A random sample will be drawn from a normal distribution, for the purpose of estimating the population mean μ. Since μ is the median as well as the mean, it seems that both the sample median m and the sample mean X̅ are reasonable estimators. This exercise is designed to determine which of these estimators has the smaller uncertainty.

a. Compute the mean m̅^∗ and the standard deviations m̅^∗ of m^∗₁, . . ., m^∗₁₀₀₀.

b. Compute the mean and standard deviation s_X̅^∗ of X̅^∗₁, . . ., X̅^∗₁₀₀₀.

c. The true value of μ is 0. Estimate the bias and uncertainty (σ_m) in m. In fact, the median is unbiased, so your bias estimate should be close to 0.

d. Estimate the bias and uncertainty (σ_X̅) in X̅. Is your bias estimate close to 0? Explain why it should be. Is your uncertainty estimate close to 1/√5? Explain why it should be.