The Central Limit Theorem

The Central Limit Theorem
Name
AP Statistics
Instructor’s Name
Date

Last Name, 2
PART 1
Introduction
As one of the most important results in statistics, Central Limit Theorem gives a way of
inferencing large samples about the population mean, when the distribution of the population is
unknown, or doesn’t need to be known.1 More so, it provides inference about a population
proportion, example while conducting surveys. Further, it approximates the distribution of the
mean for any distribution, as well as approximating other distributions.2 In this study, we are
concerned about the approximation of the mean of various distributions.
The Central Limit Theorem states that, as n → ∞ for a sequence X1 , X2 , ... , Xn of
identically, independently distributed random variables with finite mean and variance μ and
variance σ 2respectively, then, ́X−μ
σ / √n distribution approaches the standard normal distribution.
́X = 1
n ∑
i=1
n
Xi.3 The three properties of a sampling distribution are that it is normally distributed,
with mean μ, and standard deviation σ / √n . In Part 1, the validity of the theorem is going to be
assessed using the normal distribution, employing Fathom as a tool for the analysis.
Procedure
The normal distribution portrays a symmetrical (bell-shaped) density curve and is of
utmost importance while practicing, or in theoretical statistics. The distribution can be used
while modeling the distribution of student grades in a class; approximating other distributions,
1Walpole et al., Essentials of Probability and Statistics, 79.
2 Soong, Fundamentals of probability and statistics, 154.
3 Ibid., 154

Last Name, 3
example the limiting form of binomial ( n , θ ), Poisson ( μ ), among others; providing models for
sampling distributions of various statistics; while in statistical inference, most assumptions
regarding the underlying distribution are regarded as normal distribution; and acts as a building
block for other distributions, example the student’s t.4 The following graph shows the
distribution of a N (100, 15).
The normal distribution follows a probability density function defined as f ( x ) = 1
σ √ 2 π e
−1
2 ( x− μ
σ )
2
.
4 Chatfield, Statistics for technology, 89.
5
10
15
20
25
30
60 80 100 120 140 160
x
y = x    normalDensity
no data Function Plot
Figure 1: N (100, 15)