The study on genetics and fast-twitch muscles includes a sample of elite sprinters, a sample of elite endurance athletes, and a control group of nonathletes. Is there an association between genotype classification (RR, RX, or XX) and group (sprinter, endurance, control group)? Computer output is shown for this chi-square test. In each cell, the top number is the observed count, the middle number is the expected count, and the bottom number is the contribution to the chi-square statistic.

(a) What is the expected count for endurance athletes with the XX genotype? For this cell, what is the contribution to the chi-square statistic? Verify both values by computing them yourself.
(b) What are the degrees of freedom for the test? Verify this value by computing it yourself.
(c) What is the chi-square test statistic? What is the p-value? What is the conclusion of the test?
(d) Which cell contributes the most to the chisquare statistic? For this cell, is the observed count greater than or less than the expected count?
(e) Which genotype is most over-represented in sprinters? Which genotype is most overrepresented in endurance athletes?

The gene ACTN3, which encodes a protein that functions in fast-twitch muscles. People can be classified according to which genotype of this gene they have, RR, RX, or XX, and also according to which genetic allele they have, R or X.

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Total RR RX XX Control 226 130 80 436 143.76 214.15 78.09 1.316 0.655 0.047 Sprint 53 48 107 52.56 35.28 19.16 0.395 8.901 9.043 60 Endurance 194 88 46 63.96 95.29 34.75 0.246 0.558 3.645 Total 243 362 132 737 Chi-Sq = 24.805, DF = 4, P-Value = 0.000