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Sagot :
### Step-by-Step Solution:
Let's go through each part of the problem:
#### Part (a): State the Hypotheses
We need to test the claim that men and women have equal success in challenging referee calls.
- Null Hypothesis ([tex]\(H_0\)[/tex]): The success rate of men and women in challenging calls is equal. Mathematically, this can be represented as [tex]\(H_0: p_1 = p_2\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The success rate of men and women in challenging calls is not equal. This can be represented as [tex]\(H_1: p_1 \neq p_2\)[/tex].
Given these options, the correct hypothesis statements are:
[tex]\[ C. \, H_0: p_1 = p_2 \, \text{and} \, H_1: p_1 \neq p_2 \][/tex]
#### Part (b): Identify the Test Statistic
We calculate the test statistic, which is [tex]\(z\)[/tex].
Given:
[tex]\[ z = 0.16 \][/tex]
This is rounded to two decimal places.
#### Part (c): Identify the [tex]\(P\)[/tex]-value
We use the test statistic to find the [tex]\(P\)[/tex]-value, which helps us decide whether to reject the null hypothesis or not.
Given:
[tex]\[ P\text{-value} = 0.87 \][/tex]
This is rounded to three decimal places.
#### Part (d): Conclusion Based on the Hypothesis Test
To conclude, we compare the [tex]\(P\)[/tex]-value with the significance level ([tex]\(\alpha = 0.01\)[/tex]):
- If [tex]\(P\)[/tex]-value [tex]\( \leq \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\(P\)[/tex]-value [tex]\( > \alpha \)[/tex], we fail to reject the null hypothesis.
Given:
[tex]\[ P\text{-value} = 0.87 > 0.01 \][/tex]
Since the [tex]\(P\)[/tex]-value is greater than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we fail to reject the null hypothesis.
Therefore, we conclude that there is not enough evidence to warrant rejection of the claim that men and women have equal success in challenging referee calls.
### Final Conclusion
Completing the last part in the text:
"The [tex]\(P\)[/tex]-value is greater than the significance level of [tex]\(\alpha = 0.01\)[/tex], so we fail to reject the null hypothesis. There is not enough evidence to warrant rejection of the claim that women and men have equal success in challenging calls."
Let's go through each part of the problem:
#### Part (a): State the Hypotheses
We need to test the claim that men and women have equal success in challenging referee calls.
- Null Hypothesis ([tex]\(H_0\)[/tex]): The success rate of men and women in challenging calls is equal. Mathematically, this can be represented as [tex]\(H_0: p_1 = p_2\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The success rate of men and women in challenging calls is not equal. This can be represented as [tex]\(H_1: p_1 \neq p_2\)[/tex].
Given these options, the correct hypothesis statements are:
[tex]\[ C. \, H_0: p_1 = p_2 \, \text{and} \, H_1: p_1 \neq p_2 \][/tex]
#### Part (b): Identify the Test Statistic
We calculate the test statistic, which is [tex]\(z\)[/tex].
Given:
[tex]\[ z = 0.16 \][/tex]
This is rounded to two decimal places.
#### Part (c): Identify the [tex]\(P\)[/tex]-value
We use the test statistic to find the [tex]\(P\)[/tex]-value, which helps us decide whether to reject the null hypothesis or not.
Given:
[tex]\[ P\text{-value} = 0.87 \][/tex]
This is rounded to three decimal places.
#### Part (d): Conclusion Based on the Hypothesis Test
To conclude, we compare the [tex]\(P\)[/tex]-value with the significance level ([tex]\(\alpha = 0.01\)[/tex]):
- If [tex]\(P\)[/tex]-value [tex]\( \leq \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\(P\)[/tex]-value [tex]\( > \alpha \)[/tex], we fail to reject the null hypothesis.
Given:
[tex]\[ P\text{-value} = 0.87 > 0.01 \][/tex]
Since the [tex]\(P\)[/tex]-value is greater than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we fail to reject the null hypothesis.
Therefore, we conclude that there is not enough evidence to warrant rejection of the claim that men and women have equal success in challenging referee calls.
### Final Conclusion
Completing the last part in the text:
"The [tex]\(P\)[/tex]-value is greater than the significance level of [tex]\(\alpha = 0.01\)[/tex], so we fail to reject the null hypothesis. There is not enough evidence to warrant rejection of the claim that women and men have equal success in challenging calls."
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