To estimate the population proportion of voters who support the candidate, we first calculate the sample proportion (\( \hat{p} \)) and the standard error (SE) of that proportion.
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Calculate the sample proportion (\( \hat{p} \)): \[ \hat{p} = \frac{X}{n} = \frac{280}{500} = 0.56 \]
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Calculate the standard error (SE): The formula for the standard error of a proportion is: \[ SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \] Substituting in the values: \[ SE = \sqrt{\frac{0.56 (1 - 0.56)}{500}} = \sqrt{\frac{0.56 \times 0.44}{500}} = \sqrt{\frac{0.2464}{500}} \approx \sqrt{0.0004928} \approx 0.0222 \]
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Calculate the margin of error (ME) for two standard errors: \[ ME = 2 \times SE \approx 2 \times 0.0222 \approx 0.0444 \]
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Determine the confidence interval: The confidence interval is given by: \[ \hat{p} \pm ME \] In this case: \[ 0.56 - 0.0444 \quad \text{to} \quad 0.56 + 0.0444 \] \[ 0.5156 \quad \text{to} \quad 0.6044 \]
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Round to two decimal places: Rounding \( 0.5156 \) to two decimal places gives \( 0.52 \), and rounding \( 0.6044 \) gives \( 0.60 \).
Thus, the interval estimate for the population proportion of voters who support the candidate is: \[ \text{Confidence Interval: } [0.52, 0.60] \]
Final answer: \[ \leq p \leq \quad 0.52 \quad \text{to} \quad 0.60 \]