Objective: Demonstrate the Central Limit Theorem (CLT) through simulation and random sampling. Background: If a random variable X follows a discrete uniform distribution between a and b, its mean and standard deviation can be calculated as follows: a+b and o = (b-a+1)2-1 12 In Excel or Google Sheets, use function =RANDBETWEEN() to generate data that follow a discrete uniform distribution. Project Instructions: This is an experiment to illustrate the Central Limit Theorem. 1) Describe the Central Limit Theorem in your own words. 2) Use =RANDBETWEEN(1, 100) to create 250 samples of size n = 30 each by generating random integer (whole) numbers between 1 and 100. Tip: To keep the data from changing, select the random numbers you need, and press Ctrl + C to copy them, then go to select a cell you want to paste the random numbers, then right click to click Paste Special > Values(V). 3) Use the formulae provided to compute µ and o of the theoretical distribution. 4) Make a histogram of 7500 integer values you generated. Use bins 10 units wide. Describe the shape of this distribution and calculate the sample mean and standard deviation from the data you generated. Are these values similar to the theoretical values you found in (3)? 5) Now calculate the mean for each of the 250 samples. (There should be 250 sample means.) 6) Make a histogram of these 250 sample means using bins 5 units wide. 7) Discuss the histogram shape. Does the Central Limit Theorem seem to be working? 8) Use Excel or Google Sheet function to find the mean of the 250 sample means. Compare %3D
Objective: Demonstrate the Central Limit Theorem (CLT) through simulation and random sampling. Background: If a random variable X follows a discrete uniform distribution between a and b, its mean and standard deviation can be calculated as follows: a+b and o = (b-a+1)2-1 12 In Excel or Google Sheets, use function =RANDBETWEEN() to generate data that follow a discrete uniform distribution. Project Instructions: This is an experiment to illustrate the Central Limit Theorem. 1) Describe the Central Limit Theorem in your own words. 2) Use =RANDBETWEEN(1, 100) to create 250 samples of size n = 30 each by generating random integer (whole) numbers between 1 and 100. Tip: To keep the data from changing, select the random numbers you need, and press Ctrl + C to copy them, then go to select a cell you want to paste the random numbers, then right click to click Paste Special > Values(V). 3) Use the formulae provided to compute µ and o of the theoretical distribution. 4) Make a histogram of 7500 integer values you generated. Use bins 10 units wide. Describe the shape of this distribution and calculate the sample mean and standard deviation from the data you generated. Are these values similar to the theoretical values you found in (3)? 5) Now calculate the mean for each of the 250 samples. (There should be 250 sample means.) 6) Make a histogram of these 250 sample means using bins 5 units wide. 7) Discuss the histogram shape. Does the Central Limit Theorem seem to be working? 8) Use Excel or Google Sheet function to find the mean of the 250 sample means. Compare %3D
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:Objective: Demonstrate the Central Limit Theorem (CLT) through simulation and random
sampling.
Background: If a random variable X follows a discrete uniform distribution between a and b, its
mean and standard deviation can be calculated as follows:
a+b
(b-a+1)2-1
and o =
12
In Excel or Google Sheets, use function =RANDBETWEEN() to generate data that follow a
discrete uniform distribution.
Project Instructions:
This is an experiment to illustrate the Central Limit Theorem.
1) Describe the Central Limit Theorem in your own words.
2) Use=RANDBETWEEN(1, 100) to create 250 samples of size n = 30 each by generating
random integer (whole) numbers between 1 and 100. Tip: To keep the data from
changing, select the random numbers you need, and press Ctrl + C to copy them, then go
to select a cell you want to paste the random numbers, then right click to click Paste
Special > Values(V).
3) Use the formulae provided to compute µ and o of the theoretical distribution.
4) Make a histogram of 7500 integer values you generated. Use bins 10 units wide. Describe
the shape of this distribution and calculate the sample mean and standard deviation from
the data you generated. Are these values similar to the theoretical values you found in (3)?
5) Now calculate the mean for each of the 250 samples. (There should be 250 sample means.)
6) Make a histogram of these 250 sample means using bins 5 units wide.
7) Discuss the histogram shape. Does the Central Limit Theorem seem to be working?
8) Use Excel or Google Sheet function to find the mean of the 250 sample means. Compare
this result to the expected value of X. Is your sample statistic value consistent with the
theoretical expectation?
9) Use Excel or Google Sheet function to find the standard deviation of the 250 sample
means. Compare this result to the standard error of the mean, i.e. the standard deviation
of X. Are the two results consistent?
10) Write a short paragraph summarizing your results and how they support the Central Limit
Theorem.
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