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Chapter 5:
Sampling Distributions
(Part 1)
Dr. Nahid Sultana
Chapter 5:
Sampling Distributions
5.1 The Sampling Distribution of a
Sample Mean
5.2 Sampling Distributions for Counts and
Proportions
5.1 The Sampling Distribution of a
Sample Mean
 Population Distribution vs. Sampling Distribution
 The Mean and Standard Deviation of the Sample
Mean
 Sampling Distribution of a Sample Mean
 Central Limit Theorem
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Sample Mean
Displays the distribution of customer
service call lengths for a bank service
center for a month.
Take a sample of 80 calls from this
Population and calculate the mean
these 80 calls.
There are more than 30,000 calls in
this population. (omitted a few outliers)
If we take more samples of size 80, we
will get different values of x .
The population mean is μ = 173.95 sec.
This is the sampling distribution of the
values of x for 500 samples of size 80.
x of
 The sampling distribution is roughly symmetric rather than skewed.
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 The
sample means are much less spread out than the individual call lengths.
Sample Mean
Normal quantile plot of the 500 sample means
The distribution is close to Normal.
Sample Mean
This example illustrates two important facts about sample means:
FACTS ABOUT SAMPLE MEANS
1. Sample means are less variable than individual observations.
2. Sample means are more Normal than individual observations.
Mean and Standard Deviation of a
Sample Mean
Example:
Take an SRS of size 36 from a population with mean 240 and standard
deviation 18. Find the mean and standard deviation of the sampling
distribution of your sample mean.
Now repeat the previous calculations for a sample size of 144. Explain the
effect of the increase on the sample mean and standard deviation.
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The Central Limit Theorem
The shape of the distribution of x depends on the shape
of the population distribution. Here is one important case :
If individual observations have the N(µ,σ) distribution, then the sample mean
of an SRS of size n has the N(µ, σ/√n) distribution regardless of the sample
size n.
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The Central Limit Theorem
One of the most famous facts of probability theory: When the sample is
large enough, the distribution of sample means is very close to Normal, no
matter what shape the population distribution has, as long as the population
has a finite standard deviation.
Draw an SRS of size n from any population with mean μ and finite
standard deviation σ. The central limit theorem (CLT) says that when n
is large, the sampling distribution of the sample mean x is approximat ely
Normal :


x is approximat ely N  μ,
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σ 

n
The Central Limit Theorem
Example:
Take an SRS of size 144 from a population with mean 240 and standard
deviation 18.
a) According to the central limit theorem, what is the approximate sampling
distribution of the sample mean?
b) Use the 95 part of the 68–95–99.7 rule to describe the variability of this
sample mean.
c)
suppose we increase the sample size to 1296. Use the 95 part of the
68–95–99.7 rule to describe the variability of this sample mean. Compare
your results with those you found before.
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Example
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?
μ =μ = 1
x
The sampling distribution of the mean time spent working is approximately N(1, 0.12) :
If you budget 1.1 hours per unit, there is a
20% chance the technicians will not complete
the work within the budgeted time.
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A Few More Facts
Any linear combination of independent Normal random variables is also
Normally distributed.
That is, if X and Y are independent Normal random variables, and a and b are
any fixed numbers, then aX + bY is also Normally distributed.
In particular, the sum or difference of independent Normal random variables
has a Normal distribution.
The mean and standard deviation of aX + bY are found as usual from the
addition rules for means and variances.
Example:
Tom and George are playing in the club golf tournament. Their scores vary as
they play the course repeatedly. Tom’s score X has the N(110, 10) distribution,
and George’s score Y varies from round to round according to the N(100, 8)
distribution.
If they play independently, what is the probability that Tom will score lower
than George .?
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