Question

Since an instant replay system for tennis was introduced at a major​ tournament, men challenged 1426 referee​ calls, with the result that 421 of the calls were overturned. Women challenged 752 referee​ calls, and 228 of the calls were overturned. Use a 0.05 significance level to test the claim that men and women have equal success in challenging calls. Complete parts​ (a) through​ (c) below.

a. Test the claim using a hypothesis test.

Identify the test statistic.

Identify the​ P-value.

What is the conclusion based on the hypothesis​ test?

The​ P-value is ▼less than/greater than the significance level of α=0.05so▼reject/fail to reject the null hypothesis. There▼is not sufficient/is sufficient evidence to warrant rejection of the claim that women and men have equal success in challenging calls.

b. Test the claim by constructing an appropriate confidence interval.

The 95% confidence interval is

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits▼include/do not include 0, there ▼does/does not appear to be a significant difference between the two proportions. There ▼is not sufficient/is sufficient evidence to warrant rejection of the claim that men and women have equal success in challenging calls.

c. Based on the​ results, does it appear that men and women may have equal success in challenging​ calls?

Answer 1

p1cap = X1/N1 = 421/1426 = 0.2952
p1cap = X2/N2 = 228/752 = 0.3032
pcap = (X1 + X2)/(N1 + N2) = (421+228)/(1426+752) = 0.298

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.2952-0.3032)/sqrt(0.298*(1-0.298)*(1/1426 + 1/752))
z = -0.39

P-value Approach
P-value = 0.6965

As P-value >= 0.05, fail to reject null hypothesis.

b)
Here, , n1 = 1426 , n2 = 752
p1cap = 0.2952 , p2cap = 0.3032

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.2952 * (1-0.2952)/1426 + 0.3032*(1-0.3032)/752)
SE = 0.0207

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.2952 - 0.3032 - 1.96*0.0207, 0.2952 - 0.3032 + 1.96*0.0207)
CI = (-0.0486 , 0.0326)

Because the confidence interval limits include 0, there does not appear to be a significant difference between the two proportions. There is not sufficient evidence to warrant rejection of the claim that men and women have equal success in challenging calls.

c)
Yes