Find the sample size needed to estimate the percentage of Republicans among registered voters in 

California using a 1% margin of error and a 90% confidence level.

To determine the sample size needed to estimate the percentage of Republicans among registered voters in California with a 1% margin of error at a 90% confidence level, we can use the formula for sample size in proportion estimates:

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

where:
- \( Z \) is the z-score corresponding to the confidence level,
- \( p \) is the estimated proportion of Republicans among registered voters (since we don’t know the true proportion, we use \( p = 0.5 \) for maximum variability),
- \( E \) is the margin of error.

### Step 1: Identify the z-score for a 90% confidence level
For a 90% confidence level, the z-score is approximately **1.645**.

### Step 2: Substitute values into the formula
Using:
- \( Z = 1.645 \),
- \( p = 0.5 \),
- \( E = 0.01 \) (since 1% is the margin of error),

we have:

\[
n = \frac{(1.645)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.01)^2}
\]

### Step 3: Calculate
1. \( 1.645^2 = 2.7025 \)
2. \( 0.5 \cdot (1 - 0.5) = 0.25 \)
3. \((0.01)^2 = 0.0001\)

Substituting these values:

\[
n = \frac{2.7025 \cdot 0.25}{0.0001} = \frac{0.675625}{0.0001} = 6756.25
\]

Since we cannot have a fraction of a person, we round up to the nearest whole number.

### Final Answer:
The required sample size is **6757**.

Would you like more details on any part of this process, or do you have other questions?

---

Here are 5 related questions for further exploration:
1. What is the impact of using a different confidence level, such as 95%, on the sample size?
2. How does increasing the margin of error to 2% affect the required sample size?
3. What if we had an estimated proportion \( p = 0.4 \)? How would that change the sample size?
4. How would this calculation differ for estimating proportions in a smaller population, like a specific California county?
5. How is the z-score determined for other confidence levels?

**Tip:** When unsure about the population proportion, using \( p = 0.5 \) is a safe choice because it maximizes the required sample size, ensuring precision.

Find the sample size needed to estimate the percentage of Republicans among registered voters in California using a 1% margin of error and a 90% confidence level.

Learn how to calculate the sample size needed to estimate the percentage of Republicans among registered voters in California, given a 1% margin of error and a 90% confidence level. This solution includes the formula for sample size in proportion estimates and key statistical concepts like confidence intervals and the Central Limit Theorem. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation