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Example: Probability of sample mean exceeding a value

AP.STATS:
UNC‑3 (EU)
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UNC‑3.S (LO)
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UNC‑3.S.1 (EK)

Video transcript

the average male drinks two liters of water when active outdoors with the standard deviation of 0.7 liters you are planning a full-day nature trip for 50 men 50 men and will bring 110 liters of water what is the probability that you will run out of water so let's think about what's happening here so there's some distribution of how much how many litres the average man needs when they're active outdoors and let me just draw an example it might look something like this so this is so they're all going to need at least more than zero liters so this would be zero liters over here the average male so the mean of the amount of water a man needs when active outdoors is two liters so two liters would be right over here so the mean is equal to two liters it has a standard deviation of 0.7 liters or 0.7 liters so the standard deviation maybe I'll draw it this way so this distribution once again we don't know whether it's a normal distribution or not it could just be some type of crazy distribution so so maybe you know some people need almost close to well everyone needs a little bit of water but maybe some people need very very little water then you have a lot of people who need that maybe some people who need more and maybe you know no one can drink more than maybe this is like four liters of water so maybe this is the actual distribution and then one standard deviation in is going to be 0.7 liters away so this is 1.7 liters is so this would be one liter two liters three liters so one standard deviation is going to be about that far away from the mean if you go above it it'll be about that far if you go below it let me draw this is the standard deviation that right there is the standard deviation to the right that's the standard deviation to the left and we know that the standard deviation is equal to I'll write the zero in front 0.7 liters so that's the actual distribution of how much water the average man needs when active now what's interesting about this problem we are planning a full-day nature trip for 50 men and we'll bring a hundred and ten liters of water what is the probability that you will run out so the probability that you will run out let me let me write this down the probability that I will or that you will run out run out is equal or is the same thing as the probability the probability that we use more than 110 liters use more than 110 liters on our outdoor nature day whatever we're doing which is the same thing as the probability if we use more than 110 liters that means that on average because we have 50 minutes so 110 divided by 50 is what that's two points let me get the calculator out just so we don't make any so we don't make any mistakes here so this is going to be the calculator out so on average if we have 110 liter 110 liters it's going to be drunk by 50 men including ourselves I guess that means that it's the problem so that we would run out if on average more than 2.2 liters is used per man so this is the same thing as the probability of the average or maybe we should say the sample mean or let me write it this way that the average the average water use water use per man of our 50 men is greater than or we could say greater than or equal to greater let me say greater than well I'll say greater than because if we write on the money then we won't run out of water is greater than 2.2 2.2 liters per man so let's think about this we are essentially taking 50 men out of a universal sample and we got this data who knows where we got this data from that the average man drinks 2 liters and that the standard deviation is this maybe some there's some huge study and this was the best estimate of what the population parameters are that this is the mean and this is a standard deviation now we're sampling 50 men and what we need to do is figure out essentially what is the probability that the mean of this sample that the sample mean is going to be greater than 2.2 liters and to do that we have to figure out the distribution of the sampling mean and we know what that's called if the sampling distribution of the sample means and we know that that is going to net be a normal distribution we know a few of the properties of that normal distribution so this is the distribution of just all men and then if you take samples of say 50 men so this will be let me write this down so down here I'm going to draw I'm going to draw the sampling distribution the sampling distribution distribution of the sample mean of the sample mean when when n so when our sample size is equal to 50 when our sample size is 50 so this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use so let me draw that so let's say this is frequency and then here are the different values now the mean value of this the mean the mean let me write the mean of the sampling distribution of the sample mean this x-bar that's really just the sample mean right over there is equal to it's going to be equal to if we were to if we were to do this millions and millions of times if we were to plot all of them the means when we keep taking samples of 50 and there were plot them all out we would show that though this this mean of distribution is actually going to be the mean of our actual population so it's going to be the same value I want to do in that same blue it's going to be the same value as this population over here so that is going to be two liters so we still have we're still centered at two liters but what's neat about this is that the sampling distribution of the sample mean so you take 50 people find their mean plot the frequency 50 people find the mean this is actually going to be a normal distribution regardless of you know this one just has a well-defined standard deviation I mean it's not normal even though this one isn't normal this one over here will be and we've seen it in multiple videos already so this is going to be a normal distribution and the standard deviation and we saw this in the last video and we hopefully we've got a little bit of intuition for why this is true the standard DV station actually put a better way the variance the variance of the sample mean is going to be the variance so remember it's going to be this is standard deviation so it's going to be the variance of the population divided by N and if you wanted the standard deviation of this