Math 324 Project 3 The Central Limit Theorem (2)
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San Francisco State University *
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Course
324
Subject
Mathematics
Date
Jan 9, 2024
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docx
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4
Uploaded by MinisterProtonBoar36
Kareem Amin
Dylan Faulder
Math 324
Math 324 Project 3 The Central Limit Theorem
Google Sheet: https://docs.google.com/spreadsheets/d/1sAD1gBkT3Mt7tSBS2_mvOiFbCsmW2pzvD06syYGd
4No/edit?usp=sharing
Note: List of answers can also be found on row 255 as well as below.
Objective:
Demonstrate the Central Limit Theorem (CLT) through simulation and random sampling.
1) Describe the Central Limit Theorem in your own words.
Answer - The Central Limit Theorem says that if you add up, or average lots and lots of different
values, the total or average of those values plotted out will often look like a bell-shaped curve
2) Use =RANDBETWEEN(1, 100) to create 250 samples of size
=
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).
See Google Sheet attached
3) Use the formulae provided to compute
and
of the theoretical distribution.
?
?
Columns: AH & AI (on google sheet)
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)?
Note: The bin 100-105 on the right most side is storing numbers exactly equal to 100.
Answer - The Shape of the distribution see is semi-uniform with outlier at bin 90-95 being significantly higher. The mean is 50.412 and the standard Deviation for our data is 28.66.
Using this calculation we can find the answer for our Theoretical Value which is 50.5
μ = (Lowerbound + Upper bound)/2, The range from 1 to 100. μ= (1+100)/2 = 50.5
Yes, our theoretical value and sample mean are very similar. (50.5 and 50.412)
5) Now calculate the mean for each of the 250 samples. (There should be 250 sample means.)
Column: AH (on google sheet)
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?
Answer - When looking at the shape of the histogram I the central limit theorem is visible, the shape is slowly becoming more what we should be seeing as the data with 5 bins shows more of the data being in the center, having a normal distribution.
8) Use Excel or Google Sheet function to find the mean of the 250 sample means. Compare
this result to the expected value of
. Is your sample statistic value consistent with the
?
theoretical expectation?
Yes, the mean of the 250 sample means is almost equal to the theoretical expectation that we found in question 4.
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
. Are the two results consistent?
?
Answer -
σ
√
❑
The calculated standard deviation of the 250 sample means and the theoretical value of standard error are slightly apart, however this could be due to some of the variability in our data, especially seen in question 4 where our bucket from 90-95 appeared more significantly than other numbers. If we add more samples this discrepancy could become smaller.
10) Write a short paragraph summarizing your results and how they support the Central Limit Theorem.
Answer - Our results from this project illustrated the Central Limit Theorem (CLT) by generating
250 samples of size 333 with random integer values between 1 and 100. Our results supported the CLT, as the distribution of the 7500 generated values gradually conformed to a bell-shaped curve, indicative of a normal distribution. Despite initial variability, the mean and standard
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deviation are closely aligned with theoretical values. Additionally, the histogram of the 250 sample means further emphasized the CLT, revealing a distribution resembling a normal curve. Overall, our findings reinforce the reliability of the Central Limit Theorem in predicting the behavior of sample means, and can be more accurate if more data is included in the analysis.