Sampling Distribution Models
Population
Parameter
Population – all items
of interest.
Random
selection
Sample – a
few items from
the population.
Inference
Sample
Statistic
1
Proportions
 So
far we have used the
sample proportion, p̂ , to
make inferences about the
population proportion p.
 To do this we needed the
distribution of p̂ .
2
Distribution of p̂
 Shape:
Approximately Normal if
conditions are met.
The mean is p.
 Spread: The standard deviation
is
 Center:
SD p̂  
p1  p 
n
3
Categorical Variable
 When
the response variable of
interest is categorical, the
parameter is the proportion of
the population that falls in a
particular category, p.
4
Quantitative Variable
 When
the response variable of
interest is quantitative, the
parameter is the mean of the
population,  .
5
Means
 We
will use the sample mean,
y , to make inferences about
the population mean,  .
 To do this we need the
distribution of y .
6
Simulation
www.ruf.rice.edu/~lane/stat_sim/
sampling_dist/index.html
7
Simulation
 Simple
random sample of size
n=5.
 Repeat many times.
 Record the sample mean, y ,
to simulate the distribution of
y.
8
Simulation
 Different
samples will produce
different sample means.
 There is variation in the sample
means.
 Can we model this variation?
9
10
Population
 Shape:
Basically normal
 Center: Mean,   16
 Spread: Standard Deviation,
 5
11
Distribution of y
n
=5
 Shape: Normal
 Center: Mean,   16
 Spread: Standard Deviation,

5
SD y  

 2.24
n
5
12
13
Population
 Shape:
Not normal, skewed
right
 Center: Mean,   8.08
 Spread: Standard Deviation,
  6.22
14
Distribution of y
n
=5
 Shape: Approximately normal
 Center: Mean,   8.08
 Spread: Standard Deviation,

6.22
SD y  

 2.78
n
5
15
16
Population
 Shape:
Not normal, skewed
right
 Center: Mean,   8.08
 Spread: Standard Deviation,
  6.22
17
Distribution of y
n
= 25
 Shape: Approximately normal
 Center: Mean,   8.08
 Spread: Standard Deviation,

6.22
SD y  

 1.24
n
25
18
Central Limit Theorem
 When
selecting random samples
from a population with a distribution
that is not normal, the sampling
distribution of y will be
approximately normally distributed.
 The larger the sample the better
the approximation.
19
Conditions
 Random
sampling condition
– Samples must be selected at random
from the population.
 10%
condition
– When sampling without replacement,
the sample size should be less than
10% of the population size.
20
Summary
 Distribution
of y
–Shape: Approximately normal
–Center: 

–Spread: SD y  
n
21