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According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, [tex]$23\%$[/tex] of all complaints in 2007 were for identity theft (Consumer fraud and,' 2008). Assume a state had 393 complaints of identity theft out of 1650 consumer complaints. Calculate a [tex]$95\%$[/tex] confidence interval for the proportion of all complaints that are complaints of identity theft in that state.

P: Parameter

1. What is the correct parameter symbol for this problem?
- ?

2. What is the wording of the parameter in the context of this problem?
- Select an answer

A: Assumptions

Since Select an answer [tex]\theta[/tex] information was collected from each object, what conditions do we need to check? Check all that apply.

- [tex]N \geq 20n[/tex]
- [tex]\sigma[/tex] is known.
- [tex]n \geq 30[/tex] or normal population.
- [tex]n(p) \geq 10[/tex]
- [tex]n(1-p) \geq 10[/tex]
- [tex]\sigma[/tex] is unknown.

Check those assumptions:

1. [tex]n \hat{p} = \square[/tex], which is ? [tex]\square[/tex]


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%.