5'2 The Sampling Distribution of the Sample Average

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5.2The Sampling Distribution of the Sample
Average
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• Suppose a random variable X followed an unknown
  distribution with mean µ and standard deviation s
• A simple random sample taken from the population
  would look very similar to the population

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The Sample Average

• The sample average is a statistic.
• It has a mean (Expected Value).
• It has a spread (Variance).
• It has a distribution.

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Expected Value
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Variance
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Distribution

• The distribution of the sample average is
  centered at µ and has a standard deviation of
• s/vn.
• This is true for ANY sample mean from a
  population with a finite variance.

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The Central Limit Theorem (CLT)

• Draw a SRS of size n from a population with mean
  µ and finite standard deviation s. When n is
  large the sampling distribution of the sample
  average is approximately normal with mean µ and
  standard deviation s /vn, N(µ, s/vn)
• CLT Applet

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The Central Limit Theorem (CLT) part 2

• Draw a SRS of size n from a Normal population
  with mean µ and finite standard deviation s.
  When n is large the sampling distribution of
  the sample average is exactly normal with mean µ
  and standard deviation s /vn, N(µ, s/vn)

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Important Note

• The variability of an average is less than the
  variability of an individual.

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Example 1

• Suppose Y comes from an exponential distribution
  with mean ß and variance s 2. Drawing a SRS of
  size 100 from this population calculate the mean
  and standard deviation for the sample average
  y-bar.

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Example 1 Answer

• Centered at ß
• Variance s 2 / 100
• Approximately Normal

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Example 2

• Suppose 500 3rd graders were asked to pick their
  favorite number between 1 100. This is called
  a uniform distribution. Let the random variable
  X denote the number that the students selected.

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Example 2

• What is the expected value of X?
• What does the distribution of X look like?
• What is the expected value of the sample average?
  
• What does the distribution of the sample mean
  look like?

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Example 2 Answer

• What is the expected value of X?
• E(X) 50.5
• What does the distribution of X look like?
• Uniform Distribution A 100 sided die.
• What is the expected value of the sample average?
  
• E(X-bar) 50.5
• What does the distribution of the sample mean
  look like?
• Approximately Normal

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Example 3

• Aaron and Tim snuck out of the office early to
  play a round of golf. Aarons score follows a
  Normal distribution with a mean of 63 strokes and
  a standard deviation of 3 strokes. Tims score
  also follows a Normal distribution with a mean is
  64 strokes and his standard deviation is 4
  strokes.
• What is the probability that Aaron beats Tim?
  (Hint Calculate P(A T lt 0) )

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Example 3 Answer

• Let D A T where D N(63 -64, sqrt(32 42))
• D N(-1, 5)
• Calculate P(D lt 0), so z (0 -1)/5) .2
• P(Z lt .2) .5793

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Example 4

• Now suppose that Aaron and Tim are competing in a
  season long competition. They play 25 rounds of
  golf over the summer. The person with the lowest
  average score wins. Again assume that AN(63, 3)
  and TN(64, 4).
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• What is the probability that Aaron beats Tim?

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Example 4 Answer

• Let D A T
• D N(63 -64, sqrt(32/25 42/25))
• D N(-1, 1)
• Calculate P(D lt 0), so z (0 -1)/1) 11
• P(Z lt 1) .8413