Researchers during the 2004 Olympic Games monitored four sports (boxing, tae kwo

Question

Researchers during the 2004 Olympic Games monitored four sports (boxing, tae kwon do, Greco-Roman wrestling, and freestyle wrestling) and found that participants wearing red outfits won significantly more often than those wearing blue. Please use the chi-square formula to solve the following.
a. In 50 wrestling matches involving red versus blue, suppose 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.
b. In 100 wrestling matches involving red versus blue, suppose the red outfit won 62 times and lost 38. Is this result sufficient to conclude that red wins significantly more than would be expected by chance? Again, use the .05 level of significance.
c. Note that the 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 examples have an identical winning percentages, one is significant and the other is not. Explain why the two samples lead to different conclusions.

Explanation / Answer

(a) Let p denote the proportion of times red outfit wins. Ho: p=0.5 H1: p>0.5 n=50 p=3150=0.62 z=p-pp(1-p)/n=0.62-0.500.50(1-0.50)/50=1.6971 The critical region is z>1.65 Z falls in the critical region. So we reject the null hypothesis Conclusion: This is sufficient to conclude that red wins significantly more than would be expected by chance. ---------------------------------------------------------------------------------------------- (b) Let p denote the proportion of times red outfit wins. Ho: p=0.5 H1: p>0.5 n=100 p=62100=0.62 z=p-pp(1-p)/n=0.62-0.500.50(1-0.50)/100=2.4 The critical region is z>1.65 Z falls in the critical region. So we reject the null hypothesis Conclusion: This is sufficient to conclude that red wins significantly more than would be expected by chance. ----------------------------------------------------------------------------------------------- (c) The conclusions in parts a and b are different. This is because the z-scores are different. This because, the denominator of the z-formula, which is the standard error of the sample proportion decreases when n increases. Then z increases and leads to the rejection of the null hypothesis with more probability. In other words, when the sample size increases, the power of a statistical test increases.

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