Final answer:
To find a 98% confidence interval for the proportion, use the formula CI = p +- Z/2 √((p(1-p))/n). For part (b), an 80% confidence interval is smaller than a 98% confidence interval.
Step-by-step explanation:
To find a 98% confidence interval for the proportion, we can use the formula:
CI = p ± Zα/2 √((p(1-p))/n),
where CI is the confidence interval, p is the proportion of voters who indicated a preference for the candidate, Zα/2 is the critical value corresponding to the desired confidence level, and n is the sample size.
Substituting the given values, we have:
CI = (335/500) ± Zα/2 √((335/500)(165/500)/500),
Using a Zα/2 value for a 98% confidence level, which is approximately 2.33, we can calculate:
CI = (0.67) ± (2.33) √((0.67)(0.33)/500),
Calculating the values inside the square root and simplifying, we get:
CI = (0.67) ± 0.045,
Therefore, the 98% confidence interval for p is approximately (0.625, 0.715).
For part (b), without doing any calculations, we can determine that an 80% confidence interval will be smaller than a 98% confidence interval. This is because as the confidence level decreases, the critical value Zα/2 becomes smaller, resulting in a narrower confidence interval.