A random sample will be drawn from a normal distribution, for the purpose of estimating the population mean . Since is the median as well as the mean, it seems that both the sample median m and the mean X are reasonable estimators.Question a. Generate a large number (at least 1000) samples of size 5 from a N(0, 1) distribution.

Question b. Compute the sample medians m^*₁,...,m^*₁₀₀₀from the 1000 samples.

Question c. Compute the mean of the median: m^*and the standard deviation of the median: S_m^*of m^*₁,...m^*₁₀₀₀.

Question d. Compute the sample mean X^*₁,....,X^*₁₀₀₀for the 1000 samples.

Question e. Compute the mean and standard deviation of the sample means as S_X^*of X^*₁,..., X^*₁₀₀₀.

Question f. The true value of is 0. Estimate the bias and uncertainty (_m) in m. (Note: In fact, the median is unbiased, so your bias estimate should be close to 0.)

Question g. Estimate the bias and uncertainty (_x) in X, the sample mean.

(I have programmed to solve these questions. But I am not sure whether my answer to question f is correct.

My answer is for question f is:

"For a normal distribution, the population median m is equal to the populationmean . So the real value of the median m satisfies that m = = 0 for a sample that follows the normal distribution of N(0, 1). The estimated mean of every median for the1000 five-element samples is _m = -0.0196692. Therefore, the bias in m is ( _m - m) -0.01967 - 0 = -0.01967, which is close to zero. The estimated uncertainty in is 0.53127.

Regarding the estimated uncertainty, I am confused whether should I say, because for a sample following normal distribution, the sample standard deviation s and the population standard deviation satisfy thats² =²/n, =s (n),_m=s_m (n)= 0.5312677(5 ) 1.190. "