Use the margin of error E = $10, confidence level of 99%, and ó = $40 to find the minimum sample size needed to estimate an unknown population mean, .ì
Formula:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value will be 2.58 for 99% interval, sd = 40, E = 10, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.
To find the minimum sample size needed to estimate an unknown population mean, we can use the formula:
n = (Z * ó / E)^2
Where:
n = sample size
Z = z-score corresponding to the desired confidence level
ó = population standard deviation
E = margin of error
In this case, the margin of error (E) is given as $10, the confidence level is 99%, and the population standard deviation (ó) is $40.
Step 1: Find the z-score corresponding to the desired confidence level.
Since the confidence level is 99%, we need to find the z-score that corresponds to the area under the normal distribution curve of 0.99. By looking up the z-score in the standard normal distribution table or using a calculator, we find that the z-score is approximately 2.576.
Step 2: Substitute the values into the formula and solve for n.
n = (Z * ó / E)^2
n = (2.576 * $40 / $10)^2
n = (2.576 * 4)^2
n = 10.304^2
n ≈ 106
Therefore, the minimum sample size needed to estimate an unknown population mean with a margin of error of $10, a confidence level of 99%, and a population standard deviation of $40 is approximately 106.