If 1600 members of a political party were randomly sampled and 1200 said they wanted a
change in leadership, what is the 95% confidence interval for the true population
percentage for all of the members of the party who wanted a change in leadership?

To calculate the 95% confidence interval for the true population percentage of party members who want a change in leadership, we can use the formula for the confidence interval of a proportion:

\[
\text{Confidence Interval} = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

where:
- \(\hat{p}\) is the sample proportion,
- \(Z\) is the Z-score corresponding to the 95% confidence level (approximately 1.96),
- \(n\) is the sample size.

### Step 1: Calculate the Sample Proportion (\(\hat{p}\))

\[
\hat{p} = \frac{\text{number who want change}}{\text{total sample size}} = \frac{1200}{1600} = 0.75
\]

### Step 2: Plugging Values into the Formula

Using \(Z = 1.96\) for a 95% confidence level, \(n = 1600\), and \(\hat{p} = 0.75\):

\[
\text{Margin of Error} = 1.96 \cdot \sqrt{\frac{0.75 \times (1 - 0.75)}{1600}}
\]

1. Calculate \(0.75 \times (1 - 0.75) = 0.75 \times 0.25 = 0.1875\),
2. Divide by 1600: \(\frac{0.1875}{1600} = 0.0001171875\),
3. Take the square root: \(\sqrt{0.0001171875} \approx 0.01083\),
4. Multiply by 1.96: \(1.96 \times 0.01083 \approx 0.02123\).

### Step 3: Determine the Confidence Interval

\[
\text{Confidence Interval} = 0.75 \pm 0.02123
\]

Converting to percentages, we get:

\[
\text{Confidence Interval} = 75\% \pm 2.12\%
\]

### Final Answer
The 95% confidence interval for the true population percentage of party members who want a change in leadership is approximately:

\[
[72.88\%, 77.12\%]
\]

Would you like further details or have any questions?

---

Here are some related questions:
1. How would the confidence interval change if the confidence level were 99%?
2. What if the sample size were doubled to 3200 members?
3. How does the choice of confidence level affect the width of the interval?
4. What is the interpretation of a 95% confidence interval in this context?
5. How would the results differ if the sample proportion were closer to 0.5?

**Tip:** Larger sample sizes generally produce narrower confidence intervals, reflecting higher precision in the estimate of the population parameter.

If 1600 members of a political party were randomly sampled and 1200 said they wanted a change in leadership, what is the 95% confidence interval for the true population percentage for all of the members of the party who wanted a change in leadership?

Discover how to calculate the 95% confidence interval for the true population percentage of political party members wanting a leadership change, based on a sample of 1600 members, with 1200 expressing the desire for change. This guide includes detailed steps and calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation