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let's say that you're solving a problem
associated with hypothesis testing and
let's say that the null hypothesis is
that the mean is 50 and that for the
alternative hypothesis let's say someone
believes that the mean is not 50 and
decides to conduct a test how do you
know if the null hypothesis should be
rejected or not rejected for this
example we need a two-tailed test the
area shaded in blue is known as the
rejection region the area that is not
shaded is the fail to reject region now
the Z values that separate these two
regions these are the critical values
now let's say if we're dealing with a
95% confidence level you could use the
standard Z table to get the Z values so
at a 95% confidence level the critical
values will be 1.96 on both sides now in
order to know whether you should accept
or reject the null hypothesis you need
to get another Z value now calculate a Z
value and compare it to a critical value
let's call that calculate a Z value ZC
let's say if ZC is to the right of Z
then that means that you should reject
the null hypothesis if ZC is to the left
of Z that is if it's in the fail to
reject region you shouldn't reject the
null hypothesis you should keep it
so now let's talk about how we can
calculate zc so this is known as the
test statistic so first so let's talk
about if we have a population mean
versus a population proportion now
sometimes you may need to use the
t-distribution other times we need to
use the normal distribution and get the
z value so let's say that the sample
size is less than 30 and that the
population standard deviation is unknown
so if these two conditions are met in
that case we need to use the T
distribution so our T value is going to
be the sample mean minus the population
mean divided by the sample standard
deviation over the square root of n
where n is the sample size now let's say
that n is less than 30 but we know the
population standard deviation in this
case we could use a normal distribution
so we're going to calculate the z value
so our calculated z value is going to be
the sample mean minus the population
mean divided by the standard deviation
that is the population standard
deviation over the square root of n now
let's say that n is greater than 30 and
the population standard deviation is
unknown because the sample size is large
the distribution will be similar to a
normal distribution so once again we can
calculate the Z value so it's going to
be the sample mean minus the population
mean divided by the sample standard
deviation over the square root of n now
if n is large
and if we know the population standard
deviation everything is going to be the
same
instead of using s we're going to use
Sigma so those are the formulas in which
we can calculate the Z value or our T
value and compare it to the critical
value in order to determine whether we
should reject the null hypothesis or if
we should not reject it so this is when
you have when you're dealing with the
population mean now sometimes you can be
dealing with the population proportion
and so that's going to be P instead of
mu in this case we're going to use a
different formula to calculate the Z and
so the formula is it's going to be the
sample proportion minus the population
proportion divided by the square root of
P Q over N now n is still inside the
square root q is 1 minus P so just keep
that in mind so that's the formula we
need to use if we're dealing with a
proportion that's how we can get our
calculated Z value so those are the test
statistics that we need to be using when
solving problems associated with
hypothesis testing
