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Section 7.3
Second Day
Central Limit
Theorem
Quick Review
PCFS
The problem is that most
populations AREN’T normally
distributed.
Proportions
p = the proportion of
_____ who …
SRS
np≥10 and n(1-p)≥10
Population ≥ 10n
x  mean
z
std . dev
Means
Parameter
Conditions
Random
Normality
Independence
Formula
Sentence
µ = the mean …
SRS
Population ~ Normally
Population ≥ 10n
z
x  mean
std . dev
Houston,…
We have a problem…
Most
population distributions
are not Normal
Ex. - Household Incomes
So when a population
distribution is not Normal, what
is the shape of the sampling
distribution of x-bar?
Describe the shape of the sampling
distributions as n increases. What do
you notice?
Sample Means
Consider the strange population distribution from the Rice University
sampling distribution applet.
The Central Limit Theorem
 If
the population is normal, the sampling
distribution of x-bar is also normal. THIS IS
TRUE NO MATTER HOW SMALL n IS.
 As long as n is big enough, and there is a
finite population standard deviation, the
sampling distribution of x-bar will be
normal even if the population is not
distributed normally.
How big is “big enough?” We will use n > 30
 This is called the Central Limit Theorem.
and
say, “since
Memorize
it. n > 30, the CLT says the
sampling distribution of x-bar is Normal.”
PCFS
Proportions
p = the proportion of
_____ who …
Means
Parameter
Conditions
SRS
µ = the mean …
Random
SRS
np≥10 and n(1-p)≥10
Normality
Population ≥ 10n
Independence
Population ~ Normally
OR “since n> 30, the
CLT says the sampling
distribution of x-bar is
Normal.”
Population ≥ 10n
x  mean
z
std . dev
Formula
z
Sentence
x  mean
std . dev
Example: Servicing Air
Conditioners
Based on service records from the past
year, the time (in hours) that a
technician requires to complete
preventative maintenance on an air
conditioner follows the distribution
that is strongly right-skewed, and
whose most likely outcomes are close
to 0. The mean time is µ = 1 hour and
the standard deviation is σ = 1
Your company will service an SRS of 70 air
conditioners. You have budgeted 1.1 hours per unit.
Will this be enough?