To estimate the population proportion of voters who support the candidate, we first need to calculate the sample proportion (\( \hat{p} \)) and the standard error (SE) of that sample proportion.
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Calculate the sample proportion (\( \hat{p} \)): \[ \hat{p} = \frac{\text{Number of supporters}}{\text{Total sample size}} = \frac{280}{500} = 0.56 \]
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Calculate the standard error (SE) for the proportion: The standard error for a sample proportion can be calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] where \( n \) is the sample size.
Plugging in the values: \[ SE = \sqrt{\frac{0.56(1 - 0.56)}{500}} = \sqrt{\frac{0.56 \times 0.44}{500}} = \sqrt{\frac{0.2464}{500}} = \sqrt{0.0004928} \approx 0.0222 \]
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Calculate the interval using two standard errors: The interval will be \( \hat{p} \pm 2 \times SE \). \[ 2 \times SE = 2 \times 0.0222 \approx 0.0444 \]
Therefore, the interval can be calculated as follows: \[ \text{Lower bound} = \hat{p} - 2 \times SE = 0.56 - 0.0444 \approx 0.5156 \] \[ \text{Upper bound} = \hat{p} + 2 \times SE = 0.56 + 0.0444 \approx 0.6044 \]
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Round the bounds to two decimal places:
- Lower bound: \( 0.52 \)
- Upper bound: \( 0.60 \)
Putting it all together, the estimated proportion of voters who support the candidate, ranging ± two standard errors, is:
\[ 0.52 \leq p \leq 0.60 \]