Math 324 Project 3 The Central Limit Theorem
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School
San Francisco State University *
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Course
324
Subject
Mathematics
Date
Jan 9, 2024
Type
docx
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3
Uploaded by MinisterProtonBoar36
Kareem Amin
Dylan Faulder
Math 324
Intro:
Math 324 Project 3 The Central Limit Theorem
Google Sheet:
https://docs.google.com/spreadsheets/d/1sAD1gBkT3Mt7tSBS2_mvOiFbCsmW2pzvD06syYGd
4No/edit?usp=sharing
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.
?
?
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)?
Answer - The Shape of the distribution seems to not follow something specific and is mostly
uniform with outlier data at 90 being higher. The mean is 50.412 and the standard Deviation for
our data is 28.66.
μ = Lowerbound + Upper bound/2, The range from 1 to 100. μ= (1+100)/2 = 50.5
Using this calculation we can find the answer for our Theoretical Value
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.)
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?
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
√
❑
Yes, the data is consistent between the two
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 show that the values we gathered tend to follow the
Central Limit Theorem (CLT) and create a normal distribution curve as we added the data to the
graphs. The initial graph did not fully resemble our expectations; it was quite uniform, with a
higher data point outlier on one end. However, as we added more data, it changed and eventually
conformed to our expectations, with most of the data concentrated in the center, forming the
Central Limit Theorem line.
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