A legal researcher is trying to estimate the average number of years that justices have served on the Supreme Court. He would like to be within 4 years with 98% confidence. If the standard deviation is 8.6 years, how large a sample would be required?
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To determine the sample size required for estimating the average number of years served on the Supreme Court, we can use the formula:
n = ((Z * σ) / E)^2
Where:
n = sample size
Z = Z-score (associated with the desired level of confidence)
σ = standard deviation of the population
E = desired margin of error
In this case, the researcher would like to have a 98% confidence level and be within 4 years of the true average. The Z-score associated with a 98% confidence level (two-tailed test) is approximately 2.33. The standard deviation of the population is given as 8.6 years, and the desired margin of error is 4 years.
Plugging these values into the formula, we get:
n = ((2.33 * 8.6) / 4)^2
n = (19.918 / 4)^2
n = 4.98^2
n ≈ 24.8
Therefore, the researcher would need a sample size of at least 25 justices to estimate the average number of years served on the Supreme Court, with a 98% confidence level and a 4-year margin of error. Note that sample sizes are typically rounded up to the nearest whole number, so in practice, the researcher would need a sample size of 25.