A random sample of 625 registered voters was asked who they will vote for in the upcoming election and 390 of them said they will vote for candidate A. Construct the 95% confidence interval for the proportion of all registered voters who will vote for candidate A.
How would you write this confidence interval in percentage form (do not include the % symbol)?



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.

A random sample of 625 registered voters was asked who they will vote for in the upcoming election and 390 of them said they will vote for candidate A. Construct the 95% confidence interval for the proportion of all registered voters who will vote for candidate A. How would you write this confidence interval in percentage form (do not include the % symbol)?

Discover how to calculate the 95% confidence interval for the proportion of registered voters supporting candidate A, based on a sample of 625 voters with 390 in favor. This guide includes step-by-step calculations and interpretations, suitable for advanced statistics students.

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