.5% [Music] and then just to run through one of these problems which actually goes quite quickly suppose you want e to be 05 with 95 with 99% confidence and we'll take P hat to be 0.5 again how big does your sample need to be how do you do this problem yep it's just the formula so n equals and you'll get practice using it it's z^2 * P hat * 1us P hat over e^2 half okay so let's see what is it that we know what we need to figure out so P hat we know is 0.5 then we've got 1 minus .5 and we know e is 05 what's the only thing we squared what's the only thing we need to figure out Z and how do we figure out the zcore what's going to help us figure out the zcore yes so 99% and what should I do with that draw a picture so here you've got the picture and what what do I put on the picture 99 is the middle area and once we know that what else do we know right it's 001 split in half so it's 0.005 in each tail so go in here area to left is 0.005 and there's your zcore 2.57 58 now we've got our zcore we can just plug that in to the formula where we need it and if you work this out you get [Music] 664 um any questions okay you