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hello and welcome to the chapter 7 summary worksheet from Stevens's introduction to statistics the thinking to book here we're just going to go over some problems from chapter 7 and it is worth noting that these solutions as you see here are actually printed in the back of your book so we're moving along too quickly you can always check back there alright so let's take a look at this first problem we're considering the amount of coke and cans labeled as containing 12 ounces right we'll assume the actual amount of coke in such a can as a normally distributed variable alright and suppose you randomly select 25 cans and find the mean from that sample is eleven point eight five and the standard deviation is 0.3 so I have the N equals 25 that's my sample size the sample mean is eleven point eight five and the standard deviation is 0.3 right so for Part A the first question our request I guess it's construct a 95% confidence interval estimate for the mean amount of coke in all such cans right and and the big the biggest mistake that students make on this particular problem is they they go right to chapter seven point two they think oh this is a seven point two problem but that's actually not true this is a seven point four problem and the reason is the population standard deviation is unknown Sigma is unknown right remember you only know this the population standard deviation under special circumstances and you'll see a line that says the population standard deviation that wasn't in here it just said we had a sample with a mean of this and a standard deviation so we actually have the sample standard deviation not the population so Sigma is unknown so the big thing about 7.4 is that we used the T table instead of the Z table and the formula for the margin of error in calculating our confidence interval this looks similar to the one that we have whenever we use the Z table but there's a T in there and there's an S instead of a Sigma all right so now we just have to find the critical T value and that's not so bad we have n is 25 so that means my degrees of freedom is 24 right so 24 and in fact I'll put all the information down here so we have X bar is 11 point eight five the sample standard deviation is 0.3 and is 25 and the degrees of freedom 24 it's always 1 less than the sample size and this is a 95% confidence interval so really all I need is the 95% confidence level and my degrees of freedom and I have to go to my T table which is right here and actually it's after the C tables oh you know what I'm gonna go to full screen mode just to give a little more room in here okay so here we are the degrees of freedom so well we were 95% was our confidence level so we're in this middle column and the degrees of freedom was 24 so our critical values at the intersection of this row in this column so 2.06 4 so we'll go back to the worksheet I better get that into full-screen mode okay there now we have a little bit more room to work with so 2.06 for my critical value there's s the sample standard deviation and the sample size they get a margin of error point one two three eight and as with every confidence interval in this chapter we take the sample variable the sample mean in this case subtract a margin of error to get the lower limit take the sample mean add a margin of error to get the upper limit and the lower limit and upper limit create the confidence interval all right and I put a mu in there to designate my unknown population mean is between those two values and I'm 95% confident of that the second little question here all right I don't want to forget about that are you 95% confident that the mean amount of all coke cans is less than 12 ounces because look the cans has 12 ounces and your 95% confidence interval is the 11 point seven three to eleven point nine seven specifically below 12 so yes since that upper bound is below 12 you are 95% confident the true mean is below 12 ounces now suppose I want to make a 99% confidence interval and then asked the same question but the 99% confidence level so the problem remains exactly the same the only difference is that my critical value of T changes basically I go back to my T table and now instead of being at the 95% confidence level I'm at the 99 all right so I'm in this first column and then my critical value goes to two point seven nine seven so let's go back there so that's where this two point seven nine seven came from so what that means is that now we have a margin of error that's a little bigger because my confidence level went up sample mean minus the margin of error gives me my lower bound sample mean plus the margin of error gives my upper bound and my confidence interval is eleven point six eight to twelve point o2 now I ask the same question up here are you 99 percent confident the mean amount of coke in all cans is less than twelve ounces because this upper bound twelve point o2 is actually bigger than twelve it's possible that the population mean is in that interval thumbs is not less than twelve ounces so since the upper bound is above twelve I can't be or you can't be ninety nine percent confident the true mean is below twelve ounces it could be 12:01 or twelve point oh right okay so now assume and notice them all of a sudden here I'm back to chapter seven point two because I somehow know the population standard deviation that's point two so Sigma is suddenly known so that allows me to use the Z table and the margin of error is given by this formula so construct the 95% confidence