Answer :
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."
