- [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.