How large a sample should be taken if the population mean is to be estimated with 95% confidence to within $75? The population has a standard deviation of $898. (Round you answer up to the nearest whole number
To calculate the sample size needed to estimate the population mean with a specified level of confidence and margin of error, we can use the formula:
n = (Z * σ / E)²
Where:
n = sample size
Z = z-score corresponding to the desired confidence level (in this case, 95% confidence corresponds to a z-score of 1.96)
σ = population standard deviation
E = margin of error
Substituting the given values into the formula:
n = (1.96 * 898 / 75)²
Calculating this expression:
n = (1.96 * 898 / 75)²
≈ 23.387
Since we cannot have a fraction of a sample, we need to round up the value to the nearest whole number (since we're dealing with people):
n ≈ 24
Therefore, a sample size of 24 should be taken to estimate the population mean with 95% confidence to within $75.