A shipping company received an order of 1645 packages. The manager needs to determin the mean weight of the packages. (A) How many randomly selected packages would be need to weight to estimate with a 95% confidence, the mean weight within +/-1.5 pounds if prior studies indicate that the standard dev of all weights is around 8 pounds?
To estimate the mean weight of the packages with a 95% confidence interval of +/-1.5 pounds, you need to determine the sample size (number of randomly selected packages).
The formula to determine the required sample size is:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-score for desired confidence level (95% confidence level corresponds to a Z-score of approximately 1.96)
σ = standard deviation of all weights
E = desired margin of error (+/-1.5 pounds in this case)
In this case, the standard deviation (σ) is given as around 8 pounds.
Plugging the values into the formula:
n = (1.96 * 8 / 1.5)^2
n ≈ 10.389^2
n ≈ 107.82
So, you would need to randomly select approximately 108 packages to estimate the mean weight with a 95% confidence interval of +/-1.5 pounds, based on prior studies indicating a standard deviation of 8 pounds.