Reference no: EM13975824

CLT Project

In this project you will be analyzing your assigned distribution to help understand Monte Carlo simulation techniques and the Central Limit Theorem (CLT). By the end of your analysis you will determine the minimum given sample size in which the CLT is applicable for your distribution and write a short report, in which a hardcopy will be handed in.

Include in this report

• a short background about your given distribution

• what is the CLT

• all the graphs and answers to the questions described below, making sure the information are referenced somewhere (doesn't have to be in the order below) in the body of the report.

Expect that I don't have these guidelines so you need to explain what you did as well as tell me the results and analysis. Hence, you should not have sections referring the the following problems below, instead your report should be like a technical paper. If you use outside sources (such as may be neces- sary for the background of your distribuiton), remember to cite in the paper as well as at the end.

You will also need to submit your R code on collab as a script (named "LastName1_LastName2.R"). Keep your code neat and commented where applicable, otherwise points will be deducted. There is no need to include R code in your report.

Further report specifications: The main section of the report is further subject to the following limita- tions:

• Paper size: Letter (8.5 inches by 11 inches)

• Font: 12pt of either Times New Roman, Arial, or Calibri

• Line spacing: Either 1.5 lines or Double

• Total number of pages: 3 or less

• Stapled and either Double or Single sided

1. Plot the density function of your assigned distribution as well as the standard normal distribution on one graph (impose them on top of one another), in which the x-axis has a range from -10 to 10 [increment the data points by 0.1]. Also make sure to include appropriate labels and title as well as include a legend and have each line be distinct, using a combination of color and/or line type. You may need to increase the y-axis range to capture the fit of your distribution, if so try from 0 to 3. [Note - avoid using color unless you are going to use a color printer.] Compare your given distribution to the standard normal, be descriptive.

2. Now start a Monte Carlo simulations with K = 10000 iterations for your given distribution. You will be analyzing this distribution with numerous sample sizes n as you are trying to determine what sample size (if any) is large enough for the central limit theorem to be applicable. Calculate and save the mean and standard deviation of each iteration in corresponding vectors so that it can be used later. You can use a loop to cycle through the different sample sizes for your given distribution to save on tedious copying and pasting but you should not use a loop to randomly sample or calculate the mean and standard deviation of the distirbution. Hint - it might be eas- iest to save everything in lists (where each component refers to a different sample size) or 3D matrices.

n < - c(5, 20, 30, 50, 70, 100, 150, 200, 500)

3. We would expect that a normal QQ-Plot of the sampling distribution of X¯ to look like a straight line with the y-intercept being the population mean of x¯ and a slope being the corresponding population standard deviation of x¯. Graph the QQ-Plot for each X¯ (of each sample size) generated in Part 3. At the end of your analysis save the corresponding QQ-Plot for the minimum sample size in which the CLT is applicable for your report. Add corresponding titles and lines to this plot. Note - You should not be using the function qqline to graph the line and you may need to do some research to determine population mean and variance.

Also keep in mind that your plot window in RStudio may be too small and may look like the sam- ple size is large enough when it is actually not. So make so you zoom in on the graph before making your decision.

4. The CLT also has several properties that should also be satisfied. Test the following three properties before making a decision on what sample size is considered large enough for your distribution. Report these properties in a table format for all tested sample sizes.

μx' = μ, σ_x² = σ²/n, E(s²) = σ²

5. Now that you have determined what sample size is large enough for the CLT to apply to your distribution, you are going to draw some histograms to see the CLT outcome visually. For easy comparison save the following two histograms in the same window different graphs or as trans- parent histograms on top one another. The first histogram should be created from 5000 random data points drawn from your orginial distribution. The second histograms will correspond to the mean vector you created in part (2) of the sample size you determined was large enough for the CLT to apply. Assuming a large enough sample size you should notice that the second graph will look approximately normal, otherwise the CLT does not apply. Label these graphs correctly and use them in your report to support you conclusion of the CLT theorem for you determined sample size.

6. If the CLT approximations holds, we expect about 95% of the confidence intervals X¯ ± tα/2, n-1 s/√n to contain the true mean value µ.

Determine the percentage of simulated confidence intervals that contain the true mean value for the minimum sample size in which the CLT is applicable. Again at the end of your analysis save the corresponding percentage for the minimum sample size in which the CLT is applicable. Report this and what it means.

7. We can visualize the confidence intervals by plotting them vertically and making a horizontal line that represents the true value µ. Due this for the first 100 confidence intervals constructed for the minimum sample size in which the CLT is applicable. Make it so each confidence interval line is a solid line (colors can vary or be the same). Correctly label and title this graph.

Attachment:- Project-2.rar