Matlab programmer with statistics background -- 3

1. Use a random number generator and a binning function to generate bell shaped curves, and show how the fluctuation of the standard deviation for each of a series of measurements of different sample populations decreases with an increase in the number of samples. That is, if you take random numbers, ten at a time, and compute their standard deviations, do the values of standard deviation for each collection of 10 random numbers vary very much? What happens if you repeat this with several groups that each contain 100 random numbers? If you want to decide how many tests that you must make to have a given level of confidence in your answers, how do you determine how many samples that you need? If you want to decrease the uncertainty of a survey, for example, by a factor of two, do you need twice as much data or four times as much data? Does your data measuring the “fluctuations of the standard deviations” from one test to the next support sqrt(N) convergence? Answer this question in essay form. 2. Solve for a value of  using both a Monte Carlo approach and an analytic, non-stochastic approach, such as the Fourier series modelling of a square wave. Demonstrate that one approach converges as 1/N, the other as 1/(sqrt N). Explain your observations and analysis. Budget $15 Required in 2 days