### Color Red: Anger and Male Dominance

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 than those wearing blue.

1. 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 0.05 level of significance.
2. 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 the α = 0.05.
3. Note that the winning percentage for red uniforms in part (1) is identical to the percentage in part (2) (31 out of 50 is 62%, and 62 out of 100 is also 62%). Although the two samples have identical winning percentages, one is significant and the other is not. Explain why the two samples lead to different conclusions.

(1) SOLUTION:

$$df = k-1 = 2-1 = 1$$
$$f_e = 25, 25$$
$$f_o = 31, 19$$
$$\chi^2=3.841$$ with df = 1 degrees of freedom at 0.05 significance level
DECISION: Since the calculated $$\chi^2$$ is less than the critical $$\chi^2$$, we can conclude that there is no data sufficient to conclude that red wins significantly more than would be expected by chance.

(2) SOLUTION:

$$df = k-1 = 2-1 =1$$
$$f_e = 50, 50$$
$$f_o = 62, 38$$
$$\chi^2=3.841$$ with df = 1 degrees of freedom at 0.05 significance level
DECISION: Since the calculated $$\chi^2$$ is greater than the critical $$\chi^2$$, we can conclude that there is data sufficient to conclude that red wins significantly more than would be expected by chance.

(3) The two samples have different conclusions because the last sample has sufficient data or have more data in the study. If we have more data about the study in chi-square then there is big possibility that we can reject the null hypothesis compared to fewer data.