5.4.1  Errors
5.4.1  ErrorsCommitting an Error
Every time we make a decision and come to a conclusion, we must keep in mind that our decision is based on probability. Therefore, it is possible that we made a mistake.
Consider the example of the previous Lesson on whether the majority of Penn State students like the cold. In that example, we took a random sample of 500 Penn State students and found that 278 like the cold. We rejected the null hypothesis, at a significance level of 5% with a pvalue of 0.006.
 Type I Error

Rejecting \(H_0\) when \(H_0\) is really true, denoted by \(\alpha\) ("alpha") and commonly set at .05
\(\alpha=P(Type\;I\;error)\)
The significance level of 5% means that we have a 5% chance of committing a Type I error. That is, we have 5% chance that we rejected a true null hypothesis.
 Type II Error

Failing to reject \(H_0\) when \(H_0\) is really false, denoted by \(\beta\) ("beta")
\(\beta=P(Type\;II\;error)\)
If we failed to reject a null hypothesis, then we could have committed a Type II error. This means that we could have failed to reject a false null hypothesis.
Decision  Reality  

\(H_0\) is true  \(H_0\) is false  Probability Level  
Reject \(H_0\), (conclude \(H_a\))  Type I error  Correct decision  P is LESS than .05 that the null is true Small probabilities (less than .05) lead to rejecting the null 
Fail to reject \(H_0\)  Correct decision  Type II error  P is GREATER than .05 that the null is true Large probability (greater than .05) lead to not rejecting the null 
How Important are the Conditions of a Test?
In our six steps in hypothesis testing, one of them is to verify the conditions. If the conditions are not satisfied, we cannot, however, make a decision or state a conclusion. The conclusion is based on probability theory.
If the conditions are not satisfied, there are other methods to help us make a conclusion. The conclusion, however, may be based on other parameters, such as the median. There are other tests (called nonparametric) that can be used.