Near the time of an election a cable news service performs an opinion poll of 1000 probable voters Near the time of an election a cable news service performs an opinion poll of 1000 probable voters It

a. To develop a 95% confidence interval for the proportion favoring the Republican candidate, we can use the formula for confidence interval for a proportion:

CI = p̂ ± Z * √(p̂(1-p̂)/n)

Where: CI = Confidence Interval p̂ = Proportion favoring the Republican candidate (0.52 in this case) Z = Z-score for the desired confidence level (for 95% confidence level, Z = 1.96) n = Sample size (1,000 in this case)

CI = 0.52 ± 1.96 * √(0.52(1-0.52)/1000)

CI = 0.52 ± 0.0312

The 95% confidence interval for the proportion favoring the Republican candidate is (0.4888, 0.5512).

b. To estimate the probability that the Democratic candidate is actually leading, we can subtract the proportion favoring the Republican candidate from 1:

Probability = 1 - 0.52 = 0.48

Therefore, the estimated probability that the Democratic candidate is actually leading is 48%.

c. To repeat the above analysis based on a sample of 3,000 probable voters, we can use the same formulas but with the updated sample size (n = 3,000):

CI = 0.52 ± 1.96 * √(0.52(1-0.52)/3000)

CI = 0.52 ± 0.018

The 95% confidence interval for the proportion favoring the Republican candidate based on a sample of 3,000 probable voters is (0.502, 0.538).

The estimated probability that the Democratic candidate is actually leading would still be 48% since it is based on the proportion favoring the Republican candidate