Topic2-HYPOTH-error

Type I and Type II Error

The aim of any hypothesis-testing situation is to make a decision; in particular, you have to decide whether to reject the Null Hypothesis (Ho), in favour of the Alternative Hypothesis (Ha). Although you would like to be able to make a correct decision always, that are times when you might make a wrong decision. You can claim that the two means are not equal in the population when in fact they are. Or you can fail to say that there is a difference when they is really a difference. Statisticians have given names to these two types of errors as follows:

Type 1 Error: Claiming that two means are not equal when in fact they are equal.

In other words, you reject a null hypothesis when it is TRUE.

Type 2 Error: Not finding a difference between two means when in fact there is a difference.

In other words, you reject a null hypothesis when it is FALSE.

How do you remember to differentiate between the to types of errors?

Type 1 Error is the error you are likely to make when you examine your data and say that "Something is happening here!" and conclude that "There is a difference between males and females". In actual fact, there is NO difference between males and females in the population.

Type 2 Error is the error you are likely to make when you examine your data and say "Nothing is happening here!" and conclude that "There is no difference between males and females". In actual fact, there is a difference between males and females in the population.

Four Possible Situations in Testing a Hypothesis

The null hypothesis can be either true or false and you can either reject or not reject the null hypothesis. There are 4 possible situations which arise in testing a hypothesis and they are summarises as follows:

Based on your study:

You decide to Reject the Null Hypothesis (Ho:). You have a correct decision if in the real world the null hypothesis is TRUE.

You decide to Reject the Null Hypothesis (Ho:). You risk committing Type 1 Error if in the real world the hypothesis is TRUE.

You decide NOT to Reject the Null Hypothesis (Ho:). You risk committing Type 2 Error if in the real world the hypothesis is FALSE.

You decide NOT to Reject the Null Hypothesis (Ho:). You have made a correct decision if in the real world the null hypothesis is FALSE.

In other words, when you detect a difference in the sample you are studying and a difference also is detected in the population, you are OK. When there is no difference in the sample you are studying and there is no difference in the population you are OK. However, it is Not OK when you risk committing Type 1 and Type 2 error.

You can use the logic of hypothesis testing in the courtroom. A student is being tried for stealing a motorbike. The judicial system is based on the premise that a person is "innocent until proven guilty". It is the court that must prove based on sufficient evidence that the student is guilty. Thus, the null hypothesis and alternative hypotheses would be::

Ho: The student is innocent

Ha: The student is guilty.

a) Using the table above, state the four possible outcomes of the court's decision.

b) Interpret the Type 1 and Type 2 Errors in this context.