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IHE 603 Central Limit Theorem Application Project Spring 2011 This project demonstrates applications of the Central Limit Theorem and illustrates examples of Type I and Type II Errors. The project may be completed by individual students or by teams of two or three students. You may use any computer application (including computer programs that you create yourself) to generate the data tables and analyses results; recommended programs are JMP, MS Office Excel, and MatLab. Essentially, your experiments will be repeated two times by generating different populations for a Normal Distribution and any one of the other listed distributions. All populations should contain 2000 values. Since these populations consist of randomly generated numbers, all projects should use the same parametric values for the mean and variance so that we can compare results among the different outcomes. Distribution Mean Normal Exponential Uniform Triangular Bimodal μ = 130 μ = 130 μ = 130 Standard Deviation σ = 10 σ = 130 σ = 23 Comments Note: λ = 1/μ ≈ 0.007692 Minimum a = 90 Maximum b = 170 Minimum a = 90 Maximum b = 160 Mode c = 150 See Bimodal.xls The following steps apply to each of the two populations regardless of the underlying distribution: 1. Using the theoretical values listed in the table above, generate a random population (2000 data points). 2. Determine the mean μ and standard deviation σ for your randomly generated population. Notes: Since this data set is considered to be a population and not a sample, the correct symbology should be μ and σ. These computed values may not necessarily be equal to the theoretical parametric values; however they should be reasonably close. 3. Calculate the 95% tolerance interval for the population (sample size n= 40). 4. Create 200 different random samples (sample size n= 40) from the population. This may take some added ingenuity (if you are not successful in generating these samples, consult with your fellow students, and/or as at last resort, with your instructor). a. Calculate the mean X and standard deviation s for each of the 200 samples. b. Count the number of sample means that fall within the 95% population tolerance interval. c. Calculate an estimated population mean from each of your samples (use a 95% confidence interval). d. Count the number of times the population mean μ falls within the 95% confidence interval of the estimated population mean. e. Calculate the mean, the variance, and the standard deviation of the sample means. f. Plot the 200 samples means. g. Compare the Probability of Type I Error (a = 5%) to the number of sample means outside of the 95% tolerance interval for the population. h. Calculate the Probability of Type II Error for each true mean = μ + ½σ, μ + 1σ, μ + 1½σ. i. Compare the Probability of Type II Error to the number of sample means outside the intervals for μ + ½σ, μ + 1σ, μ + 1½σ. Submit a typed report with your results and comments along with computer files of your data sets including a file copy of your Results Table (see page 2); due date Friday, June 3, 2011. Page 1 of 2 IHE 603 Central Limit Theorem Application Project Submitted by: Spring 2011 Date: I (we), your name(s), certify that my(our) work on this project, project name, is an original effort on my(our) part except as noted in the references. I(we) hereby grant to instructor's name and Wright State University, Dayton Ohio, unlimited use and distribution of the contents of this report. Step 1 Store the population 2000 values data set in (an Excel Workbook, or JMP or MatLab file) with one column for each of your two distributions in a file named LastName(s)_IHE60Spring2011_Project_Population_Datasets.extension Step 4 & Step 4a & Step 4c Strongly recommend using Excel (a separate tab for each of the two different distributions) to store the sample data sets (sample size n = 40) with 200 columns (A thru GR) of 40 values each (rows 2-41), along with the sample mean (row 42), and sample standard deviation (row 43), lower bound 95% confidence interval for the estimated population mean (row 44), upper bound 95% confidence interval for the estimated population mean (row 45) in a file named: LastName(s)_IHE60Spring2011_Project_Sample_Datasets.extension Step 4f Plot Sample Means Results Table: Note: Your printed report should contain two separate Results Tables, one for each type of distribution; along with an Excel file containing the results data in two separate tabs in a file named: LastName(s)_IHE60Spring2011_Project_Result_Tables.extension Normal, Uniform, Exponential, etc. Distribution Seed Values Mean μ = Standard Deviation σ = Generated Population Actual Values Step 2 Mean μ = Standard Deviation σ = Step 3 Step 4b Lower boundary 95% tolerance interval = Upper boundary 95% tolerance interval = Number of sample means outside 95% tolerance interval = Step 4d Number of times population mean within 95% sample confidence intervals = Step 4e Mean of Sample Means = Variance of Sample Means = Standard Deviation of Sample Means = Step 4g Probability of Type I Errors = Number of sample means outside of 95% population tolerance interval = Step 4h & 4i Probability Type II Error (True Mean = μ + ½σ) = Number of sample means outside the interval for (True Mean = μ + ½σ) = Probability Type II Error (True Mean = μ + 1σ) = Number of sample means outside the interval for (True Mean = μ + 1σ) = Probability Type II Error (True Mean = μ + 1½σ) = Number of sample means outside the interval for (True Mean = μ + 1½σ) = Observations and Comments: Page 2 of 2