Report Writing

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Report Writing

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A report should be self-explanatory. It
should be capable of being read and understood
without reference to the original project
description. Thus, for each question, it should
contain all of the following
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• a statement of the problem
• (b) a full and careful description of how it is
  investigated
• (c) All relevant results, including graphical and
  numerical analyses variables should be carefully
  defined, and figures and tables should be
  properly labelled, described and referenced
• (d) relevant analysis, discussion, and
  conclusions.

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It should be written in the third person. NOT I
think the Central Limit Theorem is true for this
example because I see that the graph is
normal. INSTEAD It can be clearly seen that the
graph displays a normal distribution confirming
that the Central Limit Theorem holds.
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The Central Limit

• Theorem

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Let X1, X2. Xn be independent identically
distributed random variables with mean µ and
variance s 2. Let S X1, X2 . Xn Then
elementary probability theory tells us that E(S)
nµ and var(S) ns 2 . The Central Limit
Theorem (CLT) further states that, provided n is
not too small, S has an approximately normal
distribution with the above mean nµ, and variance
ns 2.
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In other words, S approx N(nµ, ns 2) The
approximation improves as n increases. We will
use R to demonstrate the CLT.
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Let X1,X2X6 come from the Uniform distribution,
U(0,1)
1
0
1
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For any uniform distribution on A,B, µ is equal
to and variance, s2, is equal to
So for our distribution, µ 1/2 and s2 1/12
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The Central Limit Theorem therefore states that S
should have an approximately normal distribution
with mean nµ (i.e. 6 x 0.5 3) and var ns2
(i.e. 6 x 1/12 0.5) This gives standard
deviation 0.7071 In other words, S approx
N(3, 0.70712)
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Generate 10 000 results in each of six vectors
for the uniform distribution on 0,1 in R.
gt x1runif(10000) gt x2runif(10000) gt
x3runif(10000) gt x4runif(10000) gt
x5runif(10000) gt x6runif(10000) gt
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Let S X1, X2 . X6
gt sx1x2x3x4x5x6 gt hist(s,nclass20) gt
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Consider the mean and standard deviation of S
gt mean(s) 1 3.002503 gt sd(s) 1 0.7070773 gt
This agrees with our earlier calculations
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A method of examining whether the distribution is
approximately normal is by producing a normal Q-Q
plot. This is a plot of the sorted values of
the vector S (the data) against what is in
effect a idealised sample of the same size from
the N(0,1) distribution.
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If the CLT holds good, i.e. if S is approximately
normal, then the plot should show an approximate
straight line with intercept equal to the mean of
S (here 3) and slope equal to the standard
deviation of S (here 0.707).
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gt qqnorm(s) gt
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gt qqnorm(s) gt
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gt qqnorm(s) gt
4.4 1.8 4 0.7 to 1 DP
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From these plots it seems that agreement with the
normal distribution is very good, despite the
fact that we have only taken n 6, i.e. the
convergence is very rapid!