- You've already learned about type one errors.
In this screen-cast I'm going to explain
what a type two error is.
Let's revisit what a type one error is.
Here we have a null hypothesis and an alternate hypothesis.
This is a lower tailed alternate hypothesis.
We take our alpha value and we put that into the lower tail.
Based on the null hypothesis
it would be improbable to get a sample
with an average value in the yellow range.
That probability can be termed alpha
and is represented by this yellow region.
Even if the null hypothesis is true
and the true mean is mu naught, there is a chance
that we would get a sample average in the yellow region.
In this case, we would accept the alternate hypothesis
and reject the null hypothesis.
The probability that this would occur is alpha
and if that happened, it's termed a type one error.
That is we conclude that the alternate is true
when in fact it is not.
A type one error will occur
if we are below this critical X bar xbar_c
we can determine that by taking Z alpha times sigma
divided by square root of N, and subtracting that
from mu naught.
This is essentially the lower bound
of an upper tailed confidence bound.
We can also use the NORM.INV function in Excel.
In either case
this is how we can calculate the critical sample average
that we would have to be below
in order for us to accept the alternate hypothesis.
We can do the same thing for an upper tailed test.
In this case, our critical sample average is going to be
mu naught plus Z alpha times sigma over square root of N
and we can use the NORM.INV function
but the probability is one minus alpha.
So a type one error is rejecting the null hypothesis when
in fact it is true.
In other words
we accept the alternate hypothesis when it is false.
We have talked about this example
in the previous couple of screen-casts.
You like to ride bumper cars at Elitch's.
You think that the true mean
is significantly less than three minutes.
An example of a type one error
in this case would be you incorrectly
accept the alternate hypothesis when in fact it is false.
So in other words, you have falsely accused Elitch Gardens
of having ride times of less than three minutes
when reality the meantime really is three minutes.
So that's an example of a type one error.
There's another type of error.
This is known as a type two error.
You fail to reject the null hypothesis when
in fact it is false.
So it's sort of the reverse of a type one error.
You can also think of this
as failing to accept the alternate hypothesis when
in fact the alternate hypothesis is true.
In terms of our example, the mean ride time
at Elitch Gardens really is less than three minutes
but you have failed to detect this in your sample.
You have failed to reject the null hypothesis when
in reality the alternate hypothesis is true.
The probability of committing such an error is termed beta.
Let's take a look at this graphically.
On the right here.
This solid line is our assumed distribution.
This is our null hypothesis.
Sometimes we'll put H naught next to that.
The dotted distribution
on the left is the true distribution
and we don't know that
so we're just gonna label this the alternate hypothesis
so that's true.
We're assuming that that's true
but we don't necessarily know that.
We can distribute our alpha
into the left of our null hypothesis distribution.
Now I've got a couple of questions that are gonna guide us
through understanding what a type two error is.
If we would've gotten an X bar to the left of xbar_c
then we would have accepted the alternate hypothesis
and we would've concluded
that the mean is less than mu naught.
If we would've gotten an X bar to the right of xbar_c
then we would have rejected the alternate hypothesis
and failed to reject the null hypothesis.
If the true distribution is H one the dotted distribution
we would have still needed to come up
with a sample average that's less
than xbar_c in order for us to accept
the alternate hypothesis.
If H one is true and we got an X bar that's to the right
of xbar_c that corresponds to this blue region
this is a type two error.
The alternate hypothesis is true
but we have failed to accept the alternate.
In order for us to accept the alternate hypothesis
we have to be less than xbar_c.
Anything to the right of xbar_c
underneath the true alternate hypothesis, this is beta.
This is the probability of making a type two error.
By the way, the difference between the null distribution
and the assumed alternate distribution is known
as a mean shift or a shift in the mean.
In order for us to calculate a numerical value
for beta, the type two error, that is equal
to the probability that our sample average is greater
than xbar_c, given that the mean really is mu one.
we can write something similar for an upper tailed test.
We can also standardize.
We can take our xbar_critical based
upon the alternate distribution centered about mu one
and we could convert this
into a Z value.
In the next screen-cast I'm gonna go through an example
of how we can calculate numerical values for beta.