# 9.2 Outcomes and the type i and type ii errors

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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:

STATISTICAL DECISION H 0 IS ACTUALLY...
True False
Cannot reject H 0 Correct Outcome Type II error
Cannot accept H 0 Type I Error Correct Outcome

The four possible outcomes in the table are:

1. The decision is cannot reject H 0 when H 0 is true (correct decision).
2. The decision is cannot accept H 0 when H 0 is true (incorrect decision known as a Type I error ). This case is described as "rejecting a good null". As we will see later, it is this type of error that we will guard against by setting the probability of making such an error. The goal is to NOT take an action that is an error.
3. The decision is cannot reject H 0 when, in fact, H 0 is false (incorrect decision known as a Type II error ). This is called "accepting a false null". In this situation you have allowed the status quo to remain in force when it should be overturned. As we will see, the null hypothesis has the advantage in competition with the alternative.
4. The decision is cannot accept 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 α and β represent the probabilities.

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

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

α and β should be as small as possible because they are probabilities of errors.

Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Z α (or t α ) and the alpha value determined the confidence level of the estimate because it was the probability of the interval capturing the true mean (or proportion parameter p). This alpha and that one are the same.

The easiest way to see the relationship between the alpha error and the level of confidence is with the following figure.

In the center of [link] is a normally distributed probability distribution marked Ho. This is a sampling distribution of $\stackrel{–}{X}$ and by the Central Limit Theorem it is normally distributed. The distribution in the center is marked H 0 and represents the distribution for the null hypotheses H 0 : µ = 100. This is the value that is being tested. The formal statements of the null and alternative hypotheses are listed below the figure.

The distributions on either side of the H 0 distribution represent distributions that would be true if H 0 is false, under the alternative hypothesis listed as H a . We do not know which is true, and will never know. There are, in fact, an infinite number of distributions from which the data could have been drawn if H a is true, but only two of them are on [link] representing all of the others.

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