Sampling Distributions

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Chapter 10

• Sampling Distributions

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Parameters and Statistics

• Parameter
• Is a fixed number that describes the location or
  spread of a population
• Its value is NOT known in statistical practice
• Statistic
• A calculated number from data in the sample
• Its value IS known in statistical practice
• Some statistics are used to estimate parameters
• Sampling variability
• Different samples or experiments from the same
  population yield different values of the statistic

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Parameters and statistics

• The mean of a population is called µ ? this is a
  parameter
• The mean of a sample is called x-bar ? this is
  a statistic
• Illustration
• Average age of all SJSU students (µ) is 26.5
• A SRS of 10 San Jose State students yields a mean
  age (x-bar) of 22.3
• x-bar and µ are related but are not the same
  thing!

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The Law of Large Numbers
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Example 10.2 (p. 251)
Does This Wine Smell Bad?

• Dimethyl sulfide (DMS) is sometimes present in
  wine causing off-odors
• Different people have different thresholds for
  smelling DMS
• Winemakers want to know the odor threshold that
  the human nose can detect.
• Population values
• Mean threshold of all adults ? is 25 µg/L wine
• Standard deviation ? of all adults is 7 µg/L
• Distribution is Normal

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Simulation Example 10.2 (cont.)

• Suppose you take a simple random sample (SRS)
  from the population and the first individual in
  sample has a value of 28
• The second individual has a value of 40
• The average of the first two individuals
  (2840)/2 34
• Continue sampling individuals at random and
  calculating means
• Plot the means

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Simulation law of large numbers Fig 10.1
The sample mean gets close to population mean ?
as we take more and more samples
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Sampling distribution of xbar
Key questions What would happen if we took many
samples or did the experiment many times? How
would the statistics from these repeated samples
vary?
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Case Study
Does This Wine Smell Bad?

• Recall ? 25 µg / L, ? 7 µg / L and the
  distribution is Normal
• Suppose you take 1,000 repetitions of samples,
  each of n 10 from this population
• You calculate x-bar in each sample
• You plot the x-bars as a histogram
• You study the histogram (next slide)

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Simulation 1000 sample means Example 10.4
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Mean and Standard Deviation of x-bar
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Mean and Standard Deviation of Sample Means
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Case Study
Does This Wine Smell Bad?
(Population distribution)
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Illustration Exercise 10.8 (p. 258)

• Suppose blood cholesterol in a population of men
  is Normal with m 188 and s 41. You select 100
  men at random
• (a) What is mean and standard deviation of the
  100 x-bars from these samples?
• x-barx-bar 188 (same as µ)
• sx-bar 41 / sqrt(100) 4.1 (one-tenth of s)
• (b) What is probability a given x-bar is less
  than 180?
• Pr(x-bar lt 180) ? standardize ? z (180 188)
  / 4.1 -1.95
• Pr(Z lt 1.95) ? TABLE A ? .0256

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Central Limit Theorem
No matter what the shape of the population, the
sampling distribution of xbar, will follow a
Normal distribution when the sample size is large.
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Central Limit Theorem in Action Example 10.6 (p.
259)

• Data time to perform an activity
• NOT Normal (Fig a)
• µ 1 hour
• s 1 hour
• Fig (a) is for single observations
• Fig (b) is for x-bars based on n 2
• Fig (c) is for x-bars based on n 10
• Fig (d) is for x-bars based on n 25
• Notice how distribution becomes increasingly
  Normal as n increases!

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Example 10.7 (cont.)

• Sample n 70 activities
• What is the distribution of x-bars?
• x-barx-bar 1 (same as µ)
• sx-bar 1 / sqrt(70) 0.12 (by formula)
• Normal (central limit theorem)
• What of x-bars will be less than 0.83 hours?
• Pr(x-bar lt 0.83) ? standardize ? z (0.83 1)
  / 0.12 -1.42
• Pr(Z lt - 1.42) ? TABLE A ? 0.0778
• Notice that if Pr(x-bar lt 0.83 0.0778, then
  Pr(Z gt 0.83) 1 0.0778 0.9222

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Statistical process control
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