Stat 101 – Lecture 31
Sampling Distribution Models
Population Parameter?
Population
Inference
Sample 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
sampling distribution of p̂ .
2
Sampling Distribution of p̂
• Shape: Approximately Normal
if conditions are met.
• Center: The mean is p.
• Spread: The standard deviation
is
p (1 − p )
n
3
Stat 101 – Lecture 31
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, μ .
5
Means
• We will use the sample, y, to
make inferences about the
population mean, μ .
• To do this we needed the
sampling distribution of y .
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Stat 101 – Lecture 31
Sampling Distributions
Quantitative/Numerical variable
Population Parameter: μ
Population
Sample
Distribution of
Sample Mean
y
7
Example
• Population? Stat 101 students in
Sections P&Q.
• Variable? Number of children
in your family.
• Type of variable? Numerical or
Quantitative.
8
Example
• Population
–All Stat 101 students in Sections
P&Q.
• Population Parameter
–The mean number of children in
a family of a Stat 101, Sections
P&Q, student.
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Stat 101 – Lecture 31
Example
• Sample
–5 randomly selected students.
• Sample Statistic
–The sample mean number of
children in the 5 students’
families.
10
Random Samples
• First Sample
–Sample mean number of children.
• Second Sample
–Sample mean number of children.
• Third Sample
–Sample mean number of children.
11
What have we learned?
• Different samples produce
different sample means.
• There is variation among sample
means.
• Can we model this variation?
– What is a model for the distribution
of the sample mean?
12
Stat 101 – Lecture 31
Simulation
http://onlinestatbook.com
/stat_sim/sampling_dist/
index.html
13
Simulation
• Simple random sample of size
n=5.
• Repeat many times.
• Record the sample mean, y , to
simulate the sampling
distribution of y .
14
Simulation
• Different samples will produce
different sample means.
• There is variation in the sample
means.
• Can we model this variation?
15
Stat 101 – Lecture 31
16
Population
• Shape: Basically normal
• Center: Mean
–
μ = 16
• Spread: Standard Deviation
–
σ =5
17
Sampling Distribution of y
• n=5
• Shape: Normal
• Center: Mean
μ = 16
• Spread: Standard Deviation
5
σ
SD( y ) =
n
=
5
= 2.24
18