Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
### Solution
To find the [tex]$95\%$[/tex] confidence interval for the proportion of identity theft complaints in the state, we need to follow these steps:
#### 1. Identify the Parameter
The parameter we are analyzing is the proportion of all complaints that are identity theft complaints.
- Symbol: Let [tex]\( p \)[/tex] represent the true proportion of identity theft complaints in the state.
- Wording of the Parameter: The true proportion of all consumer complaints that are identity theft complaints in the state.
#### 2. Calculate the Sample Proportion ([tex]\( \hat{p} \)[/tex])
Given:
- Total consumer complaints ([tex]\( n \)[/tex]): 1650
- Identity theft complaints ([tex]\( x \)[/tex]): 393
The sample proportion [tex]\( \hat{p} \)[/tex] is calculated using:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{393}{1650} = 0.2381818181818182 \][/tex]
#### 3. Check Assumptions
To use the normal approximation for the confidence interval, the following conditions need to be satisfied:
- [tex]\( n \hat{p} \geq 10 \)[/tex]
- [tex]\( n (1 - \hat{p}) \geq 10 \)[/tex]
Check these:
[tex]\[ n \hat{p} = 1650 \times 0.2381818181818182 = 393 \][/tex]
[tex]\[ n (1 - \hat{p}) = 1650 \times (1 - 0.2381818181818182) = 1257 \][/tex]
Both conditions are satisfied because:
[tex]\[ 393 \geq 10 \][/tex]
[tex]\[ 1257 \geq 10 \][/tex]
#### 4. Calculate the Standard Error (SE)
The standard error of the sample proportion is computed as:
[tex]\[ \text{SE} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \][/tex]
[tex]\[ \text{SE} = \sqrt{\frac{0.2381818181818182 \times (1 - 0.2381818181818182)}{1650}} \][/tex]
[tex]\[ \text{SE} = 0.01048667956430909 \][/tex]
#### 5. Find the Critical Value ([tex]\( z^ \)[/tex]) for a 95% Confidence Interval
For a 95% confidence interval, the critical value [tex]\( z^ \)[/tex] corresponds to the z-value that captures the central 95% of the standard normal distribution:
[tex]\[ z^ = 1.959963984540054 \][/tex]
#### 6. Calculate the Margin of Error (ME)
The margin of error is calculated by:
[tex]\[ \text{ME} = z^ \times \text{SE} \][/tex]
[tex]\[ \text{ME} = 1.959963984540054 \times 0.01048667956430909 = 0.020553514263458 \][/tex]
#### 7. Determine the Confidence Interval
The confidence interval is calculated as:
[tex]\[ \text{Lower bound} = \hat{p} - \text{ME} \][/tex]
[tex]\[ \text{Lower bound} = 0.2381818181818182 - 0.020553514263458 = 0.2176283039183602 \][/tex]
[tex]\[ \text{Upper bound} = \hat{p} + \text{ME} \][/tex]
[tex]\[ \text{Upper bound} = 0.2381818181818182 + 0.020553514263458 = 0.2587353324452762 \][/tex]
Therefore, the 95% confidence interval for the true proportion of identity theft complaints in the state is:
[tex]\[ (0.2176283039183602, 0.2587353324452762) \][/tex]
### Summary
- Parameter Symbol: [tex]\( p \)[/tex]
- Wording of the Parameter: The true proportion of all consumer complaints that are identity theft complaints in the state.
- Assumptions Validated: [tex]\( n \hat{p} = 393 \geq 10 \)[/tex] and [tex]\( n (1 - \hat{p}) = 1257 \geq 10 \)[/tex]
- Calculated Confidence Interval: [tex]\( (0.2176283039183602, 0.2587353324452762) \)[/tex]
This interval means that we are 95% confident that the true proportion of identity theft complaints in the state lies between approximately 21.76% and 25.87%.
To find the [tex]$95\%$[/tex] confidence interval for the proportion of identity theft complaints in the state, we need to follow these steps:
#### 1. Identify the Parameter
The parameter we are analyzing is the proportion of all complaints that are identity theft complaints.
- Symbol: Let [tex]\( p \)[/tex] represent the true proportion of identity theft complaints in the state.
- Wording of the Parameter: The true proportion of all consumer complaints that are identity theft complaints in the state.
#### 2. Calculate the Sample Proportion ([tex]\( \hat{p} \)[/tex])
Given:
- Total consumer complaints ([tex]\( n \)[/tex]): 1650
- Identity theft complaints ([tex]\( x \)[/tex]): 393
The sample proportion [tex]\( \hat{p} \)[/tex] is calculated using:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{393}{1650} = 0.2381818181818182 \][/tex]
#### 3. Check Assumptions
To use the normal approximation for the confidence interval, the following conditions need to be satisfied:
- [tex]\( n \hat{p} \geq 10 \)[/tex]
- [tex]\( n (1 - \hat{p}) \geq 10 \)[/tex]
Check these:
[tex]\[ n \hat{p} = 1650 \times 0.2381818181818182 = 393 \][/tex]
[tex]\[ n (1 - \hat{p}) = 1650 \times (1 - 0.2381818181818182) = 1257 \][/tex]
Both conditions are satisfied because:
[tex]\[ 393 \geq 10 \][/tex]
[tex]\[ 1257 \geq 10 \][/tex]
#### 4. Calculate the Standard Error (SE)
The standard error of the sample proportion is computed as:
[tex]\[ \text{SE} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \][/tex]
[tex]\[ \text{SE} = \sqrt{\frac{0.2381818181818182 \times (1 - 0.2381818181818182)}{1650}} \][/tex]
[tex]\[ \text{SE} = 0.01048667956430909 \][/tex]
#### 5. Find the Critical Value ([tex]\( z^ \)[/tex]) for a 95% Confidence Interval
For a 95% confidence interval, the critical value [tex]\( z^ \)[/tex] corresponds to the z-value that captures the central 95% of the standard normal distribution:
[tex]\[ z^ = 1.959963984540054 \][/tex]
#### 6. Calculate the Margin of Error (ME)
The margin of error is calculated by:
[tex]\[ \text{ME} = z^ \times \text{SE} \][/tex]
[tex]\[ \text{ME} = 1.959963984540054 \times 0.01048667956430909 = 0.020553514263458 \][/tex]
#### 7. Determine the Confidence Interval
The confidence interval is calculated as:
[tex]\[ \text{Lower bound} = \hat{p} - \text{ME} \][/tex]
[tex]\[ \text{Lower bound} = 0.2381818181818182 - 0.020553514263458 = 0.2176283039183602 \][/tex]
[tex]\[ \text{Upper bound} = \hat{p} + \text{ME} \][/tex]
[tex]\[ \text{Upper bound} = 0.2381818181818182 + 0.020553514263458 = 0.2587353324452762 \][/tex]
Therefore, the 95% confidence interval for the true proportion of identity theft complaints in the state is:
[tex]\[ (0.2176283039183602, 0.2587353324452762) \][/tex]
### Summary
- Parameter Symbol: [tex]\( p \)[/tex]
- Wording of the Parameter: The true proportion of all consumer complaints that are identity theft complaints in the state.
- Assumptions Validated: [tex]\( n \hat{p} = 393 \geq 10 \)[/tex] and [tex]\( n (1 - \hat{p}) = 1257 \geq 10 \)[/tex]
- Calculated Confidence Interval: [tex]\( (0.2176283039183602, 0.2587353324452762) \)[/tex]
This interval means that we are 95% confident that the true proportion of identity theft complaints in the state lies between approximately 21.76% and 25.87%.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
