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9.1 Sampling Distribution
◦ Know the difference between a statistic and a
parameter
◦ Understand that the value of a statistic varies
between samples
◦ Be able to describe the shape, center and spread
of a given sampling distribution
◦ Understand how bias and variability of a statistic
affects the sampling distribution


Parameter a number that describes the population
(usually it’s unknown)
Statistic a number computed from the sample data

Is the boldfaced number a parameter or a
statistic?
1. 60,000 members of the labor force were
interviewed of whom 7.2% were unemployed
statistic
2. A lot of ball bearings has a mean diameter
of 2.5003 cm. A 100 bearings are selected
from the lot and have a mean diameter of
2.5009 cm.
2.5003- parameter
2.5009- statistic

3. A telemarketing firm in Los Angeles
randomly dials telephone numbers. Of the
first 100 numbers dialed 48% are unlisted.
This is not surprising because 52% of all Los
Angeles residential phones are unlisted.
48%- statistic
52%- parameter
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
Sample proportion:
p̂ (“p hat”)
Example:
A poll found that 1650 out of 2500 randomly
selected adults agreed with the statement
that shopping is frustrating. What is the
proportion of the sample who agreed?
p̂=1650/2500


Sampling variability –
the value of a statistic varies with repeated
sampling
Applet:
http://www.rossmanchance.com/applets/Ree
ses/ReesesPieces.html
the distribution of values taken by the statistic
in all possible samples of the same size from
the same population.

1. the overall shape is symmetric (normal)

2. there are no outliers or other important
deviations from the overall pattern

3. the center of the distribution is the true
value p

4. the values of p̂ have a large spread

a statistic is unbiased if the mean of the
sampling distribution is equal to the true
value of the parameter being estimated



1. Is described by the spread of its sampling
distribution.
2. This spread is determined by the sampling
design and the size of the sample.
3. Larger samples give smaller spread.
High bias;
low variability
Low bias;
high variability
High bias;
high variability
Low bias;
low variability
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

60% of people find clothes shopping
frustrating.
Find the proportion of people that fall within
2 standard deviations of the mean for
samples of size
n =100
(0.502,0.698)

n = 2500
(0.5804 and 0.6196)

Why does the size of the population have
little influence on how statistics from a
random sample behave?
The larger the sample size, the smaller the
standard deviation.
Learning Objectives:
◦ Know the characteristics of the sampling
distribution of p̂
◦ Know when to use the normal approximation for p̂
◦ Be able to solve problems using the normal
approximation for p̂
p̂


Choose an SRS of size n from a large
population, then:
1. the sampling distribution of p̂is
approximately normal. (closer to a normal
dist. when n is large)

2. the mean of the sampling dist. is exactly p

3.

Assumption 1
p̂ can only be
The standard deviation p for __
used when the population is at least 10 times
as large as the sample
Assumption 2
We can say that the sampling distribution of
is approximately normal when np>10 and
n(1-p)>10.
*(some books use np>5)


There are 1.7 million first-year college
students of those, 1500 first-year college
students are asked whether they applied for
admission to any other college. In fact 35%
of all first-year students applied to a college
other than the one they are attending. What is
the probability that your sample will give a
result within 2 percentage points of this true
value?
Assumptions:
-random sample
-population is at least 10X the sample
-(1500)(0.35)> 10
525>10
-(1500)(0.65)>10
975>10

Assumptions:
-SRS
-population is at least 10X the sample
-(1540)(0.15)> 10
231>10
-(1540)(0.85)>10
1309>10
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