To calculate the 95% confidence interval for the true population proportion, 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 confidence level (for 95%, $Z = 1.96$),
• $n$ is the sample size.

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

In this case: $\hat{p} = \frac{1200}{1600} = 0.75$

Step 2: Plug into the Confidence Interval Formula

Now, we can calculate the standard error and then find the margin of error.

Standard Error

$\text{Standard Error} = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} = \sqrt{\frac{0.75 \cdot (1 - 0.75)}{1600}} = \sqrt{\frac{0.75 \cdot 0.25}{1600}}$

Calculating this value: $\text{Standard Error} = \sqrt{\frac{0.1875}{1600}} = \sqrt{0.0001171875} \approx 0.01083$

Margin of Error

$\text{Margin of Error} = Z \cdot \text{Standard Error} = 1.96 \cdot 0.01083 \approx 0.0212$

Step 3: Determine the Confidence Interval

Now we can find the upper and lower bounds of the confidence interval.

$\text{Lower bound} = \hat{p} - \text{Margin of Error} = 0.75 - 0.0212 = 0.7288$ $\text{Upper bound} = \hat{p} + \text{Margin of Error} = 0.75 + 0.0212 = 0.7712$

Conclusion

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

$(0.7288, 0.7712)$

This means we are 95% confident that the true percentage of all party members who want a change in leadership is between 72.88% and 77.12%.

Would you like more details on any part of this calculation or have any questions?

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Here are some related questions to explore:

1. How would the confidence interval change if the sample size were smaller?
2. What if the confidence level were increased to 99%? How would that affect the interval?
3. How does the margin of error relate to the sample size?
4. What would happen if the proportion wanting change was higher, say 0.85?
5. How would we calculate a confidence interval if we were given the population standard deviation instead?

Tip: The confidence level affects the width of the interval; a higher confidence level results in a wider interval.