Sampling Distribution of the Mean

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Chapter 13

• Sampling Distribution of the Mean

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Sampling Distribution of the Mean

• Sampling Distribution of the Mean FAQ
• What is the Sampling Distribution of the Mean?
• Frequency polygon of all possible samples that
  can be taken from a population.
• The possible values that can be selected when we
  go a sampling!
• What is m??
• The mean of the sampling distribution.
• The central tendency of all of the possible
  samples that can be collected.
• Typical ?

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Sampling Distribution of the Mean

• Sampling Distribution of the Mean FAQ
• What is s??
• The standard error of the mean.
• A measure of the variability amongst all possible
  samples that can be collected.
• Standard deviation of the sampling distribution.
• Why call it an error?
• This comes from the chapter on estimation.
• When we use ? to estimate m, s? represents about
  how far off our estimate is likely to be (its
  error)

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Sampling Distribution of the Mean

• Sampling Distribution of the Mean FAQ
• How do we know what the values of m? s? are?

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Sample Sampling Distribution

• Make up a population Take all possible samples
  Find out!

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Sample Sampling Distribution
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Shape Rectangular m 3.5 s 1.12
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Sample Sampling Distribution
3
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Lets Go Sampling n2
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? 4.5
Sample
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Sample Sampling Distribution
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Lets Go Sampling n2
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? 3.5
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Sample Sampling Distribution
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Lets Go Sampling n2
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Sample Sampling Distribution
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Lets Go Sampling n2
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? 2.5
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Sample Sampling Distribution
3
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Lets Go Sampling n2
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? 3
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Sample Sampling Distribution
3
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Lets Go Sampling n2
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? 2
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Sample Sampling Distribution
3
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Lets Go Sampling n2
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Sample Sampling Distribution
3
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Lets Go Sampling n2
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? 3
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Sample Sampling Distribution
3
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Lets Go Sampling n2
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? 3.5
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Sample Sampling Distribution
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Lets Go Sampling n2
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Sample Sampling Distribution
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Lets Go Sampling n2
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? 2
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Sample Sampling Distribution
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Lets Go Sampling n2
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? 3.5
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Sample Sampling Distribution
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Lets Go Sampling n2
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Sample Sampling Distribution
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Lets Go Sampling n2
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Sample Sampling Distribution
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Sample Sampling Distribution
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Sample Sampling Distribution
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The Sampling Distribution of the (Sample)
Mean(s) n2
Sample
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Sample Sampling Distribution
Shape Normal m? 3.5 s? .79
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The Central Limit Theorem

• Describing the properties of the sampling
  distribution of the mean.

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Central Limit Theorem
From the Sample Sampling Distribution
Population Parameters
Sampling Distribution of the Mean Parameters
Shape Rectangular m 3.5 s 1.12
Shape Normal m? 3.5 s? .79
s / ?n 1.12 / ?2 1.12 / 1.41 .79
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Central Limit Theorem

• The Central Limit Theorem
• The sampling distribution of the mean has
• Normal Shape (with sufficient sample size)
• m? m
• s? s / ?n
• Regardless of the properties of the Population,
  the sampling distribution has the following
  characteristics

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Central Limit Theorem

• Why does the Central Limit Theorem work?
• Why does the sampling distribution of the mean
  have a Normal Shape (with sufficient sample size)
  m? m
• Most common samples will have low, medium high
  values
• Rare samples will have all low, or all high,
  values
• The relationship between n
• As n goes up, s? goes down (distribution narrows)
• s? s / ?n
• Why?
• As take larger samples, the sample to sample
  overlap increases
• Variability down.

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Central Limit Theorem

• Why does the Central Limit Theorem work?
• The relationship between s? n
• As n goes up, s? goes down (distribution narrows)
• s? s / ?n

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Central Limit Theorem
Standard Error, Sample Size, Hypothesis Tests
Small Sample Size Large Standard Error
Large Sample Size Small Standard Error
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Central Limit Theorem

• Why does the Central Limit Theorem work?
• The relationship between s? n
• As n goes up, s? goes down (distribution narrows)
• s? s / ?n
• Why?
• As take larger samples, the sample to sample
  overlap increases
• Variability amongst samples down.