A researcher wishes to estimate the mean amount of money spent per month on food by households in a certain neighborhood. She desires a margin of error of $30. Past studies suggest that a population standard deviation of $248 is reasonable. Estimate the minimum sample size needed to estimate the population mean with the stated accuracy.
A. 274
B. 284
C. 264
D. 272
My answer is C
To estimate the minimum sample size needed to estimate the population mean with a desired margin of error, we can use the formula:
n = (Z * σ / E)²
Where:
n = sample size
Z = Z-value for a desired level of confidence (usually found in a Z-table or calculated based on the desired confidence level)
σ = population standard deviation
E = margin of error
In this case, the researcher desires a margin of error of $30 and the past studies suggest a population standard deviation of $248. However, we need to determine the Z-value for the desired level of confidence.
Let's assume a 95% level of confidence, which corresponds to a Z-value of approximately 1.96. Now we can plug the values into the formula:
n = (1.96 * 248 / 30)²
Calculating this expression:
n ≈ (484.48 / 30)²
n ≈ 16.15²
n ≈ 261.12
Since we're looking for the minimum sample size, we round up to the next whole number, which gives us a minimum sample size of 262.
Thus, the correct answer is not listed among the options provided.