hello everyone we're going to be talking about a specific type of confidence interval called the one sample Z interval for p this is a follow-up to our previous lesson so it's going to involve the same calculations and um other things that we write up about confidence intervals now we're just going to put it all together so the first thing is is there's quite a few things that you have to check off when we're doing these type of questions and they're always going to say something like construct and interpret a 95% confidence interval for the true proport of something um it doesn't have to be 95% could be any sort of percent um that the confidence level is set at there's a few things we want to check off that we make sure we're doing we want to make sure we're naming the procedure so in this case here we have the name it's the one sample Z interval for p we want to Define what the parameter is so the true proportion of whatever we're doing in context of the problem we want to check our three conditions remember it's the random 10% and large count conditions as we did before in our previous lesson perform our calculations either by hand or we can use the calculators commands and I'll show you both ways um just a little bit warning on the calculator commands um you're likely not to get partial credit using those so um you you got to make sure you know what you're inputting on there if you want to get um full credit and then for your conclusion is always going to be the interpretation of the confidence interval so we are 95% confident that the true and then we finish the rest of that interpretation so let's look at the a problem in practice um remember on our formula sheet you are given this you are not given this you are provided the Standard air formula but not the entire interval so you have to know what we're kind of putting around it when we're doing our calculations uh so these type of questions usually have one or two parts um in this case here let's read through the examples so we can see the two parts um a recent poll of 738 randomly selected cell phone users found that 170 of the respondents admitted to walking into something or someone while talking on their cell phone and part A says instruct and interpret a 95% confidence interval for the proportion of all cell phone users who admit to walking into something or someone while talking on their cell phone Part B says interpret the confidence level the part A is the bulk of your work here when it says construct and interpret this is where you're following those procedures to get all the things that checked off the list here the last part's going to be usually be a followup question um in this case it's interpreting the level we have a we have a um template for that as well so let's talk about the first thing we do we want to name the procedure well the name of this procedure is one sample Z interval for p it's the only thing we've studied so far so this is kind of the only thing you have to work with and then what is the parameter we're trying to um Define so we're talking about the true what this is what what we're trying to estimate here so it's the true proportion of cell phone users who admit to walking into something or some while on the phone you can paraphrase that but make sure you use context on that so I'm going to write context here and don't use P hat use P because we're estimating the true proportion so it's the parameter not the statistic parameter not the statistic so uh it's going to be the true proportion not the sample proportion sometimes this part's called the state so if you open a textbook it might say like the state part so we're naming our procedure and our parameter I want to make sure that we identify procedure and parameter so I'm putting those both as p and p the next part we want to do is check our conditions we've done this before in our previous lesson so it's the random the 10% and the large counts so we have our random condition we just want to make sure we repeat that so we can paraphrase 738 randomly selected cell phone users as stated in the problem we're going to assume in this case that that's going to be less than 10% of all cell phone users we're not told how many cell phone users users there are but I'm sure there are billions in the world and the large counts well we want to make sure our large counts our number of successes and failures is at least 10 remember this satisfies the approximately normal condition and allows us to use a um a normal curve to make these estimates now remember that we have to have the number of successes n times P hat well we're not given P hat we're given the number of successes from this problem so that's literally 170 so 170 of the respondents admitted to walking into something that's going to be your emps P hat which means that your number of failures would be the difference between the 738 here and the 170 that's going to be how many are left over now as you can see both of those are going to be equal to 10 so we've met our large count conditions but we got to write that out and sometimes the textbooks will call this the plan part so this is the conditions the last part are usually the easiest part and we just want to do the calculations I say the easiest it's going to look difficult at first because I'm not going to show you any shortcuts but for our calculations remember we're only given the confidence interval generic formula and so that first thing that statistic is our sample proportion well what are we looking at we're looking at 170 out of 738 so we're going to divide those in our calculator we'll get a decimal you can route it to a few decimal places if you want I'm going to say that's so close to 23% that I'll just say 023 next thing is is our um our critical value and that has to do with your confidence level and remember in our previous lesson we are showed you how to get the critical value there so if you're not sure follow the previous lesson and or the videos posted and you'll see how to get that 1.96 the standard err formula on your formula sheet is provided remember it's right over here it's p hat 1us p hat Over N all under the square root so since we have our P hat and we have our n remember n is our sample size that's going to be 738 we can plug all of those into the formula so I just rounded at 23% so23 * 1 -23 over 738 under the square root and carefully use your calculator you're going to get about 1.55% so 0155 so the next thing we're going to do is put it back all together the formula and so you'll have your statistic 23 plus or minus your critical value which is going to be that 1.96 times your standard error which is 0.0155 next I want to calculate the um the critical value times the standard erir and that's going to give you your margin of err which is about 3% um give or take so it's a little bit off of 3% so I'm rounding to 3% here and so your final interval will be um 23 plus or minus 3% so um 20% to 26% approximately that's your calculation now let me show you how you can get those values using your step menu in your calculator you can get this interval and a lot of times on the scoring rubric they're only asking for the correct interval now if you make some minor calculation ears in here you might be able to e out partial credit if you show your work on your calculator you're not going to get partial credit but if you know the correct inputs you can get out everything in there correctly so what you'll do is you'll open up your calculator and you're going to go to stat go to tests and your calculator calls this the one Prop z interval it's one sample uh z z interval for p so one proportion Z interal so the X represents the number of successes in this case here it's going to be 170 so we're going to put 170 on top and then n is the sample size so our sample size is 738 so we'll put the 738 there and then you'll set your confidence level and the default level on your calculator is 95% which is the most common one so since ours is 95% we don't have to change it go down to calculate and it'll give you the interval so it'll be rounded to sometimes three four five decimal places depending on how the calculator rounds it um in this case here I said about 20 to 26% and our calculator says 19 .99 7% to 26.7 3% so really close um by hand that we got it um this pad is your sample proportion at about 23% and it gives you the um sample size so everything is in there and if you write all of this information down seeing you used your calculator one proportions Z interval you can get credit for this part but you have to put it incorrectly and so here are the screenshots again if you want to take a write those down or take a screenshot of that and finally the conclusion remember we have a template for that so we're always going to interpret the confidence interval in context of the problem so remember we have this template we are blank percent confident that the interval from this to this captures the true parameter in context so in this case here we have our 95% our interval was about 20 to 26% so 0. 