Handout 11A: Central limit theorem, estimating X bar

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7 Terms

1
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What is X bar (x̄)?

A symbol for the SAMPLE mean → average of set of data points in sample

  • Sampling without replacement: removing from population once drawn

<p>A symbol for the SAMPLE mean → average of set of data points in sample</p><ul><li><p>Sampling without replacement: removing from population once drawn</p></li></ul><p></p>
2
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What is Y?

Sum of the sample values → summary

  • just the sum, not the average! The average before divided by # of values

<p>Sum of the sample values → summary</p><ul><li><p>just the sum, not the average! The average before divided by # of values</p></li></ul><p> </p>
3
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What is the expected value, variance, and standard deviation of Y and x̄?

n = the sample size

Y = the sum

x̄ = the average

<p>n = the sample size</p><p>Y = the sum</p><p>x̄ = the average</p>
4
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What is the shape probability distribution function of x̄ and Y?

If the sample size (n) is large & the sample is random (the observations is independent)

→ It follows a normal model (bell-curve)

  • Summaries (x̄ & Y) will follow a normal model

<p>If the sample size (n) is large &amp; the sample is random (the observations is independent)</p><p>→ It follows a normal model (bell-curve)</p><ul><li><p>Summaries (x̄ &amp; Y) will follow a normal model</p></li></ul><p></p>
5
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What is the central limit theorem?

The middle values of a distribution is more common because there are multiple paths to get to the middle values!

  • only 1 path to get to maximum/minimum values

  • that’s why it’s shaped like a bell (huddled in the middle)

6
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What are some things to note about the CLT (central limit theorem)?

  • The spread in Y increases with increasing sample size (n)

  • The spread in x̄ decreases with increasing sample size (n) → the SD would grow smaller as the denominator (n) grows larger

→ as n increases, the shape looks more like a normal model

  • can turn any sample (if skewed) into a central limit distribution (normal model, less skewed)

<ul><li><p>The spread in Y increases with increasing sample size (n)</p></li></ul><ul><li><p>The spread in&nbsp;x̄ decreases with increasing sample size (n) → the SD would grow smaller as the denominator (n) grows larger</p></li></ul><p>→ as n increases, the shape looks more like a normal model</p><ul><li><p>can turn any sample (if skewed) into a central limit distribution (normal model, less skewed)</p></li></ul><p></p>
7
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What is a sample distribution & a sampling distribution?

  • Sampling distribution (of x̄ & Y)

<ul><li><p>Sampling distribution (of x̄ &amp; Y)</p></li></ul><p></p>