1) Determine manually the lower and upper limits on sample standard deviations, using your own population's (μ, σ) for (1- α) = 0.95 and sample sizes of n = 5, 20. 80. and 320.

2) Use your own random (μ, σ) to generate four sets of (normal-distribution based) random numbers (i.e., samples) of sizes n = 5, 20, 80, and 320, respectively.

3) Calculate the means and standard deviations for the 4 samples you generated in Point-2 above. Ensure that these are within the limits found in Point-1, otherwise, repeat Point-2 and replace the data set(s).

The Normal Distribution - Estimation Theory

4) Now, assume that (μ, σ) are unknown. Estimate the four pairs of different 95% confidence intervals, for the population's statistics (Ica). using the four samples' data in Point-3 above, respectively.

5) Plot your results found in Point-4 above as (p versus n) and (a versus n), (i.e., two limit points for every n value), and include reference lines at the true population statistics values. Explain, in two or less sentences, your observations. (Note that your own (μ, σ) selected in Point-1 above are the true values for the population statistics).

The Normal Distribution - Statistical Process Control

6) Use all the 80 points of the n = 80 data set in Point-2 above to create 20 sub-set samples of size n = 4 each.

7) Determine the ±3σ control limits, both for sample-mean and -standard-deviation (i.e., X-Bar and S) charts, using the sample data in Point-6 above.

8) To simulate a potentially out-of-control process, change your own original μ and σ (Point-1 above) as (μ* =K₁xμ, σ* =K₂xσ), i.e., calculate a new set of population statistics, where K₁ and K₂ are two separate randomly generated values: K₁ is between 2.1-2.3, and K₂ is between 1.5-2.0, respectively.

9) Generate a new set of 20 samples all of size n = 4, using the new population statistics (Point 8). μ* and σ*.

10) Calculate and plot the mean and standard-deviation data for the 20 new samples (Point-9 above) on the X-Bar and S charts using the control limits calculated for the population with the original (μ, σ) values (Point-7 above). Explain, in two or less sentences, your observations.