21 6.6 Hypothesis Tests In-Depth
[latexpage]
Establishing the parameter of interest, type of distribution to use, the test statistic and p-value can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when interpreting the results.
Rare Events
Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. (Remember that your assumption is just an assumption—it is not a fact and it may or may not be true. But your sample data are real and the data are showing you a fact that seems to contradict your assumption.)
For example, Didi and Ali are at a birthday party of a very wealthy friend. They hurry to be first in line to grab a prize from a tall basket that they cannot see inside because they will be blindfolded. There are 200 plastic bubbles in the basket and Didi and Ali have been told that there is only one with a \$100 bill. Didi is the first person to reach into the basket and pull out a bubble. Her bubble contains a \$100 bill. The probability of this happening is [latex]\frac{1}{200}[/latex] = 0.005. Because this is so unlikely, Ali is hoping that what the two of them were told is wrong and there are more \$100 bills in the basket. A “rare event” has occurred (Didi getting the \$100 bill) so Ali doubts the assumption about only one \$100 bill being in the basket.
Errors in Hypothesis Tests
When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H0 and the decision to reject or not. The outcomes are summarized in the following table:
| H0 IS ACTUALLY | ||
|---|---|---|
| ACTION | True | False |
| Do not reject H0 | Correct Outcome | Type II error |
| Reject H0 | Type I Error | Correct Outcome |
The four possible outcomes in the table are:
- The decision is not to reject H0 when H0 is true (correct decision).
- The decision is to reject H0 when H0 is true (incorrect decision known as a Type I error).
- The decision is not to reject H0 when, in fact, H0 is false (incorrect decision known as a Type II error).
- The decision is to reject H0 when H0 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.
The power of a test is 1 – β.
Ideally, α and β should be as small as possible because they are probabilities of errors, but rarely are they zer