For the population of farm workers in New Zealand, suppose that weekly income has a distribution that is skewed to the right with a mean of μ = +500 (N.Z.) and a standard deviation of σ = +160. A researcher, unaware of these values, plans to randomly sample 100 farm workers and use the sample mean annual income x to estimate μ.
a. Show that the standard deviation of x equals 16.0.
b. Explain why it is almost certain that the sample mean will fall within $48 of $500.
c. The sampling distribution of x provides the probability that x falls within a certain distance of μ, regardless of the value of μ. Show how to calculate the probability that x falls within $20 of μ for all such workers. (Using the standard deviation, convert the distance 20 to a z -score for the sampling distribution.)