To determine the sample size needed for a poll with a specified margin of error and confidence level, we can use the formula for sample size in proportion studies. The formula is given by:
\[ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) \]
Where:
- \( n \) is the sample size,
- \( Z \) is the Z-score corresponding to the desired confidence level,
- \( p \) is the estimated proportion of the population (if unknown, 0.5 is often used for maximum variability),
- \( E \) is the margin of error.
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Identify the values:
- For a 95% confidence level, the Z-score (for a two-tailed test) is approximately 1.96.
- The margin of error \( E \) is 10%, or 0.10.
- Since we do not know the population proportion, we will use \( p = 0.5 \) for maximum sample size.
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Plug values into the formula: \[ n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.10)^2} \right) \] Simplifying this: \[ n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot 0.5}{0.01} \right) \] \[ n = \left( \frac{3.8416 \cdot 0.25}{0.01} \right) \] \[ n = \left( \frac{0.9604}{0.01} \right) \] \[ n = 96.04 \]
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Round up to the nearest whole person: Since sample size must be a whole number, we round up \( 96.04 \) to \( 97 \).
Therefore, the required sample size for the poll, rounded to whole people, is 97.