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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%.
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