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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

- [Tutor] What I wanna do in this video is give a primer, I'm thinking about when to use a 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 are 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, no, the population proportion is greater than that or less than that or it's 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 suggest 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've 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 zero, 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 wanna divide it by what's often known as 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 our 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 one 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 mu one and then there's going to be an alternative hypothesis, that maybe your population mean is not equal to mu one and you're gonna do something very simple, you take your population, you take a sample of size n and instead of calculating a sample proportion, you calculate a sample mean and actually 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 in the null hypothesis, so that would be this right over here, that's what that zero 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 by an estimate of this, 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 will see, we will then look this up in a t table and this will give us a better sense of the probability.
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