Sampling Distribution Models
Population
Parameter
Population – all items
of interest.
Random
selection
Sample – a
few items from
the population.
Inference
Sample
Statistic
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Example





Who? Students who took Stat 101
and filled out a questionnaire.
What? Number of siblings.
When? Today.
Where? In class.
Why? To find out what proportion
of students’ have no siblings.
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Example

Population


Stat 101 students who filled out
questionnaire
Population Parameter

Proportion of all Stat 101 students
who filled out questionnaire who
have no siblings.
3
Example

Sample


100 randomly selected students.
Sample Statistic

The proportion of the 100
students who have exactly one
sibling, p̂ .
4
Demonstration

Sample 1

Sample 2

Sample 3
p̂ 
p̂ 
p̂ 
5
What have we learned?
Different samples produce
different sample proportions.
 There is variation among
sample proportions.
 Can we model this variation?

6
Simulation

Population

Reeses Pieces
www.rossmanchance.com/applets/Reeses/ReesesPieces.html
statweb.calpoly.edu/chance/applets/Reeses/ReesesPieces.html

Population Parameter

Proportion of Orange Reese’s Pieces
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8
Simulation
Simple random sample of size
n=25.
 Repeat several times.
 Record the sample proportion of
orange Reese’s Pieces.

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Sampling Distribution of p̂

Shape: Approximately Normal
Center: The mean is p.
 Spread: The standard deviation
is
p1  p 

n
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Reese’s Pieces

Sampling distribution of p̂
Shape: Approximately Normal.
 Center: The mean is 0.45
 Spread: The standard deviation is

p1  p 

n
0.450.55
 0.0995
25
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Conditions
The sampled values must be
independent of each other.
 The sample size, n, must be
large enough.

13
Conditions

10% Condition
When sampling without replacement,
the sample size should be less than
10% of the population size.
 Reese’s Pieces – the number of pieces
in the machine is much greater than
250.

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Conditions

Success/Failure Condition


The sample size must be large
enough so that np and n(1- p) are
both bigger than 10.
Reeses Pieces – np = 11.25 and
n(1- p) = 13.75 which are both
greater than 10.
15
Comment


To be able to use these results you
need to know what the value of the
population parameter, p, is.
This is no problem in the Reese’s
Pieces simulation because we can
choose the proportion of Orange
pieces.
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17
Inference
For most populations we don’t
know p, the population
proportion.
 We can use the sampling
distribution of p̂ to help us make
inferences about the reasonable
or plausible value of p.

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