Question

Since an instant replay system for tennis was introduced at a major​ tournament, men challenged 14301430 referee​ calls, with the result that 416416 of the calls were overturned. Women challenged 740740 referee​ calls, and 219219 of the calls were overturned. Use a 0.050.05 significance level to test the claim that men and women have equal success in challenging calls. Complete parts​ (a) through​ (c) below.

Answer 1

For men, we have that the sample size is N1​=1430, the number of favorable cases is X1​=416, so then the sample proportion is

For women, we have that the sample size is N2​=740, the number of favorable cases is X2​=219, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho : p1​=p2​

Ha : p1​̸ p2​

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

The significance level is α=0.05, and the critical value for a two-tailed test is zc​=1.96.

The rejection region for this two-tailed test is R = { z : ∣z∣ > 1.96 }

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣ = 0.244 ≤ zc ​= 1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.8069, and since p = 0.8069 ≥ 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p1​ is different than p2​, at the 0.05 significance level.

Hence we can conclude that men and women have equal success in challenging calls.