Welcome to a lesson on outcomes
and the Type I and Type II errors of hypothesis testing.
When you perform a hypothesis test,
there are four possible outcomes,
depending on the truth or falseness of the null hypothesis
and the decision to reject or not.
The outcomes are summarized in the table below.
And the four possible outcomes are number one,
the decision is not to reject the null hypothesis
when the null hypothesis is true,
this would be a correct decision.
Two, the decision to reject the null hypothesis
when the null hypothesis is true.
This is an incorrect decision known as a Type I error,
as well as a false positive.
Three, the decision is not to reject the null hypothesis
when in fact the null hypothesis is false.
This is another incorrect decision
known as a Type II error, as well as a false negative.
And then four,
the decision to reject the null hypothesis
when the null hypothesis is false.
This is a correct decision whose probability
is called the power of the test.
Each of the errors occurs with a particular probability.
The Greek letters alpha and beta represent
the probabilities.
Alpha is equal to the probability of a Type I error,
which is the probability of rejecting the null hypothesis
when the null hypothesis is true.
Beta is equal to the probability of a Type II error,
which is the probability of not rejecting
the null hypothesis when the null hypothesis is false.
Alpha and beta should be as small as possible
because they are probabilities of errors.
However, they are rarely zero.
The power of the test is one minus beta,
where beta is the probability of a type two error.
Ideally, we want a high power that is
as close to one as possible.
Increasing the sample size
can increase the power of the test.
Now let's take a look at some examples.
For the first example,
we will suppose the null hypothesis
is Frank's rock-climbing equipment is safe.
So again the Type I error would be the decision
to reject the null hypothesis
when the null hypothesis is true.
So for this situation,
the Type I error would be Frank thinks
that his rock-climbing equipment may not be safe
when in fact it really is safe.
Again the decision to reject the null hypothesis
when the null hypothesis is true.
And now for the Type II error,
which is a decision to not reject the null hypothesis
when, in fact, the null hypothesis is false.
In this situation,
the Type II error would be Frank thinks
that his rock-climbing equipment may be safe
when, in fact, it is not safe.
Again this is because we are not rejecting
the null hypothesis when, in fact,
the null hypothesis is false.
From here it follows that alpha,
the probability of the Type I error
is a probability that Frank thinks
his rock-climbing equipment may not be safe
when, in fact, it really is safe.
And beta is the probability of the Type II error,
which is the probability that Frank thinks
his rock-climbing equipment may be safe
when, in fact, it is not.
Notice, in this case,
the error with a greater consequence is the Type II error.
If Frank thinks his rock-climbing equipment is safe,
he will go ahead and use it when, in fact, it is not safe,
which may put him in danger.
Let's take a look at one more example.
Suppose the null hypothesis is the victim
of an automobile accident is alive
when the victim arrives at the emergency room of a hospital.
Again the Type I error is the decision
to reject the null hypothesis
when the null hypothesis is true.
In this situation,
the Type I error is the emergency crew thinks
that the victim is dead when, in fact, the victim is alive.
Again this is because the decision
is to reject the null hypothesis
when the null hypothesis is true.
The Type II error again is the decision
to not reject the null hypothesis
when, in fact, the null hypothesis is false.
Which means the Type II air here is
the emergency crew thinks that the victim may be alive
when, in fact, the victim is dead.
Again because the decision is to not reject
the null hypothesis when, in fact,
the null hypothesis is false.
From here it follows that alpha,
the probability of the Type I error
is the probability that the emergency crew thinks
that the victim is dead when, in fact,
the victim is really alive.
And beta, the probability of the Type II error
is equal to the probability that the emergency crew thinks
that the victim may be alive when, in fact,
the victim is dead.
Notice in this situation,
the error with the greater consequence
is the Type I error,
which again is when they think the victim is dead
when, in fact, the victim is alive.
If the emergency crew thinks the victim is dead,
they will not treat the victim.
And therefore, the victim could go on to die.
I hope you found this helpful.