# Test of Independence for a Table

There are many goodness-of-fit'' type tests. We will look at a simple one which tests for independence in a table.

Consider a professional basketball player. Suppose he has two free throws. Is the result of the second free throw independent of the first? This has perplexed society for ages. Our hypotheses are:

Complements of Kitchens (1998, Exploring Statistics, Pacific Grove, CA: Duxbury Press), we have data on two basketball players: Larry Bird and Rick Robey. Consider the data for Larry Bird (taken during the season 1980-1981).

 Second Shot Hit Miss Total First Hit 251 34 285 shot Miss 48 5 53 Total 299 39 338

The test is extremely simple. The above table are the observed frequencies. Simply form the expected frequencies under H0 and obtain the statistic discussed above, (1). Our rejection rule changes slightly to:

 (3)

How do we get the expected frequencies under H0? Under H0,

What good does this do? Well for one thing, I can estimate the right-side by

Now .6498 is an estimate of the left-side provided H0 is true; otherwise, it is not. Hence an estimate of the expected frequency under H0 is

where the subscript 11 stands for the cell in row 1 and column 1. Notice how close this is to the observed frequency. To complete this table we need the other 3 expected frequencies. But wait, these expected frequencies must add to the margin frequencies; hence, we can just subtract. Once we do we get the following table with expected frequencies in parentheses:
 Second Shot Hit Miss Total First Hit 251 34 285 (252.11) (32.88) shot Miss 48 5 53 (46.88) (6.12) Total 299 39 338
The computation of is

Because , we fail to reject H0. It seems that Larry Bird's (at least in the season 1980-81) first and second freethrows were independent of one another.

EXERCISES

0.0.2   Below are similar data for the basketball player Rick Robey; complements of Kitchens (1998, Exploring Statistics, Pacific Grove, CA: Duxbury Press), Obtain the test of independence for this data.
 Second Shot Hit Miss Total First Hit 54 37 91 shot Miss 49 31 80 Total 103 68 171