interval well that's one of my popular confidence level so I go to my Z table let me put this here Z table okay and again this is pretty nice it's one of the popular levels 0.95 my critical value is 1.96 we see a lot so that's where this 1.96 comes from that is now the population standard deviation going the same place that the sample standard deviation used to be and the sample size underneath the square root so now we get a margin of error point zero seven eight four take the sample mean subtract the margin of error get the lower bound sample mean plus the margin of error get the upper bound and my confidence interval is eleven point seven seven to eleven point nine okay and Part D continue to assume to assume that you know the population standard deviation so Sigma is known what size sample would be required to be 95% confident that the sample mean is within point zero five ounces of the true population mean and this is still from 7.2 but it's a sample size question and if you go back to 7.2 you'll notice we have let me get that here oh wait I'm gonna get a full screen there you go we have this that's the formula for um unnecessary sample size give any maximum tolerance on a margin of error right so I rewrote that that's on that's back here so that's this formula right here right and in this case now let's see I need Z sub alpha over so I need the critical value of Z but again at the 95% confidence level that was 1.96 we found that in the last problem submit Sigma is given to be 0.2 and the maximum margin of error we are willing to tolerate is 0.05 so when you take these three values stick it into that formula that's 61.5 and again we always round up so you need to sample at least 62 cans of soda so that's sort of all the variations on a confidence interval about a mean the next problem asks you a confidence interval about a proportion right and that is chapter 7.3 all right so here we go a travel agent wants to estimate the proportion of vacationers who plan to travel outside of the United States in the next 12 months a random sample of 150 vacationers revealed that 45 had plans for foreign travel in that time right so I want to construct the 95% confidence interval estimate for the population proportion and from Chapter 7.3 the margin of error looks like this but I need P hats and Z's valve over 2 and n so let's work on P hat P hat in this case there's the number of successes over the number of trials or it's the sample proportion of vacationers who plan to travel outside of the country in the next 12 months so it's really just 45 over 150 which is 0.3 that makes Q hat 0.7 my sample size is the 150 vacationers and my critical value of Z this is the 95% confidence level so again the popular confidence level and a popular critical value 1.96 again right so that's my critical value right there P hat Q hat and so my margin of error 0.073 365 I keep a lot of extra decimals around and then I'll round to the appropriate three significant digits at the very end okay so I get my actual confidence interval I need my lower limit so I take my sample proportion P hat minus the margin of error you get 0.227 for the upper limit I take P hat plus the margin of error the only 0.3 7/3 so my 95% confidence interval 0.227 is less than P is less than 0.3 7/3 so I'm 95% confident that the proportion of all vacationers planning to travel outside of the US and over the next 12 months is between say like 20 and 37% right okay now we're gonna do let's see this is a sample size problem suppose if the 95% confidence level you need to have a margin of error no more than four percentage points how many vacationers when you have to sample and in this case use the sample proportion you calculated in Part A as an estimate of P hat so specifically we're going to say alright P hat is 0.3 now if you go back to and again this is chapter 7.3 if you go back to chapter 7.3 you'll see that there's two formulas this is P hat known this is P hat unknown since I know P hat I'll use this one seven point six alright so there's that formula right there I have P hat Q hat is 0.7 the maximum tolerated margin of error this ye down here is given right here four percentage points so e equals point O four equals point O four remember to change the 4% to a point O four decimal otherwise your answer will be way off and then my critical value at the 95% confidence level will once again be 1.96 from the Z table there's P hat Q hat Yi plug it all in I get five hundred four point two one again we always round up because we need at least five hundred four point two one in our sample that means we need at least five hundred and five people all right now we're looking at the same confidence level you need to have a margin of error no more than four percentage points again so again e goes 0.04 but in this case you have no estimate of P hat so what that means we have to use that second formula from Chapter seven point three P hat is not known I'm going to use this one and all that does is change the P hat times Q hat here into 0.25 so we'll go back to the worksheet here there's that formula again at the 95% confidence level the critical value is 1.96 found from the z table 0.25 replaces P at times Q hat there's my max margin of error and when you do that you get 600 point two five got a little bit bigger because I didn't know P hat or Q hat so and again rounding up I don't need at least six hundred and one people in your sample I believe that wraps us up for Chapter seven so um I'll catch up with you come chapter eight thanks see you later