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# Z-statistics vs. T-statistics

## Video transcript

I want to use this video to kind of make sure we intuitively and and otherwise and understand the difference between a Z this is Z statistic Z statistic something I have trouble saying and a and a T statistic T statistic so in a lot of what we're doing in this inferential statistics we're trying to figure out what is the probability of getting a certain sample mean so what we've been doing especially when we have a large sample size so let me just draw a sampling distribution here so let's say we have a sampling distribution of the sample mean right here it has some assumed mean value it has some assumed mean value in some standard deviation and what we want to do is we any results that we get any result that we get let's say we get some sample mean out here we want to figure out the probability of getting a result at least as Extreme as this so what you can either figure out the probability of getting a result below this and subtract that from 1 or just figure out this area right over there and to do that we're essentially we've been figuring out how many standard deviations above the mean we actually are the way we figure that out is we take our we take our sample mean we subtract from that our mean itself we subtract from that what we assumed the mean should be or maybe we don't know what this is we the mean should be and then we divide that we divide that by the standard deviation of the sampling distribution we divide that by the standard deviation of the sampling distribution this is how many standard deviations we are above the mean that is that distance right over there now we usually don't know what this is either we normally don't know what that is either and the central limit theorem told us that assuming that we have a sufficient sample size this thing right here this thing is going to be the same thing as the sample is going to be the same thing as the standard deviation of our population the standard deviation of our population divided by the square root of our sample size so this thing right over here can be rewritten can be re-written as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here divided by our population mean divided by the square root of our sample size and this is essentially our best sense of how many standard deviations away from the actual mean we are and this thing right here we've learned it before is a z-score or when we're dealing with an actual statistic when it's derived from the sample mean statistic we call this a Z statistic Z statistic and then we could look it up in a in a Z table or in a normal distribution table to say what's the probability of getting a value of this Z or greater so that would give us that probability so what's the probability of getting that extreme of a result now normally when we've done this in the last few videos in the last few videos we also don't know what the we also do not know what the standard deviation of the population is so in order to approximate that in order to approximate that we say that the z-score is approximately or the Z statistic is approximately going to be so let me just write the numerator over again over we estimate this using our sample standard deviation we estimate it using our let me just in a new color with using our sample standard deviation using our sample standard deviation and this is okay this is okay if if our sample size is greater than 30 or another way to think about it is this will be normally distributed normally distributed normally distributed if our sample size is greater than 30 even this approximation will be approximately normally distributed now if your sample size is less than 30 especially it's a good bit less than 30 all of a sudden this expression will not be normally distributed so let me rewrite the expression over here sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size we just said if this thing is well over 30 or at least 30 then this is this value right here this statistic is going to be normally distributed if it's not if this is small if that is small then this is going to have a T distribution this is going to have a T distribution and then you're going to do the exact same thing you did here but now you would assume that the Bell is no longer a normal distribution so in this example it was normal it was normal all of the Z's are normally distributed over here in a t distribution and this will actually be a normalized t-distribution right here because we subtracted out the mean so in a t distribution in a normalized t-distribution you're going to have a mean of zero and what you're going to do is you want to figure out the probability of getting a t-value this at least this extreme so this is some this is your T value you would get and then essentially figure out the area under the curve right over there and so a very easy rule of thumb is calculate this quantity either way calculate this quality quantity either way if if you have more than 30 samples if your sample size is more than 30 this is going your sample standard deviation is going to be a good approximator for your population standard deviation and so this whole thing is going to be approximately normally distributed and so you can use a z table to figure out the probability of getting a result at least that extreme if your sample size is small if your sample size is small then this statistic this quantity this is going to be is going to have a T distribution and then you're going to have to use a t table to figure out the probability of getting a t-value at least this extreme and we're going to see this in an example a couple of videos from now anyway hopefully that helped clarify some things in your head about when to use a Z to statistic or when to use a T statistic