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.