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.
not my thing, hope stats teacher happens by
189
To estimate the minimum sample size needed to estimate the population mean with the stated accuracy, we need to use the formula for margin of error:
Margin of Error = (Z * Standard Deviation) / sqrt(n)
Where:
- Margin of Error is $10,000
- Z is the Z-score corresponding to the desired level of confidence (typically a 95% confidence level corresponds to a Z-score of approximately 1.96)
- Standard Deviation is $70,000
- n is the sample size
Rearranging the formula, we can solve for the sample size (n):
n = (Z * Standard Deviation / Margin of Error)^2
Plugging in the values:
n = (1.96 * $70,000 / $10,000)^2
Calculating the expression inside the parenthesis:
(1.96 * $70,000 / $10,000) = 13.72
Squaring this result:
13.72^2 = 188.7424
Rounding up to the nearest whole number:
n = 189
Therefore, the minimum sample size needed to estimate the population mean with a margin of error of $10,000 is 189.
To estimate the minimum sample size needed to estimate the population mean with a desired margin of error, we can use the formula for sample size calculation:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score (corresponding to the desired confidence level)
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 a population standard deviation of $70,000.
To determine the Z-score, we need to determine the desired confidence level. Let's assume we want a 95% confidence level, which corresponds to a Z-score of 1.96 (for a large sample size).
Plugging the values into the formula, we get:
n = (1.96 * 70,000 / 10,000)^2
n = (137,200 / 10,000)^2
n = 13.72^2
n ≈ 188.14
Since the sample size must be a whole number, we round up the value to the nearest whole number.
n = 189
Therefore, the minimum sample size needed to estimate the population mean with a margin of error of $10,000 is 189.