Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, she has garnered 51% support. Construct a 95% confidence interval on the population proportion for the support of candidate A in the following election

To construct a 95% confidence interval for the population proportion of support for Candidate A, we can use the formula for the confidence interval for a proportion:

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

Where:
- \(\hat{p}\) is the sample proportion.
- \(Z\) is the Z-score corresponding to the desired confidence level.
- \(n\) is the sample size.

### Step 1: Identify the values
- Sample size (\(n\)) = 300
- Sample proportion (\(\hat{p}\)) = 51% = 0.51
- Z-score for a 95% confidence level ≈ 1.96 (from Z-tables).

### Step 2: Calculate the standard error (SE)
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.51 \times (1 - 0.51)}{300}} = \sqrt{\frac{0.51 \times 0.49}{300}} \approx \sqrt{\frac{0.2499}{300}} \approx \sqrt{0.000833} \approx 0.0289
\]

### Step 3: Calculate the margin of error (ME)
\[
ME = Z \cdot SE = 1.96 \cdot 0.0289 \approx 0.0567
\]

### Step 4: Construct the confidence interval
\[
CI = \hat{p} \pm ME = 0.51 \pm 0.0567
\]
- Lower limit: \(0.51 - 0.0567 \approx 0.4533\)
- Upper limit: \(0.51 + 0.0567 \approx 0.5667\)

### Final Result
The 95% confidence interval for the population proportion of support for Candidate A is approximately:

\[
(0.4533, 0.5667)
\]

### Summary
This means we are 95% confident that the true proportion of support for Candidate A in the population lies between approximately 45.33% and 56.67%.

Would you like more details on this calculation or have any questions? Here are some related questions you might consider:
1. How would the confidence interval change with a larger sample size?
2. What is the effect of a higher or lower Z-score on the confidence interval?
3. How do you interpret the confidence interval in practical terms for the election?
4. What assumptions are made when using this method for confidence intervals?
5. How would you perform this calculation for a different confidence level, such as 90%? 

**Tip:** Always check that the sample size is sufficiently large (np and n(1-p) both greater than 5) to use the normal approximation for proportions.

Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, she has garnered 51% support. Construct a 95% confidence interval on the population proportion for the support of candidate A in the following election.

Learn how to construct a 95% confidence interval for Candidate A's support in a mayoral election poll, where she received 51% support from 300 residents. This guide covers the calculation of standard error and margin of error.

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