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9.2: Outcomes and the Type I and Type II Errors

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 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:

  1. The decision is not to reject \(H_{0}\) when \(H_{0}\) is true (correct decision).
  2. The decision is to reject \(H_{0}\) when \(H_{0}\) is true (incorrect decision known as aType I error).
  3. The decision is not to reject \(H_{0}\) when, in fact, \(H_{0}\) is false (incorrect decision known as a Type II error).
  4. 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.)

Exercise \(\PageIndex{1}\)

Suppose the null hypothesis, \(H_{0}\), is: the blood cultures contain no traces of pathogen \(X\). State the Type I and Type II errors.

Answer

Type I error: The researcher thinks the blood cultures do contain traces of pathogen \(X\), when in fact, they do not.

Type II error: The researcher thinks the blood cultures do not contain traces of pathogen \(X\), when in fact, they do.

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.)

Exercise \(\PageIndex{2}\)

Suppose the null hypothesis, \(H_{0}\), is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?

Answer

The error with the greater consequence is the Type II error: the patient will be thought well when, in fact, he is sick, so he will not get treatment.

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.

Exercise \(\PageIndex{3}\)

“Red tide” is a bloom of poison-producing algae–a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence.

Answer

In this scenario, an appropriate null hypothesis would be \(H_{0}\): the mean level of toxins is at most \(800 \mu\text{g}\), \(H_{0}: \mu_{0} \leq 800 \mu\text{g}\).

  • Type I error: The DMF believes that toxin levels are still too high when, in fact, toxin levels are at most \(800 \mu\text{g}\). The DMF continues the harvesting ban.
  • Type II error: The DMF believes that toxin levels are within acceptable levels (are at least 800 μg) when, in fact, toxin levels are still too high (more than \(800 \mu\text{g}\)). The DMF lifts the harvesting ban. This error could be the most serious. If the ban is lifted and clams are still toxic, consumers could possibly eat tainted food.

In summary, the more dangerous error would be to commit a Type II error, because this error involves the availability of tainted clams for consumption.

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.

 

Exercise \(\PageIndex{4}\)

Determine both Type I and Type II errors for the following scenario:

Assume a null hypothesis, \(H_{0}\), that states the percentage of adults with jobs is at least 88%.

Identify the Type I and Type II errors from these four statements.

  1. Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%
  2. Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.
  3. Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.
  4. Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%.

Answer

Type I error: c

Type I error: b

Chapter Review

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.

 

Exercise 9.3.5

The mean price of mid-sized cars in a region is $32,000. A test is conducted to see if the claim is true. State the Type I and Type II errors in complete sentences.

Answer

Type I: The mean price of mid-sized cars is $32,000, but we conclude that it is not $32,000.

Type II: The mean price of mid-sized cars is not $32,000, but we conclude that it is $32,000.

Exercise 9.3.6

A sleeping bag is tested to withstand temperatures of –15 °F. You think the bag cannot stand temperatures that low. State the Type I and Type II errors in complete sentences.

Exercise 9.3.7

For Exercise 9.12, what are \(\alpha\) and \(\beta\) in words?

Answer

\(\alpha =\) the probability that you think the bag cannot withstand -15 degrees F, when in fact it can

\(\beta =\) the probability that you think the bag can withstand -15 degrees F, when in fact it cannot

Exercise 9.3.8

In words, describe \(1 - \beta\) For Exercise 9.12.

Exercise 9.3.9

A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, \(H_{0}\), is: the surgical procedure will go well. State the Type I and Type II errors in complete sentences.

Answer

Type I: The procedure will go well, but the doctors think it will not.

Type II: The procedure will not go well, but the doctors think it will.

Exercise 9.3.10

A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, \(H_{0}\), is: the surgical procedure will go well. Which is the error with the greater consequence?

Exercise 9.3.11

The power of a test is 0.981. What is the probability of a Type II error?

Answer

0.019

Exercise 9.3.12

A group of divers is exploring an old sunken ship. Suppose the null hypothesis, \(H_{0}\), is: the sunken ship does not contain buried treasure. State the Type I and Type II errors in complete sentences.

Exercise 9.3.13

A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, \(H_{0}\), is: the sample does not contain E-coli. The probability that the sample does not contain E-coli, but the microbiologist thinks it does is 0.012. The probability that the sample does contain E-coli, but the microbiologist thinks it does not is 0.002. What is the power of this test?

Answer

0.998

Exercise 9.3.14

A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, \(H_{0}\), is: the sample contains E-coli. Which is the error with the greater consequence?

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

  • 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.