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5'2 The Sampling Distribution of the Sample Average

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Bert and Ernie like to go mini golfing on their days off. ... It is assumed that both Bert and Ernie's scores follow a normal distribution. ... – PowerPoint PPT presentation

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Title: 5'2 The Sampling Distribution of the Sample Average


1
5.2The Sampling Distribution of the Sample
Average
2
  • 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

3
The Sample Average
  • The sample average is a statistic.
  • It has a mean (Expected Value).
  • It has a spread (Variance).
  • It has a distribution.

4
Expected Value
5
Variance
6
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.

7
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

8
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)

9
Important Note
  • The variability of an average is less than the
    variability of an individual.

10
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.

11
Example 1 Answer
  • Centered at ß
  • Variance s 2 / 100
  • Approximately Normal

12
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.

13
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?

14
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

15
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) )

16
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

17
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).
  •  
  • What is the probability that Aaron beats Tim?

18
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
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