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 *H*_{0}
and obtain the
statistic discussed above, (1).
Our rejection rule changes slightly to:

How do we get the expected frequencies under *H*_{0}?
Under *H*_{0},

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

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 |

Because
,
we fail to reject *H*_{0}.
It seems that Larry Bird's (at least in the season 1980-81) first
and second freethrows
were independent of one another.

EXERCISES

Second Shot | ||||

Hit | Miss | Total | ||

First | Hit | 54 | 37 | 91 |

shot | Miss | 49 | 31 | 80 |

Total | 103 | 68 | 171 |