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# Statistical Issues

• conditional probability
• stationarity
• serial correlation
• runs test
• chi-square test

Tversky and Gilovich state that, in statistical terms, the concensus of fans is that the shooting sequences exhibit non-stationarity and serial dependence. They perform four sets of calculations for each 76-er player's sequence of hits and misses:

1. The long-run probability of a hit was computed. This was compared with conditional probability of a hit following one, two and three hits, and following one, two or three misses.

2. The serial correlation (first-order correlation coefficient) was computed.

3. A Wald-Wolfowicz Runs Test.

4. To check stationarity, chi-square test was performed on successive blocks of four shots. Each block was classified according to hit rate as ``high" (3-4 H's). ``medium" (2 H's) or ``low" (0-1 H's). The null cell probabilities are those for tosses of a fair coin.

A few of these tests show significance, as might be expected running a large number of tests over this many players.

Tversky and Gilovich make the comment that in repeated tosses a fair coin, four heads in a row is ``quite likely in even relatively small samples." A simple recursion for computing these probabilities is given in Schilling (1989). In 10 tosses, the chance that the longest run of heads is four or longer is .245; for 20 tosses, it is .479.

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laurie.snell@chance.dartmouth.edu