Download Central Limit Theorem Application Project

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.
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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:
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