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Let's break down the problem and go step-by-step:

### (a) Finding a 99% confidence interval for \( p \)
We are given the following dataLet's break down the problem and go step-by-step:

### (a) Finding a 99% confidence interval for \( p \)
We are given the following 
- \( n = 403 \) (total number of voters sampled)
- \( X = 40 \) (number of voters who said they would vote for Candidate A)
- Confidence level = 99%, corresponding to \( Z = 2.576 \) (Z-value for 99% confidence level from Z distribution)

The formula for the confidence interval (CI) for proportion \( p \) is:

\[
\hat{p} = \frac{X}{n}
\]

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

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

\[
\hat{p} = \frac{40}{403} \approx 0.0993
\]

2. **Step 2**: Plug into the confidence interval formula. First, calculate the standard error (SE):

\[
SE = \sqrt{\frac{0.0993 \times (1 - 0.0993)}{403}} \approx 0.01467
\]

3. **Step 3**: Now, calculate the margin of error (ME):

\[
ME = Z \times SE = 2.576 \times 0.01467 \approx 0.0378
\]

4. **Step 4**: Finally, compute the lower and upper bounds for the confidence interval:

\[
\text{Lower Bound} = 0.0993 - 0.0378 = 0.0615
\]
\[
\text{Upper Bound} = 0.0993 + 0.0378 = 0.1371
\]

Thus, the 99% confidence interval for \( p \) is approximately \( [0.0615, 0.1371] \).

### (b) Interpretation of the Confidence Interval
The interpretation of the confidence interval is as follows:

We are 99% confident that the true proportion of all voters who will vote for Candidate A lies between 6.15% and 13.71%.

### (c) Using Bootstrapping in R-Studio
The R code snippet is provided to perform bootstrapping to estimate the 99% confidence interval for \( p \). The code does the following:
- It sets a random seed for reproducibility.
- It performs 1000 resamples of the voters and calculates the proportion of voters choosing Candidate A in each resample.
- Finally, it computes the 99% confidence interval based on the resampled proportions.

You can use this code in R-Studio to simulate the sampling process.

Do you need more details or have any questions?

Here are five related questions for deeper understanding:
1. What is the difference between a confidence interval and a margin of error?
2. How does increasing the sample size affect the confidence interval?
3. Why do we use the Z-distribution instead of the t-distribution in this case?
4. What assumptions are made when calculating a confidence interval for a proportion?
5. How would the confidence interval change if the confidence level were reduced to 95%?

**Tip**: Confidence intervals provide a range of plausible values for a population parameter, but they do not guarantee that the parameter falls within that range for every sample.

A poll of n voters is taken in an attempt to predict the outcome of a by-election in a certain riding. Specifically, we are interested in the proportion of voters who will vote for Candidate A. 403 voters have been randomly chosen, and 40 have indicated that they will vote for Candidate A. We are tasked with finding a 99% confidence interval for the proportion of all voters who will vote for Candidate A using both the Z-distribution and bootstrapping in R-Studio.

Learn how to calculate a 99% confidence interval for the proportion of voters supporting Candidate A using a sample size of 403 and a count of 40 votes. This guide covers using the Z-distribution formula and bootstrapping methods in R-Studio. Ideal for university students studying statistics or probability.

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