distribution right here you just take the square root of both sides so if you take the square root of both sides of that we have the standard deviation of the sample mean is going to be equal to the square root of this side over here is going to be equal to the standard deviation of the population divided by the square root of N and what's this going to be in our case we know what the standard deviation of the population is it is 0.7 0.7 0.7 and what is n we have 50 men so 0.7 over the square root of 50 now let's figure out what that is with the calculator so we have we have 0.7 divided by the square root of 50 and we have 0.09 well I'll say 0.098 was pretty close to 0.99 so I'll just write that down so it's this is equal to 0.09 9 that's going to be the standard deviation of this so it's going to have a lower standard deviation so it's going to look the distribution it's going to be normal it's going to look something like this so this is this is three liters over here this is one liter one liter the standard deviation is almost a tenth so it's going to be a much narrower much narrower distribution it's going to look something I'm drum I try my best to draw it it's going to look something like something like this something like this you get the idea where the standard deviation right now is almost point one so it's point zero nine almost a tenth so it's going to be you know something one standard deviation away is going to look something like that so we have our distribution it's a normal distribution and now let's go back to our question that we're asking we want to know the probability that our say simple we'll have an average that our sample will have an average greater than 2.2 so this is this is the distribution of all of the possible samples the means of all of the possible samples now to be greater than two point two two point two is going to be right around here two point two is going to be right around here so we want to essentially are asking we will run out if our sample mean falls into this bucket over here so we essentially need to figure out what is you can even view it as what's this area under this curve there and to figure that out we just have to figure out how many standard deviations above the mean we are which is going to be our z-score and then we could use a Z table to figure out what this area right over here is so if we are so we want to know when we are above 2.2 liters so 2.2 liters 2.2 we could even do it in our head 2.2 liters is what we care about that's right over here that is our mean is 2 so we are point two above the mean above the mean above the mean and if we want that in terms of in terms of standard deviations we just divide this by the standard deviation of this distribution over here and we figured out what that is the standard deviation of this distribution is 0.09 9 so if we take and you'll see a formula where you take this value minus the mean and divide it by the standard deviation that's all we're doing we're just figure out how much how many standard deviations above the mean we are so you just take this number right over here divided by the standard deviation so zero point zero nine nine or zero point zero nine nine and then we get let's get our calculator and actually had the exact number over here so we can just take point two we could just take this point two point two divided by this value over here and on this calculator when I press second answer just means the last answer so I'm taking point two divided by this value over there and I get two point zero two zero so this means so that means that this value or actually right this probability is the same probability is the same probability of being 2.0 two standard deviations standard deviations oh maybe I should write it this way more than more than let me write it down here where I have more space so this is this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select remember if we take a bunch of samples of 50 and plot all of them we'll get this whole distribution for the 150 the group of 50 that we happen to select the probability of running out of water is the same thing as the probability of the mean of those people will be more than two point zero two zero standard deviations standard deviations above the mean above the mean of this distribution which they're actually the same the same the same distribution so what is that going to be and here we just have to look up our Z table remember this two point zero two is just this value right here 0.2 divided by zero point zero nine I just have to pause the video because there's some type of fighter jet outside or something anyway hopefully they won't they won't come back but anyway so we need to figure out the probability that the sample mean will be more than two point zero two standard deviations above the mean and to figure that out we go to a Z table and you can find this pretty much anywhere usually it's in any stat book or on the internet wherever and so essentially we want to know the probability the Z table will tell you how much area is below this value so if you go to Z of two point zero two that was the value that we were dealing with right you have two point zero two it was so you go for the first digit we go to two point zero and it was two point zero two point zero two two point zero two is right over there right so we had two point zero and then the next digit you go up here so it's two point zero two is right over there so this point nine seven eight three let me write it down over here this point nine seven eight three I want to be very careful zero point nine seven eight that Z table did not that's not this value here this point nine seven eight three on the z table that is giving us this whole area over here it's giving us the probability that we are below that value that we are less than we are less than 2.0 two standard deviations above the mean so it's giving us that value over here so to answer our question to answer this probability we just have to subtract this from one because these will all add up to one so we just take our calculator back out and we just take one minus one minus 0.9 seven eight three is equal to point zero two one seven so this right here is point zero two one seven or another way you could say it it is a two point one seven percent probability that we will run out of water and we are done let me sure I got make sure I got that number right so that number it was Y at point two zero one seven right so it's a two point one seven percent chance we run out of water
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