Chi-Square Test: Do Olympic participants wearing red win more often?

The color red is often associated with anger and male dominance. Based on this observation, Hill and Barton (2005) monitored the outcome of four combat sports (boxing, tae kwan do, Greco-Roman wrestling, and freestyle wrestling) during the 2004 Olympic games and found that participants wearing red outfits won significantly more often than those wearing blue.

a. In 50 wrestling matches involving red versus blue, suppose that the red outfit won 31 times and lost 19 times. Is this result sufficient to conclude that red wins significantly more than would be expected by chance? Test at the .05 level of significance. Since no expected proportions are given, assume 50% each for red and blue. Then multiply by 50 to get the expected frequencies. Use the formula to calculate Chi-square. Compare with the tabulated value.

b. In 100 matches, suppose red won 62 times and lost 38. Is this sufficient to conclude that red wins significantly more than would be expected by chance? Again, use α = .05. Same as above. With different number of participants.

c. Note that winning percentage for red uniforms in part a is identical to the percentage in part b (31 out of 50 is 62%, and 62 out of 100 is also 62%). Although the two samples have an identical winning percentage, one is significant and the other is not. Explain why the two samples lead to different conclusions.