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   will be around 0.57. Round up to the nearest whole integer.

To determine the required sample size \( n \) to achieve a specified margin of error \( E \) for a proportion at a given confidence level, we can use the formula:

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

Where:
- \( Z \) is the z-score corresponding to the desired confidence level,
- \( p \) is the estimated proportion,
- \( E \) is the margin of error.

Given:
- Confidence level = 95% → \( Z \approx 1.96 \)
- Estimated proportion \( p = 0.57 \)
- Margin of error \( E = 0.015 \)

Now substituting the values into the formula:

\[
n = \left( \frac{(1.96)^2 \cdot 0.57 \cdot (1 - 0.57)}{(0.015)^2} \right)
\]

Calculating step by step:

1. Calculate \( 1 - p = 1 - 0.57 = 0.43 \).
2. Calculate \( (1.96)^2 \approx 3.8416 \).
3. Calculate \( E^2 = (0.015)^2 = 0.000225 \).
4. Now substitute these values into the formula:

\[
n = \frac{3.8416 \cdot 0.57 \cdot 0.43}{0.000225}
\]

5. Calculate the numerator:

\[
3.8416 \cdot 0.57 \cdot 0.43 \approx 0.100728
\]

6. Finally, calculate \( n \):

\[
n = \frac{0.100728}{0.000225} \approx 447.88
\]

Since we need a whole number, we round up:

\[
n \approx 448
\]

Thus, the required sample size is **448**.

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

1. How does the margin of error affect the required sample size?
2. What are the implications of underestimating or overestimating the sample size?
3. Can the sample size formula be used for different confidence levels?
4. How does the proportion affect the sample size calculation?
5. What other factors should be considered when designing a survey?

**Tip:** Always round up your sample size to ensure that the desired margin of error is met.

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 will be around 0.57. Round up to the nearest whole integer.

Learn how to calculate the sample size needed to estimate the proportion of voters in California supporting a ballot measure with a margin of error of 0.015 at a 95% confidence level. This guide includes detailed calculations using an estimated proportion of 0.57 and the required sample size of 448.

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