One needs to understand the theory behind
the hypothesis testing and how do
you reject a null hypothesis or otherwise.
There are rules of thumb.
That is in case of a two-tail test,
one can use 1.96 as the calculated threshold for
either Z or T statistics to reject
a null hypothesis or for a one-tail test,
absolute value of 1.64 to reject a null hypothesis.
But what does it mean,
how do you get to these 1.64 or 1.96?
There is some theory to it.
It involves statistical distributions and
now perhaps is a good time to,
to learn about those.
Now imagine if the mean values
of two variables is the same.
That is, we are assuming that the difference
between the two means is essentially zero.
Let's say the mean of
variable A and the mean of Variable B,
we assume that they are equal,
that is mu_A equals mu_B.
And if that were to be the case,
the difference between the two should be equal to zero.
So the alternative hypothesis could
be that the differences mean is not equal to zero.
So, and you would say that mu_A is not equal to mu_B,
or the difference is greater than zero.
That is, mu_A is greater than mu_B,
or the difference is less than zero,
where mu_A is less than mu_B.
And in these three circumstances,
the rejection region or the how do
you reject the null hypothesis
means three different things.
And for this, we were to revert
to the normal distribution curve.
Imagine that you're conducting a Z-test using
a normal distribution and
the shape of the curve would be very similar.
This image will be very similar if we were to do
a two-tailed test for T-distribution.
But let's assume that we are working with
normal distribution and you have a rejection region
that is to the left and to the right of the curve,
as you saw the normal distribution curve.
Let's say our alternative hypothesis is
that the mean difference is not equal to zero.
It could be greater than or less than zero,
but it's not equal to zero.
So we will call this a two-tail test because we are not
making an assumption of
the difference being greater or less than zero.
Then we have to define
the rejection region in both tails,
that is the left tail and
the right tail of the normal distribution.
Remember we only consider 5 percent of the area under
the normal curve to define the rejection region
and for the two-tail tests that 5 percent gets divided
into half of it goes into
the left tail and the other half
goes into the right tail,
so two and half percent under the curve in each tail.
Graphically you can see this
again as the same as we saw earlier,
that this is a normal distribution curve
and two-and-a-half percent is
in the left tailed into the other
two-and-a-half percent is in the right tail.
If the test statistic is 1.96,
if the absolute value of
the test statistic is greater
than 1.96 or less than 1.96,
it falls in the rejection region
and you can safely reject the null.
The null would be that the difference of mean equals to
zero or in common parlance,
what we are saying is
that the two means are not the same.
Now let us work with the assumption of the situation
where we're testing if
the difference of mean is less than zero,
we are only interested in the left tail.
Our alternative hypothesis is
that the difference of mean is less than zero.
In this case, the entire rejection region,
that is 5 percent of the rejection region is
to the left and in any situation in,
for a one-tailed test,
if we were to get the T-stat of 1.64 or less,
we would reject the null that the mean is greater than
zero in favor of
the alternative that the difference is less than zero.
The exact opposite to
this would be the right tailed test.
Where the alternative hypothesis is
that the mean is greater than zero.
And if you get the T-test statistics
of greater than 1.64,
for a right-tailed test,
you reject the null in favor of the alternative that
the mean difference is greater than zero.