9.2: Type I and Type II Errors

Last updated
Save as PDF

Page ID: 26093

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis \(H_{0}\) and the decision to reject or not. The outcomes are summarized in the following table:

ACTION	\(H_{0}\) is Actually True	\(H_{0}\) is Actually False
Do not reject \(H_{0}\)	Correct Outcome	Type II error
Reject \(H_{0}\)	Type I Error	Correct Outcome

The four possible outcomes in the table are:

The decision is not to reject \(H_{0}\) when \(H_{0}\) is true (correct decision).
The decision is to reject \(H_{0}\) when \(H_{0}\) is true (incorrect decision known as aType I error).
The decision is not to reject \(H_{0}\) when, in fact, \(H_{0}\) is false (incorrect decision known as a Type II error).
The decision is to reject \(H_{0}\) when \(H_{0}\) 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 \(\alpha\) and \(\beta\) represent the probabilities.

\(\alpha =\) probability of a Type I error \(= P(\text{Type I error}) =\) probability of rejecting the null hypothesis when the null hypothesis is true.
\(\beta =\) probability of a Type II error \(= P(\text{Type II error}) =\) 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. They are rarely zero.

The Power of the Test is \(1 - \beta\). 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. The following are examples of Type I and Type II errors.

Example \(\PageIndex{1}\): Type I vs. Type II errors

Suppose the null hypothesis, \(H_{0}\), is: Frank's rock climbing equipment is safe.

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

\(\alpha =\) probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe.

\(\beta =\) probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.

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 \(\PageIndex{2}\)

Suppose the null hypothesis, \(H_{0}\), 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 thinks that the victim is dead when, in fact, the victim is alive.
Type II error: The emergency crew does not know if the victim is alive when, in fact, the victim is dead.

\(\alpha =\) probability that the emergency crew thinks the victim is dead when, in fact, he is really alive \(= P(\text{Type I error})\).

\(\beta =\) probability that the emergency crew does not know if the victim is alive when, in fact, the victim is dead \(= P(\text{Type II error})\).

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.)

Example \(\PageIndex{3}\)

It’s a Boy Genetic Labs claim to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, \(H_{0}\), is: It’s a Boy Genetic Labs has no effect on gender outcome.

Type I error: This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It’s a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, \(\alpha\).
Type II error: This results when we fail to reject a false null hypothesis. In context, we would state that It’s a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, \(\beta\).

The error of greater consequence would be the Type I error since couples would use the It’s a Boy Genetic Labs product in hopes of increasing the chances of having a boy.

Example \(\PageIndex{4}\)

A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context. Which error is the more serious?

Type I: A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%.
Type II: A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%.

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option.

Summary

In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected. The probabilities of these errors are denoted by the Greek letters \(\alpha\) and \(\beta\), for a Type I and a Type II error respectively. The power of the test, \(1 - \beta\), quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.

Formula Review

\(\alpha =\) probability of a Type I error \(= P(\text{Type I error}) =\) probability of rejecting the null hypothesis when the null hypothesis is true.
\(\beta =\) probability of a Type II error \(= P(\text{Type II error}) =\) probability of not rejecting the null hypothesis when the null hypothesis is false.

WeBWorK Problems

Query \(\PageIndex{1}\)

Query \(\PageIndex{2}\)

Query \(\PageIndex{3}\)

Query \(\PageIndex{4}\)

Glossary

Type 1 Error: The decision is to reject the null hypothesis when, in fact, the null hypothesis is true.

Type 2 Error: The decision is not to reject the null hypothesis when, in fact, the null hypothesis is false.

Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.