Answer:
Explanation:
To construct a 99.8% confidence interval for the proportion of adults who believe that economic conditions are getting better, the pollster needs to determine the sample size required for a specific margin of error.
(a) When an estimate of the proportion is available, the sample size can be calculated using the formula:
n = (Z * sqrt(p * (1 - p))) / E)^2
Given that the estimated proportion is 0.34 and the desired margin of error is 0.03, we can calculate the sample size as follows:
n = (Z * sqrt(p * (1 - p))) / E)^2
n = (2.967 * sqrt(0.34 * (1 - 0.34))) / 0.03)^2
n ≈ (2.967 * sqrt(0.34 * 0.66)) / 0.03)^2
n ≈ (2.967 * sqrt(0.2244)) / 0.03)^2
n ≈ (2.967 * 0.473) / 0.03)^2
n ≈ (1.401) / 0.03)^2
n ≈ 46.7^2
n ≈ 2,179.89
Therefore, a sample size of approximately 2,180 is needed using the estimated proportion of 0.34.
(b) When no estimate of the proportion is available, a conservative estimate of p = 0.5 can be used to maximize the required sample size. Using the same formula, we can calculate the sample size:
n = (Z * sqrt(p * (1 - p))) / E)^2
n = (2.967 * sqrt(0.5 * (1 - 0.5))) / 0.03)^2
n ≈ (2.967 * sqrt(0.5 * 0.5)) / 0.03)^2
n ≈ (2.967 * sqrt(0.25)) / 0.03)^2
n ≈ (2.967 * 0.5) / 0.03)^2
n ≈ (1.484) / 0.03)^2
n ≈ 49.47^2
n ≈ 2,447.2
Therefore, a sample size of approximately 2,448 is needed when no estimate of the proportion is available.
It's important to note that these calculations assume a simple random sample and other factors, such as the desired confidence level and population size, can also influence the required sample size.