## Hypothesis Testing with One Sample

Read this section on the two types of errors in hypothesis testing and some examples of each.

### Consequences of Type I and Type II Errors

Both types of errors are problems for individuals, corporations, and data analysis. A false positive (with null hypothesis of health) in medicine causes unnecessary worry or treatment, while a false negative gives the patient the dangerous illusion of good health and the patient might not get an available treatment. A false positive in manufacturing quality control (with a null hypothesis of a product being well made) discards a product that is actually well made, while a false negative stamps a broken product as operational. A false positive (with null hypothesis of no effect) in scientific research suggest an effect that is not actually there, while a false negative fails to detect an effect that is there.

Based on the real-life consequences of an error, one type may be more serious than the other. For example, NASA engineers would prefer to waste some money and throw out an electronic circuit that is really fine (null hypothesis: not broken; reality: not broken; test find: broken; action: thrown out; error: type I, false positive) than to use one on a spacecraft that is actually broken. On the other hand, criminal courts set a high bar for proof and procedure and sometimes acquit someone who is guilty (null hypothesis: innocent; reality: guilty; test find: not guilty; action: acquit; error: type II, false negative) rather than convict someone who is innocent.

Minimizing errors of decision is not a simple issue. For any given sample size the effort to reduce one type of error generally results in increasing the other type of error. The only way to minimize both types of error, without just improving the test, is to increase the sample size, and this may not be feasible. An example of acceptable type I error is discussed below.

Type I Error: NASA engineers would prefer to waste some money and throw out an electronic circuit that is really fine than to use one on a spacecraft that is actually broken. This is an example of type I error that is acceptable.