Transcript Chapter 11 Sampling and Sampling Distributions

Lecture 7
How certain are we?
Sampling and the normal distribution
Chapter 11 – 1
Lecture 6: Summary
• If we take a simple random sample
– from a well-defined population
• we expect
– that the sample mean
– is “probably” “close” to the population mean
• By “close” we mean “within ~2 standard errors”
Lecture 7: Preview
• Today, we’ll learn that “probably” means
– in 95% of all samples
Chapter 11 – 2
Overview
• Review of sampling distributions
• Sampling distributions have a “normal” shape
• Properties of the “normal” distribution, e.g.:
– In 95% of all samples,
• the sample mean
• is within 1.96 standard errors
• of the population mean
Chapter 11 – 3
Repeated sampling
Population: All US households
mY=1.75
Y Y Y Y Y YY
Y
Y Y Y Y Y Y Y
Y Y
Y Y Y Y Y Y Y Y Y Y Y
Y Y Y Y Y Y Y Y Y Y Y
Y Y Y Y Y Y Y Y Y Y
Y Y Y Y Y Y Y Y
Y
Y
All possible samples
Y Y
Y Y
Y Y
Y Y
N=4
Y  1.25
N=4
Y  2.50
…
Each Y represents the number of children in a household
Chapter 11 – 4
Notation
Mnemonics:
Population measures are called Parameters.
Sample measures are called Statistics.
The P words and S words go together.
Population parameters use Greek letters
Sample statistics use Roman letters
m=Greek m
p=Greek p
s=Greek s
The population is the source of the sample.
Greek culture was the source of Roman culture.
Chapter 11 – 5
Population
Population
Variable
population mean
population standard deviation
US households
Y (# of children)
mY=1.75
sY=1.62
Chapter 11 – 6
Sample
CHILDS
1
0
2
2
sample mean
1.25
Within sample…
Sample size
Variable
sample mean
sample standard deviation
N=4
Y (# of children)
Y  1.25
sY=.92
Chapter 11 – 7
Population of samples
CHILDS
1
0
2
2
sample mean
1.25
Y
CHILDS
2
4
0
4
sample mean
2.50
Y
Mean
Standard error
# of samples: infinite
but just N=4 adults per sample
Across samples…
“Variable”
…
Y
mY  mY
here 1.75
sY  sY / N
here 1.62 / 41/2 = 1.62 / 2 = 0.81
(Std. dev. of sample means)
Chapter 11 – 8
As sample size (N) grows…
…standard error shrinks!
…shape of sampling distribution gets closer to “normal”!
1.75
.81
1.75
.405
1.75
.2025
Chapter 11 – 9
Normal distribution
• symmetric
• bell-shaped
• very specific numeric properties
Chapter 11 – 10
“Margin of error”
In your course binder,
find the “z (standard normal)…table”.
Look for this line.
Confidence z
94%
1.88
95%
1.96
96%
2.05
This means:
In 95% of all samples,
the sample mean is
within 1.96 standard errors
of the population mean.
+/- 1.96 (or 2) standard errors often called “margin of error”Chapter 11 – 11
Example 1
Again “population”: US adults
Variable: Y: “How many children have you ever had?”
mY=1.75, sY=1.62.
Consider samples of size N=16.
95% of all sample means Y are within 1.96 standard errors of pop. mean
—i.e., in
95%
mY  Zs Y
 mY  1.96s Y
 mY  1.96s Y / N
 1.75  1.96(1.62 / 16)
 1.75  1.96(1.62 / 4)
 1.75  .79
 0.96 to 2.54
Chapter 11 – 12
More on sampling error
Look for this line.
Confidence z
98%
2.33
99%
2.58
99.9%
3.29
This means: In 99% of all samples,
the sample mean
is within 2.58 standard errors
of the population mean.
(1% of samples have means that are further away.)
Chapter 11 – 13
Example 2
Variable: Y: “How many children have you ever had?”
mY=1.75, sY=1.62.
Consider samples of size N=45.
99% of all sample means are within 2.58 standard errors
of the population mean
—i.e., in
99%
mY  2.58s Y
 mY  2.58s Y / N
 1.75  2.58(1.62 / 45)
 1.75  .62
 1.13 to 2.37
Chapter 11 – 14
Sampling error: Exercise
Complete the following:
90% of all samples have means within _______
SE’s of the population mean.
Complete the following:
If researchers take samples of 100 US adults,
90% of the time the sample will average between
_______ and _________ children.
Chapter 11 – 15
Summary:
Central Limit Theorem (CLT)
• The sampling distribution of Y
– has mean mY  mY
– and standard error s Y
 sY / N
• As the sample size N gets larger,
– the standard error gets smaller
– and the sampling distribution gets closer to “normal.”
• So
– larger samples give
• closer
• more predictable
– approximations to the population mean
Chapter 11 – 16
Summary
• Lecture 6 (Law of Large Samples)
– If we take simple random samples
• from a well-defined population
– we expect
• that the sample means
• is “usually” “close” to the population mean
• Lecture 7 (Central Limit Theorem)
– If by “close”
• we mean “within 1.96 standard errors”
– then by “usually”
• we mean “in 95% of all samples”
– For other definitions of “close” and “usually,”
• see the “z (standard normal)…table” in your course binder
Chapter 11 – 17
Teaser:
Lecture 8 (Confidence intervals)
• So if we take
– just one sample
• we can guess
– that the population statistic is “close”
• and we’ll “usually” be right
Chapter 11 – 18