Get expert advice and community support on IDNLearn.com. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.
Sagot :
To address the question, we need to go through the steps of the hypothesis testing method for comparing two proportions. Let's do this step-by-step.
### Step 1: State the Hypotheses
We are testing the claim that men and women have equal success in challenging calls. This sets up our hypotheses as follows:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
From the provided options, the correct hypotheses setup is:
C. [tex]\( H_0: p_1 = p_2 \)[/tex] and [tex]\(H_1: p_1 \neq p_2\)[/tex]
### Step 2: Calculate the Sample Proportions
We have the following data:
- Men: [tex]\( n_1 = 1420 \)[/tex], [tex]\( x_1 = 416 \)[/tex]
- Women: [tex]\( n_2 = 739 \)[/tex], [tex]\( x_2 = 215 \)[/tex]
The sample proportions are calculated as:
[tex]\[ \hat{p}_1 = \frac{x_1}{n_1} = \frac{416}{1420} \approx 0.29296 \][/tex]
[tex]\[ \hat{p}_2 = \frac{x_2}{n_2} = \frac{215}{739} \approx 0.29093 \][/tex]
### Step 3: Calculate the Combined Proportion
The combined proportion [tex]\(\hat{p}\)[/tex] is calculated as:
[tex]\[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{416 + 215}{1420 + 739} \approx 0.29226 \][/tex]
### Step 4: Calculate the Standard Error
The standard error (SE) for the difference in proportions is calculated using the combined proportion:
[tex]\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \approx \sqrt{0.29226(1 - 0.29226) \left(\frac{1}{1420} + \frac{1}{739}\right)} \approx 0.02063 \][/tex]
### Step 5: Calculate the Test Statistic
The test statistic (z) for the difference in proportions is:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \approx \frac{0.29296 - 0.29093}{0.02063} \approx 0.098 \][/tex]
### Step 6: Determine the P-value
For a two-tailed test, the p-value is calculated as:
[tex]\[ p\text{-value} = 2 \times (1 - \Phi(|z|)) \][/tex]
Given our test statistic [tex]\( z \approx 0.098 \)[/tex], the corresponding p-value is approximately [tex]\( 0.9218 \)[/tex].
### Step 7: Conclusion
At a [tex]\( \alpha = 0.05 \)[/tex] significance level, we compare the p-value with [tex]\( \alpha \)[/tex]:
- If [tex]\( p\text{-value} < \alpha \)[/tex], we reject [tex]\( H_0 \)[/tex].
- If [tex]\( p\text{-value} \geq \alpha \)[/tex], we fail to reject [tex]\( H_0 \)[/tex].
Since [tex]\( p\text{-value} \approx 0.9218 \)[/tex] is much greater than [tex]\( 0.05 \)[/tex], we fail to reject the null hypothesis. This means that there is not enough evidence to suggest that the success rates of men and women in challenging referee calls are different.
### Summary
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
- Test Statistic ([tex]\(z\)[/tex]): [tex]\(\boxed{0.10}\)[/tex]
Thus, we conclude that the success rates for men and women in challenging calls are statistically equal in this context.
### Step 1: State the Hypotheses
We are testing the claim that men and women have equal success in challenging calls. This sets up our hypotheses as follows:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
From the provided options, the correct hypotheses setup is:
C. [tex]\( H_0: p_1 = p_2 \)[/tex] and [tex]\(H_1: p_1 \neq p_2\)[/tex]
### Step 2: Calculate the Sample Proportions
We have the following data:
- Men: [tex]\( n_1 = 1420 \)[/tex], [tex]\( x_1 = 416 \)[/tex]
- Women: [tex]\( n_2 = 739 \)[/tex], [tex]\( x_2 = 215 \)[/tex]
The sample proportions are calculated as:
[tex]\[ \hat{p}_1 = \frac{x_1}{n_1} = \frac{416}{1420} \approx 0.29296 \][/tex]
[tex]\[ \hat{p}_2 = \frac{x_2}{n_2} = \frac{215}{739} \approx 0.29093 \][/tex]
### Step 3: Calculate the Combined Proportion
The combined proportion [tex]\(\hat{p}\)[/tex] is calculated as:
[tex]\[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{416 + 215}{1420 + 739} \approx 0.29226 \][/tex]
### Step 4: Calculate the Standard Error
The standard error (SE) for the difference in proportions is calculated using the combined proportion:
[tex]\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \approx \sqrt{0.29226(1 - 0.29226) \left(\frac{1}{1420} + \frac{1}{739}\right)} \approx 0.02063 \][/tex]
### Step 5: Calculate the Test Statistic
The test statistic (z) for the difference in proportions is:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \approx \frac{0.29296 - 0.29093}{0.02063} \approx 0.098 \][/tex]
### Step 6: Determine the P-value
For a two-tailed test, the p-value is calculated as:
[tex]\[ p\text{-value} = 2 \times (1 - \Phi(|z|)) \][/tex]
Given our test statistic [tex]\( z \approx 0.098 \)[/tex], the corresponding p-value is approximately [tex]\( 0.9218 \)[/tex].
### Step 7: Conclusion
At a [tex]\( \alpha = 0.05 \)[/tex] significance level, we compare the p-value with [tex]\( \alpha \)[/tex]:
- If [tex]\( p\text{-value} < \alpha \)[/tex], we reject [tex]\( H_0 \)[/tex].
- If [tex]\( p\text{-value} \geq \alpha \)[/tex], we fail to reject [tex]\( H_0 \)[/tex].
Since [tex]\( p\text{-value} \approx 0.9218 \)[/tex] is much greater than [tex]\( 0.05 \)[/tex], we fail to reject the null hypothesis. This means that there is not enough evidence to suggest that the success rates of men and women in challenging referee calls are different.
### Summary
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
- Test Statistic ([tex]\(z\)[/tex]): [tex]\(\boxed{0.10}\)[/tex]
Thus, we conclude that the success rates for men and women in challenging calls are statistically equal in this context.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is your go-to source for dependable answers. Thank you for visiting, and we hope to assist you again.
