Question

The color red is often associated with anger and made dominance. Based on this observation, Hill and Barton (2005) monitored the outcome of 4 combat sports (boxing, tae kwan do, Greco-Roman wrestling, and freestyle wrestling) during 2004 Olympic Games and found that participates wearing red won significantly more than those wearing blue.

In wrestling matches inwolving red versus bloc, 50 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. a. b. In 100 matches, suppose red won 62 times and lost 38. Is this sufficient to conc?ude that red wins sig- nificantly more tham would be expected by chance? Again, use ? .05. 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 samples have an identical winning percentages, one is significant and the other is not. Explain why the two samples lead to different conclusions.

Answer 1

(a) H₀ : p_red= 0.5

H_a : p_red > 0.5

Here sample proportion p^ = 31/50 = 0.62

Here p - value = Pr(T > = 31 ; 50 ; 0.5) = 1 - BIN (T < 31 ; 50 ; 0.5) = 1 - 0.9405 = 0.0595

so here alpha > 0.0595 so we will say the significance is there.

(b) Here p- value = Pr(T >= 62 ; 100 ; 0.5) = 1 - BIN (T < 62 ; 100 ; 0.5) = 1 - 0.9895 = 0.0105

so here p - value is less than alpha so we will say the result is significant here.

(c) Here the two samples lead to different conclusion due to different sample size. As we increase sample size, margin of error tend to decrease and that will lead to rejection of null hypothesis more often.