Week 8, Sessions one and two, Sampling Distribution Models (Chapter 18)
Professor Esfandiri
In this lecture we will go through the Pennies Lab on Fathom in lecture in order
to…
•
Show you the sampling distribution of the mean (also called the distribution of
sample means); i.e. distribution of
X,
•
Show how it differs from the distribution of a quantitative random variable such
as X in the sample and population, and show you how to calculate the standard
deviation and mean of the distribution of sample means,
•
Discuss the Central Limit Theorem (CLT)
•
Illustrate the effect of the sample size
on the shape of the distribution of sample
means.
•
Show you the distribution of sample proportions (p^) and the standard deviation
of the (p^) distribution.
Mean and standard deviation of a quantitative random variable in the
population
Example: Let us say that you have a population of 100,000 adults who are enrolled in a
health plan.
If we draw the histogram of the amount of money that the health plan spends on these
100,000 individuals annually, we will get the histogram for the population. We cannot
predict the shape of this histogram. It could be normal, skewed right, skewed left, too tall
(leptokurtic), too flat (playkurtic), etc. The mean and standard deviation for this
population are calculated as follows:
100,000
μ
=
∑
Xi
/ 100,000
(population mean)
i=1
100,000
σ
^2 =
∑
(Xi -
μ
)^2 / 99,000 (population standard deviation)
i=1
sample
Example: Suppose that now we pick a random sample of 100 individuals enrolled in this
health plan. If we draw the histogram of the amount of money that the health plan spends
on these 100 individuals annually, we will have the histogram of the sample. The
histogram could have any shape. The mean and standard deviation in this sample are
calculated as follows:
100
X =
∑
/ 100 (Sample mean)
i=1

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S^2
=
∑
(Xi -
X)^2 / 99 (Sample Standard deviation)
i=1
Mean and standard deviation for the distribution of sample means
Now suppose we go to the population described above and pick 100 random
samples of size 80 and compute the mean for each including
X1,
X2,…….,
X100. The
mean for sample one and the other 99 samples will be computed as follows:
80
X =
∑
Xi /80
i=1
100
μ
=
∑
X / 100
i=1
(the mean of the sampling distribution or the mean or the mean of the
sample means)
Variance of the distribution of the sample means or the variance of the
sampling distribution of the sample means is computed as follows:
100
σ
^2 (
X)
∑
(
X -
μ
)^2 / 99
i = 1
Standard deviation of the sampling distribution of the mean or the
standard deviation of the distribution of sample means
σ
(
X) =
σ
/
N
However, in the majority of cases we do not have
σ
or the standard deviation in the
population and thus we use the standard deviation of the sample, or S (X), as an estimate
of
σ
. Example of cases in which we have the standard deviation for the population are
SAT scores, GRE scores LSAT scores, MCAT scores, weight of newborns and similar
situations.

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