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Type I and Type II Errors
For type I and type II errors, we must know the null and
alternate hypotheses.
For our example to follow:
H0: µ = 40 The mean of the population is 40.
Ha: µ ≠ 40 The mean of the population is not 40.
We can define:
P(Type I error) = α
P(Type II error) = β
Type I error occurs when the null hypothesis is true, but our
sample mean is extreme, leading us to believe that the null
hypothesis is false.
We establish our planned amount of Type I error when we
choose α. Of course, we expect to be correct in failing to
reject the null hypothesis 1-α, when the null hypothesis is
true.
When we choose a level of significance we set α. Our sampling
distribution has a mean of 40 and standard deviation of 10, based
on population standard deviation of 80 with a sample size of 64.
We get a Type I error when our distribution is centered at 40,
but our sample mean happens to be larger than 59.6 or smaller
than 20.4.
The red regions represent regions of Type I error and have
probability α. In this example α is 5%.
If we change the significance level to 90%, we change α. Here
α is 10%.
As you can see, we have increased the probability of Type I
error.
Type II error occurs only when the null hypothesis is false. It
cannot occur if the null hypothesis is true, by our very
definition.
When we speak of Type II error we must know that the null
hypothesis is not true.
Let’s start with our hypothetical distribution:
Now we add an alternate distribution. Our sample mean will
come from this sampling distribution, N(68,10), instead of the
hypothetical sampling distribution.
The region shaded pink is our probability of Type I error,
here 5%.
The region shaded blue is the probability of Type II area.
Notice that it is under the alternate (blue) distribution.
We make a Type II error whenever the null hypothesis is false,
but we get a sample mean that falls into a range that will cause
us to fail to reject the null hypothesis.
Let’s take a closer look:
Sample means between 20 and 60 will “look good” to us, we
will not reject the null hypothesis.
Now we check the alternate distribution. Are there times
when sample means from this distribution will give us
values between 20 and 60?
In fact, there are.
To find the probability of Type II area we find the area under
the curve.
60 - 68 æ
æ
P(x < 60) = P æZ <
= P(Z < -.8) = .2118
æ
æ
10 æ
So the probability of Type II error is 21%.
That is, when the true mean is 68, there is a 21% probability
that we will fail to reject the null hypothesis.
How can we reduce the probability of Type II error?
Examine the following figures:
Can you see that β is less now, but α is greater?
56.4485 - 68 æ
æ
P(x < 56.4) = P æZ <
=
æ
æ
æ
10
= P(Z < -1.1551) = .1240
Here the probability of Type II error is 12.4%
Increasing α does result in a decrease in β.
This does not necessarily get you very far ahead.
Suppose we could compare to a different alternate distribution.
Suppose we could make it have a larger mean, perhaps 72
instead of 68. Would this change β?
Now we have a new alternate distribution N(72,10) and so a
new probability.
60 - 72 æ
æ
P(x < 60) = P æZ <
=
æ
æ
10 æ
= P(Z < -1.2) = .1150
So we now have 11.5% Type II error. While moving the
alternate distribution further away reduces Type II error, we
cannot always do this.
Another approach is to decrease standard deviation. Any
way we can accomplish this will have the same effect.
Usually you can change sample size.
If our sampling distributions are now N(40,8) and N(68,8) we
can find the effect on probability of Type II error.
This also shows a reduction in Type II error. Increasing sample
size will be our most effective way to minimize Type II error.
55.6 - 68 æ
æ
P(x < 55.6) = P æZ <
=
æ
æ
æ
8
P(Z < -1.55) = .06057
With a decrease in the standard deviation we see the
probability of Type II error decrease to 6%. Decreasing the
standard deviation reduces the amount of overlap between
the two distributions, thus reducing the Type II error.
We have seen the difference between Type I and Type II
errors.
We set the probability of Type I error when we choose a level
of significance.
The probability of Type II error can be reduced by increasing
α, by reducing the standard deviation (perhaps by increasing
sample size), or by increasing the distance between the
hypothetical and alternate means.
THE END