To construct a 95% confidence interval for the proportion of registered voters who will vote for candidate A, we can use the formula for the confidence interval for a population proportion:

$\text{CI} = \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

where:

• $\hat{p}$ is the sample proportion,
• $z$ is the z-value corresponding to the desired confidence level,
• $n$ is the sample size.

Step 1: Calculate the sample proportion ($\hat{p}$)

$\hat{p} = \frac{390}{625} = 0.624$

Step 2: Determine the z-value for a 95% confidence level

For a 95% confidence interval, the z-value is approximately 1.96.

Step 3: Calculate the standard error (SE)

$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.624(1 - 0.624)}{625}} = \sqrt{\frac{0.624 \times 0.376}{625}} \approx 0.020$

Step 4: Calculate the confidence interval

$\text{Margin of Error} = z \times SE = 1.96 \times 0.020 \approx 0.0392$ $\text{CI} = 0.624 \pm 0.0392$ Calculating the bounds:

• Lower bound: $0.624 - 0.0392 \approx 0.5848$
• Upper bound: $0.624 + 0.0392 \approx 0.6632$

Step 5: Convert to percentage form

• Lower bound: $0.5848 \times 100 \approx 58.48$
• Upper bound: $0.6632 \times 100 \approx 66.32$

Final Result

The 95% confidence interval for the proportion of all registered voters who will vote for candidate A, in percentage form, is approximately 58.48 to 66.32.

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the importance of the confidence interval in statistical analysis?
2. How would the interval change if the sample size were larger?
3. What assumptions are made when constructing a confidence interval for proportions?
4. How do you interpret the results of a confidence interval in a practical context?
5. What other statistical methods can be used to analyze voter preferences?

Tip: Always check that your sample is random to ensure that the confidence interval is valid.