Use the R command x1mat= matrix(rweibull(50000, shape=1, scale=1), nrow=500) to simulate random Weibull distributed values with = 1 and = 1. The matrix x1mat has 500 rows, each of which is a sample of size n = 100 from the Weibull distribution. Therefore, x1mat contains 500 replications of Weibull distributed data. Then calculate mean for the histograms in the questions below with x1matmn = apply(x1mat[,-1], 1, mean).

1. On the histogram, overlay the density (PDF as a line graph) of 500 randomly drawn values from the normal distribution with estimated sample mean and sample variance from x1matmn. Give the estimated mean and standard deviation. (Note these are the model-free estimates). Is the normal distribution a good estimate for the distribution of the means? Explain.

2. On the histogram, overlay the density (PDF as a line graph) of 500 randomly drawn values from the normal distribution with the mean and variance of the mean from the Weibull distribution (Hint: Use the mean and variance for the exponential distribution since = 1). Give the mean and standard deviation of the mean. (Note these are the model-based estimates). Is the normal distribution a good estimate for the distribution of the means? Explain.

3. Interpret the histogram with the overlayed densities. Add a legend to the plot so that the models can be identified. Give the plot. Which distribution provides the best estimate to the distribution of the means? Explain.