2 to 26 captures the true and remember we're talking about cell phone users who admit to walking in a something or someone while talking on the cell phone so we're going to put that context in there and that's going to be your conclusion and then the textbook will call this the conclude so the textbook says state plan do conclude I'm a procedure parameter conditions calculations conclusions so I like to use those five things to remind us to have everything on there A lot of times there's a follow-up questions to these so in this case it says interpret the confidence level remember level's different than interval we have level template we used in our last lesson here's a reminder what that looks like um if we were to select many random samples of the same size from the same population and construct a blank percent confidence interval using each Sample about um same percentage of those intervals would capture the true parameter and context I do paraphrase this a little bit I say if we were to repeat this process many many times about blank percent of the confidence intervals we construct would capture the true parameter and context so a little bit less writing because it's already kind of a lot to write the level so it's 95% and then um 95% of those confidence intervals we construct would capture the true proportion of cell phone users that admit to walking into something while on the phone so it's a lot going on on this um so you have one check your understanding and you're going to want to pause this video and try it on your own so let's read through the question sleep awareness week begins in the spring with the release of the National sleep foundation's annual poll of us sleep habits and ends with the beginning of daylight savings time when most people lose an hour of sleep excuse me in the foundation's random sample of 10,29 us adults 48% reported that they often or always got enough sleep during the past seven nights part A says construct an interpret a 95 a 90% confidence interval for the proportion of all us adults who would report often or always getting enough sleep during the past seven nights and then Part B is the follow-up question in this case it's asking if interval provides convincing evidence so convincing evidence that fewer than half of all us adults would report they often are always got enough sleep during the past seven nights justify your answer so try this question on your own follow those those five steps and we're going to go through it in three two one here we go so the first thing is we want to make sure we list out the things we're going to do so we're going to name the procedure identify the parameter check the conditions the calculations and find our do our conclusion so anytime you see construct and interpret that's what you're doing over here so the procedure is the one sample Z interval for p this is the only thing we've learned so far the parameter is the true proportion of what are we measuring us adults who report getting enough sleep during the past seven nights in our conditions random 10% and large counts so our random sample of 1,29 adults that's right here our 10% is we're going to assume 10% of of less than 10% of all us adults were sampled and which is almost always going to be the case here in our large counts um we don't have the number of successes we're actually given the sample proportion so this 48% is actually your phap I'm G write that down here so since we're trying to calculate that remember we do n * P hat and we do n * 1 minus P hat so we actually have to do 1,00 29 * 0.48 and 1029 * 1us 0.48 don't worry about rounding those um you can round up or down if you want but um we just want to make sure that those are at least 10 which they are so we've checked all three of these we've met our conditions and the last two steps are the calculations and the conclusion so the calculations remember our sample proportion is 48% our Z star is a 90% confidence interval so you can calculate that at 1.645 um plug it into your standard error formula so the square OT 48 * 1 -48 over your sample size 1029 you'll get about 1.56% or 0156 and then you can put it all into the formula so 048 plus or minus 1.645 * 0156 use your calculator and you'll get about uh 2.57% for your margin of error and then you can go below and above it to get your final interval which is about 45.4% to 50.6% notice I highlighted the interval over here because that is how you want to write your interval at the end your conclusion is going to be we are and what's our percentage well we're 90% confident so 90% confident that the interval from this number to this number captures the true parameter and context so 90% confident that the interval from 0454 to 0506 captures the true and we're talking about proportion of all us adults who report report always getting enough sleep during the past seven nights often or always but I paraphrase it a little bit so proportion of us adults who report getting enough sleep during the past seven nights and that takes care of that part here's your calculations again again I'll walk you through the calculations on this one this is a little bit trickier you got to remember some things about the calculator on this one so remember that when you do your inputs on this we're not putting a percentage we're actually putting in the counts so it's like our our top number is going to be your number of successes so it's like 48% of 1029 well you can do that by multiplying 048 time 1029 and you get 4 93.9 to you remember your large Counts from before uh that's one of those large counts now your total sample size is 1029 so we put that over here and if you do your confidence level at 90% I believe it was set at 90% if you do it like this it's going to give you an air um because it just it doesn't allow you to put decimals into there so you have to go back and round to the nearest whole number so 493 is not going to be allowed you have to put 494 and it's not going to do the rounding for you so put them in like that and then you get your confidence interval 45.4% to 50.6% which is exactly what we got well actually it's more precise than what we got to be honest because the calculator will give you U no rounding erors so it'll be that'd be the preferred number to put in using your calculator just don't make any input mistakes and it'll be fine so you can use your calculator if you want for the due and then finally does the interval provide convincing evidence that fewer than half of all us adults would report they always got enough sleep during the past seven nights well what is fewer than half that's going to be less than 50% right so if you look at the interval yeah there are plenty of numbers less than 50% but also there's a number uh at 50% and above you know 50.1% 50.2% so I don't have convincing evidence that fewer than half because there are values at 50% and above in the interval so the interval does not provide convincing evidence in this case all right that's going to conclude this video the next uh video we'll talk about how to uh adjust the margin of error and find a sample size thanks for watching