Question

since an instant replay system for tennis was introduced at a major tournament, men challenged 1394 referee calls, with the result that 417 of the calls were overturned. women challenged 779 referee calls, and 216 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.
consider the first sample to be the sample of male tennis players who challenged referee calls and the second sample to be the sample of female tennis players who challenged referee calls. what are the null and alternative hypotheses for the hypothesis test?
a. $h_0: p_1 \geq p_2$ $h_1: p_1 \
eq p_2$
b. $h_0: p_1 = p_2$ $h_1: p_1 > p_2$
c. $h_0: p_1 \
eq p_2$ $h_1: p_1 = p_2$
d. $h_0: p_1 = p_2$ $h_1: p_1 \
eq p_2$
e. $h_0: p_1 \leq p_2$ $h_1: p_1 \
eq p_2$
f. $h_0: p_1 = p_2$ $h_1: p_1 < p_2$
identify the test statistic.
$z = 1.08$ (round to two decimal places as needed.)
identify the p - value.
p - value = (round to three decimal places as needed.)

Explanation:

Part a: Hypotheses

The claim is that men and women have equal success in challenging calls, so the null hypothesis \( H_0 \) should state that the proportions \( p_1 \) (men's success proportion) and \( p_2 \) (women's success proportion) are equal. The alternative hypothesis \( H_1 \) for a two - tailed test (since we are testing for equality, and the alternative is that they are not equal) is \( p_1
eq p_2 \). Looking at the options, option D has \( H_0:p_1 = p_2 \) and \( H_1:p_1
eq p_2 \), which matches our reasoning.

The test statistic \( z = 1.08 \) is already given in the problem.

Step 1: Recall the distribution

For a two - tailed z - test, the P - value is the probability that \( Z < -|z| \) or \( Z > |z| \), where \( Z \) follows the standard normal distribution \( N(0,1) \). Here, \( |z|=1.08 \).

Step 2: Calculate the probability for one tail

First, find \( P(Z > 1.08) \). Using the standard normal table or a calculator, \( P(Z\leq1.08)=0.8599 \), so \( P(Z > 1.08)=1 - 0.8599 = 0.1401 \).

Step 3: Calculate the two - tailed P - value

Since it is a two - tailed test, we multiply the one - tailed probability by 2. So the P - value \(=2\times0.1401 = 0.2802\approx0.280\) (rounded to three decimal places).

Answer:

D. \( H_0:p_1 = p_2 \), \( H_1:p_1
eq p_2 \)

Part: Test Statistic