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## Tests about a population mean

# Example calculating t statistic for a test about a mean

AP.STATS:

VAR‑7 (EU)

, VAR‑7.E (LO)

, VAR‑7.E.1 (EK)

## Video transcript

- [Tutor] Rory suspects that
teachers in his school district have less than five years
of experience on average. He decides to test his null hypothesis is that the mean number of years
of experience is five years and his alternative hypothesis
is that the true mean of years of experience
is less than five years, using a sample of 25 teachers. His sample mean was four years and his sample standard
deviation was two years. Rory wants to use these sample data to conduct a t test on the mean. Assume that all conditions
for inference have been met. Calculate the test
statistic for Rory's test. So I always just like to remind
ourselves what's going on, so you have your null hypothesis here, that the mean number
of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of
experience is less than five for teachers in the district. So if this represents all
the teachers in the district, the population, then what
he did is he took a sample and says he used a sample of 25 teachers, so n here is equal to 25
and then from that sample, he was able to calculate some statistics, he was able to calculate the sample mean, so that sample mean was four years, the sample mean was four years and then he was also able to calculate the sample standard deviation, the sample standard deviation
was equal to two years. Now the whole point that we do, or the main thing we do when
we do significance tests is we say alright, if we assume
the null hypothesis is true, what's the probability of getting a sample mean this low or lower and if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative, but in order to figure
out that probability, we need to figure out a test statistic, sometimes we use a z test, if we're dealing with proportions, but when we deal with means,
we tend to use a t test and the reason why is if you wanted to figure out a z statistic, what you would do is you
would take your sample mean, subtract from that the assumed mean from the null hypothesis, so mu and I'll just put a little
zero, sub zero there, so this is the assumed mean
from the null hypothesis and then you would want to divide by the standard deviation of
the sampling distribution of the sample mean, so
you'd wanna divide by that, but this, we don't know and so that's why instead,
we do a t statistic, in which case, we take the difference between our sample mean and
our assumed population mean, the population parameter
and we try to estimate this and we estimate that with
our sample standard deviation divided by the square
root of our sample size and so if you're inspired,
I encourage you right, even if you're not inspired, (laughs) I encourage you to pause this video and try to calculate this t statistic. Well, this is going to
be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five, our sample standard deviation is two, all of that over the square
root of the sample size, all of that over the square root of 25, so this is going to be equal,
our numerator is negative one, so it's negative one
divided by two over five, which is equal to negative
one times five over two and so this is going to be equal to, equal to negative five
over two or negative 2.5 and then what we would do
in this, what Rory would do is then look this t value up on a t table and say, so if look at a
distribution of a t statistic, something like that and say, okay, we are negative 2.5 below the mean, so negative 2.5 and so what he would wanna do
is figure out this area here, 'cause this would be
the probability of being that far below the mean or
even further below the mean and so that would give us our p value and then if that p value is below some preset significance level, that Rory should have set, maybe 5% or 1%, then he'll reject the null hypothesis, which would suggest his suspicion that the true mean of years of experience for the teachers in his
district is less than five. Now another really important
thing to keep in mind is they told us that assume all conditions for inference have been
met and so that's the, assuming that this was
truly a random sample, that each of the individual observations are either truly independent
or roughly independent, then maybe he observed
either with replacement or it's less than 10% of the population and he feels good that
the sampling distribution is going to be roughly normal and we've talked about
that in other videos.