Lecture 7 How certain are we Sampling and the normal distribution

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Lecture 7How certain are we?Sampling and the
normal distribution
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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, well learn that probably means
• in 95 of all samples

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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

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Repeated sampling
Population All US households
All possible samples
mY1.75
N4
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 Y Y
Y Y Y Y
Y
N4

Each Y represents the number of children in a
household
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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 mGreek m pGreek p sGreek s The
population is the source of the sample. Greek
culture was the source of Roman culture.
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Population
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Sample
Within sample
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Population of samples

of samples infinite
but just N4 adults per sample
Across samples
here 1.75
here 1.62 / 41/2 1.62 / 2 0.81
(Std. dev. of sample means)
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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
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Normal distribution

• symmetric
• bell-shaped
• very specific numeric properties

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Margin of error
In your course binder, find the z (standard
normal)table. Look for this line.
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
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Example 1
Again population US adults Variable Y How
many children have you ever had? mY1.75,
sY1.62. Consider samples of size N16. 95 of
all sample means are within 1.96 standard
errors of pop. mean i.e., in
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More on sampling error
Look for this line.
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.)
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Example 2
Variable Y How many children have you ever
had? mY1.75, sY1.62. Consider samples of size
N45. 99 of all sample means are within 2.58
standard errors of the population mean i.e., in
99
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Sampling error Exercise
Complete the following 90 of all samples have
means within _______ SEs 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.
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Summary Central Limit Theorem (CLT)

• The sampling distribution of
• has mean
• and standard error
• 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

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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

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Teaser Lecture 8 (Confidence intervals)

• So if we take
• just one sample
• we can guess
• that the population statistic is close
• and well usually be right