Experience the convenience of getting your questions answered at IDNLearn.com. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.
Sagot :
Sure! Let's go through the detailed solution step by step:
### Step 1: Calculate the sample proportions
First, we need to calculate the sample proportions of overturned calls for both men and women.
- For men:
- Total challenges: [tex]\(1416\)[/tex]
- Overturned calls: [tex]\(426\)[/tex]
- Proportion: [tex]\( p_1 = \frac{426}{1416} = 0.3008474576271186 \approx 0.301 \)[/tex]
- For women:
- Total challenges: [tex]\(755\)[/tex]
- Overturned calls: [tex]\(225\)[/tex]
- Proportion: [tex]\( p_2 = \frac{225}{755} = 0.2980132450331126 \approx 0.298 \)[/tex]
### Step 2: Compute the difference in sample proportions
The difference in sample proportions is calculated as follows:
[tex]\[ \text{Difference} = p_1 - p_2 = 0.301 - 0.298 = 0.0028342125940060137 \approx 0.003 \][/tex]
### Step 3: Calculate the standard error of the difference
Using the sample proportions and their counts, the standard error is calculated with the formula:
[tex]\[ \text{Standard Error (SE)} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}} \][/tex]
Upon calculation:
[tex]\[ \text{SE} = \sqrt{\frac{0.301 \times (1 - 0.301)}{1416} + \frac{0.298 \times (1 - 0.298)}{755}} = 0.02063084715277985 \approx 0.021 \][/tex]
### Step 4: Determine the critical value for a 99% confidence interval
For a 99% confidence level, the critical value (z-value) from the standard normal distribution is:
[tex]\[ z = 2.5758293035489004 \approx 2.576 \][/tex]
### Step 5: Margin of Error
The margin of error (MOE) is calculated as:
[tex]\[ \text{MOE} = z \times \text{SE} \][/tex]
Upon calculation:
[tex]\[ \text{MOE} = 2.576 \times 0.021 = 0.05314154065316873 \approx 0.053 \][/tex]
### Step 6: Construct the confidence interval
The confidence interval for the difference in proportions [tex]\( (p_1 - p_2) \)[/tex] is:
[tex]\[ \text{CI} = (\text{Difference} - \text{MOE}, \text{Difference} + \text{MOE}) \][/tex]
Upon calculation:
[tex]\[ \text{CI} = (0.003 - 0.053, 0.003 + 0.053) = (-0.050307328059162715, 0.05597575324717474) \approx (-0.050, 0.056) \][/tex]
### Step 7: Conclusion based on the confidence interval
The 99% confidence interval for the difference in proportions [tex]\((p_1 - p_2)\)[/tex] is [tex]\(-0.050 \text{ to } 0.056\)[/tex].
#### Conclusion:
- Since the interval [tex]\(-0.050 \text{ to } 0.056\)[/tex] contains zero, there does not appear to be a significant difference between the two proportions.
- Therefore, there is not enough evidence to warrant rejection of the claim that men and women have equal success in challenging calls.
So, the completed statement would be:
- "Because the confidence interval limits contain 0, there does not appear to be a significant difference between the two proportions. There is not evidence to warrant rejection of the claim that men and women have equal success in challenging calls."
### Step 1: Calculate the sample proportions
First, we need to calculate the sample proportions of overturned calls for both men and women.
- For men:
- Total challenges: [tex]\(1416\)[/tex]
- Overturned calls: [tex]\(426\)[/tex]
- Proportion: [tex]\( p_1 = \frac{426}{1416} = 0.3008474576271186 \approx 0.301 \)[/tex]
- For women:
- Total challenges: [tex]\(755\)[/tex]
- Overturned calls: [tex]\(225\)[/tex]
- Proportion: [tex]\( p_2 = \frac{225}{755} = 0.2980132450331126 \approx 0.298 \)[/tex]
### Step 2: Compute the difference in sample proportions
The difference in sample proportions is calculated as follows:
[tex]\[ \text{Difference} = p_1 - p_2 = 0.301 - 0.298 = 0.0028342125940060137 \approx 0.003 \][/tex]
### Step 3: Calculate the standard error of the difference
Using the sample proportions and their counts, the standard error is calculated with the formula:
[tex]\[ \text{Standard Error (SE)} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}} \][/tex]
Upon calculation:
[tex]\[ \text{SE} = \sqrt{\frac{0.301 \times (1 - 0.301)}{1416} + \frac{0.298 \times (1 - 0.298)}{755}} = 0.02063084715277985 \approx 0.021 \][/tex]
### Step 4: Determine the critical value for a 99% confidence interval
For a 99% confidence level, the critical value (z-value) from the standard normal distribution is:
[tex]\[ z = 2.5758293035489004 \approx 2.576 \][/tex]
### Step 5: Margin of Error
The margin of error (MOE) is calculated as:
[tex]\[ \text{MOE} = z \times \text{SE} \][/tex]
Upon calculation:
[tex]\[ \text{MOE} = 2.576 \times 0.021 = 0.05314154065316873 \approx 0.053 \][/tex]
### Step 6: Construct the confidence interval
The confidence interval for the difference in proportions [tex]\( (p_1 - p_2) \)[/tex] is:
[tex]\[ \text{CI} = (\text{Difference} - \text{MOE}, \text{Difference} + \text{MOE}) \][/tex]
Upon calculation:
[tex]\[ \text{CI} = (0.003 - 0.053, 0.003 + 0.053) = (-0.050307328059162715, 0.05597575324717474) \approx (-0.050, 0.056) \][/tex]
### Step 7: Conclusion based on the confidence interval
The 99% confidence interval for the difference in proportions [tex]\((p_1 - p_2)\)[/tex] is [tex]\(-0.050 \text{ to } 0.056\)[/tex].
#### Conclusion:
- Since the interval [tex]\(-0.050 \text{ to } 0.056\)[/tex] contains zero, there does not appear to be a significant difference between the two proportions.
- Therefore, there is not enough evidence to warrant rejection of the claim that men and women have equal success in challenging calls.
So, the completed statement would be:
- "Because the confidence interval limits contain 0, there does not appear to be a significant difference between the two proportions. There is not evidence to warrant rejection of the claim that men and women have equal success in challenging calls."
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
