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# When to use z or t statistics in significance tests

AP.STATS:
VAR‑7 (EU)
,
VAR‑7.E (LO)
,
VAR‑7.E.1 (EK)

## Video transcript

what I want to do in this video is give a primer I'm thinking about when to use the Z statistic versus a T statistic when we are doing significance tests so there's two major scenarios that we will see in an introductory statistics class one is when we are dealing with proportions so I'll write that on the left side right over here and the other is when we are dealing with means in the proportion case when we're doing our significance test we will set up some null hypothesis that usually deals with the population proportion we might say it is equal to some value let's just call that P sub one and then maybe you have an alternative hypothesis that well know the population proportion is greater than that or less than that or is just not equal to that so let me just go with that one it's not equal to P sub one and then what we do to actually test to actually do the significance test is we take a sample from the population it's going to have a sample size of n we need to make sure that we feel good about making the inference we've talked about the conditions for inference in previous videos multiple times but from this we calculate the sample proportion and then from this we calculate the p-value and the way that we do the p value remember the p value is the probability of getting a sample proportion at least this extreme and if it's below some threshold we reject the null hypothesis and it suggests the alternative and over here the way we do that is well we find an associated z value for that p for that sample proportion and the way that we calculate it we say okay look our z is going to be how many of the sampling distributions standard deviations are we away from the mean and remember the mean of the sampling distribution is going to be the population proportion so here we got this sample statistic this sample proportion the difference between that and the assumed proportion remember when we do these significance tests we try to figure out the probability assuming the null hypothesis is true and so when we see this P sub 0 this is the assumed proportion from the null hypothesis so that's the difference between these two the sample proportion and the assumed proportion and then you'd want to divide it by what's often known the city the standard error of the statistic which is just the standard deviation of the sampling distribution of the sample proportion and this works out well for proportions because in proportions I can figure out what this is this is going to be equal to the square root of the assumed population proportion times 1 minus the assumed population proportion all of that over N and then I would use this Z statistic to figure out the p-value and in this case I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion now with means there's definitely some similarities here you will make a null hypothesis maybe you assume the population mean is equal to mu1 and then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu1 and you're going to do something very simple you take your population take a sample of size n instead of calculating a sample proportion you calculate a sample mean unless you can calculate other things like a sample standard deviation but now you have an issue you say well ideally I would use a Z statistic and you could if you were able to say well I could take the difference between my sample mean and the assumed mean and the null hypothesis so that would be this right over here that's what that 0 means the assumed mean from the null hypothesis and I would then divide by the standard error of the mean which is another way of saying the standard deviation of the sampling distribution of the sample mean but this is not so easy to figure out in order to figure out this this is going to be the standard deviation of the underlying population divided by the square root of n we know what n is going to be if we conducted a sample but we don't know what the standard deviation is so instead what we do is we estimate this and so we'll take the sample mean we subtract from that the assumed population mean from the null hypothesis and we divide it by an estimate of which is going to be our sample standard deviation divided by the square root of n but because this is an estimate we actually get a better result instead of saying hey this is an estimate of our Z statistic we will call this our T statistic and as we'll see by it will then look this up in a t-table and this will give us a better sense of the probability
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