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Hypothesis Testing of Single Mean and Single Proportion: Outcomes and the Type I and Type II Errors

Module by: Dr. Barbara Illowsky, Susan Dean

When you perform a hypothesis test, there are four outcomes. The outcomes are summarized in the following table:

HoHo = the null hypothesis
Action True False
Do not reject HoHo Correct Outcome Type II error
Reject HoHo Type I Error Correct Outcome

The four outcomes in the table are:

  • The decision is to not reject HoHo when, in fact, HoHo is true (correct decision).
  • The decision is to reject HoHo when, in fact, HoHo is true (incorrect decision known as a Type I error).
  • The decision is to not reject HoHo when, in fact, HoHo is false (incorrect decision known as a Type II error).
  • The decision is to reject HoHo when, in fact, HoHo is false (correct decision whose probability is called the Power of the Test).

Each of the errors occurs with a particular probability. The Greek letters αα and ββ represent the probabilities.

αα = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

ββ = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

αα and ββ should be as small as possible because they are probabilities of errors. They are rarely 0.

The Power of the Test is 1-β1-β. Ideally, we want a high power that is as close to 1 as possible.

The following are examples of Type I and Type II errors.

Example 1

Suppose the null hypothesis, HoHo, is: Frank's rock climbing equipment is safe.

Type I error: Frank concludes that his rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Frank concludes that his rock climbing equipment is safe when, in fact, it is not safe.

αα = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is. ββ = probability that Frank thinks his rock climbing equipment is safe when, in fact, it is not.

Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)

Example 2

Suppose the null hypothesis, HoHo, is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.

Type I error: The emergency crew concludes that the victim is dead when, in fact, the victim is alive. Type II error: The emergency crew concludes that the victim is alive when, in fact, the victim is dead.

αα = probability that the emergency crew thinks the victim is dead when, in fact, he is really alive = P(Type I error)P(Type I error). ββ = probability that the emergency crew thinks the victim is alive when, in fact, he is dead = P(Type II error)P(Type II error).

The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.)

Glossary

Type 1 Error:
The decision is to reject Null hypothesis, when, in fact, Null hypothesis is true.
Type 2 Error:
The decision is not to reject Null hypothesis, when, Null hypothesis is false.

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