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Types of Statistical Error: Basic Statistics Lecture Series Lecture #12

As promised last time, I will cover types of statistical error this time.  Knowing the magnitude and the type of error is important to convey with any hypothesis test.  This also happens to be why, in science, it is said that nothing can ever truly be proven; only disproven.



First, it is important to understand that error typing is an integral part of hypothesis and no other part of statistics, similar to the human brain and the person it's in.  The human brain cannot fit into any other species, and it is necessary for humans to live with it.  The same concept applies with these types of errors and hypothesis; it cannot fit anywhere else, and is necessary for the success of hypothesis testing.

So what specifically is hypothesis testing?  It is the chances that the conclusion is incorrect, namely the chances of the null hypothesis is rejected when it's true (Type I Error, false positive) and the chances of failing to reject the null hypothesis when it's false (Type II Error, false negative).  In statistical hypothesis testing, the alternative hypothesis will always be a positive result.  In the case of statistical analysis, a positive result doesn't mean the same as it does in common language; it means that the result which is different than the proposed mean is the true value.

That's no baby, sir.
How would we find how big the errors would be?  For type I errors, that would be easy.  The chances are simply the level of α we've chosen to compare the p-value to.  Type II errors are are a little bit more complicated to calculate, but we've already calculated it before.  What you do is obtain the p-value, which is to calculate the difference between the sample mean and the proposed population mean, and divide it by the standard error, and obtain the p-value from the resultant z-value.  Remember from the hypothesis testing posts that the value for z is given by $z_{calc}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.  Here's a table which contains the types of errors and their calculations:

True Statement
Null Hypothesis True Alternative Hypothesis True
Statistical Conclusion Null Hypothesis Rejected Type I Error, α Correct Decision
Null Hypothesis Not Rejected Correct Hypothesis Type II Error, p from $$z_{calc}= β = \frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$$

So that's the types of error and how to find them.  If you have any questions, please leave a comment.  Next time, I'll start on regression analysis.  Until then, stay curious.
K. "Alan" Eister has his bacholers of Science in Chemistry. He is also a tutor for Varsity Tutors.  If you feel you need any further help on any of the topics covered, you can sign up for tutoring sessions. here
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