A government survey conducted to estimate the mean price of houses in a metropolitan area is designed to have a margin of error of $10,000. Pilot studies suggest that the population standard deviation is $70,000. Estimate the minimum sample size needed to estimate the population mean with the stated accuracy.
To estimate the minimum sample size needed to estimate the population mean with the stated accuracy, we can use the formula for determining sample size for estimating a population mean:
n = (Z * σ / E)^2
Where:
n = sample size
Z = z-score corresponding to the desired level of confidence
σ = population standard deviation
E = desired margin of error
In this case, the desired margin of error is $10,000, and the pilot studies suggest that the population standard deviation is $70,000.
To determine the z-score corresponding to the desired level of confidence, we need to specify the level of confidence. Let's assume a 95% confidence level, which corresponds to a z-score of approximately 1.96 (obtained from a standard normal distribution table).
Now, we can substitute the known values into the formula:
n = (1.96 * 70,000 / 10,000)^2
Simplifying:
n = (1.96 * 7)^2 (dividing 70,000 by 10,000)
n = (13.72)^2 (multiplying 1.96 by 7)
n = 188.4224
Rounding up to the nearest integer, the minimum sample size needed is approximately 189.