You wish to estimate the proportion of all voters in California who plan to vote in favor of a certain ballot measure. Obtain a sample size that will ensure a margin of error of .015 for a 95% confidence interval. From a pilot study we know that ^p  will be around 0.57. Round up to the nearest whole integer.

To calculate the required sample size \( n \) for estimating the proportion of voters with a specified margin of error, we can use the formula for the sample size in proportion estimation:

\[
n = \left( \frac{Z^2 \cdot p(1 - p)}{E^2} \right)
\]

Where:
- \( Z \) is the Z-value corresponding to the desired confidence level.
- \( p \) is the estimated proportion from the pilot study.
- \( E \) is the margin of error.

Given:
- Confidence level = 95%, so \( Z \approx 1.96 \) (from Z-tables).
- Estimated proportion \( \hat{p} = 0.57 \).
- Margin of error \( E = 0.015 \).

Now, substitute these values into the formula:

1. Calculate \( p(1 - p) \):
   \[
   p(1 - p) = 0.57 \times (1 - 0.57) = 0.57 \times 0.43 = 0.2451
   \]

2. Substitute into the sample size formula:
   \[
   n = \frac{(1.96^2) \cdot 0.2451}{(0.015^2)}
   \]

   \[
   n = \frac{(3.8416) \cdot 0.2451}{0.000225}
   \]

   \[
   n = \frac{0.9429}{0.000225} \approx 4186.67
   \]

3. Round up to the nearest whole integer:
   \[
   n = 4187
   \]

Thus, the required sample size is **4187**.

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You wish to estimate the proportion of all voters in California who plan to vote in favor of a certain ballot measure. Obtain a sample size that will ensure a margin of error of .015 for a 95% confidence interval. From a pilot study we know that ^p will be around 0.57. Round up to the nearest whole integer.

Discover how to calculate the required sample size for estimating the proportion of voters in California supporting a ballot measure with a margin of error of .015 at a 95% confidence level. Based on a pilot study proportion of 0.57, learn the steps leading to a required sample size of 4187.

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