Download 11:Sampling distributions

Chapter 11
Sampling Distributions
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Sampling Distributions
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Parameters and Statistics
• Parameter ≡ a constant that describes a
population or probability model, e.g., μ from a
Normal distribution
• Statistic ≡ a random variable calculated from a
sample e.g., “x-bar”
• These are related but are not the same!
• For example, the average age of the SJSU
student population µ = 23.5 (parameter), but the
average age in any sample x-bar (statistic) is
going to differ from µ
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Sampling Distributions
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Example: “Does This Wine
Smell Bad?”
• Dimethyl sulfide
(DMS) is present in
wine causing off-odors
• Let X represent the
threshold at which a
person can smell DMS
• X varies according to
a Normal distribution
with μ = 25 and σ = 7
(µg/L)
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Sampling Distributions
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Law of Large Numbers
This figure shows
results from an
experiment that
demonstrates the law
of large numbers (will
be discussed in class)
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Sampling Distributions
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Sampling Distributions of Statistics
 The
sampling distribution of a statistic predicts
the behavior of the statistic in the long run
 The next slide show a simulated sampling
distribution of mean from a population that has
X~N(25, 7). We take 1,000 samples, each of n
=10, from population, calculate x-bar in each
sample and plot.
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Sampling Distributions
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Simulation of a Sampling
Distribution of xbar
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Sampling Distributions
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μ and σ of x-bar
x-bar is an unbiased
estimator of μ
Square root law
x 
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Sampling Distributions

n
7
Sampling Distribution of Mean
Wine tasting example
Population
X~N(25,7)
Sample n = 10
By sq. root law,
σxbar = 7 / √10 = 2.21
By unbiased property,
center of distribution = μ
Thus
x-bar~N(25, 2.21)
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Illustration of Sampling Distribution:
Does this wine taste bad?
What proportion of samples based on n = 10
will have a mean less than 20?
(A)State: Pr(x-bar ≤ 20) = ?
Recall x-bar~N(25, 2.21) when n = 10
(B)Standardize: z = (20 – 25) / 2.21 = -2.26
(C)Sketch and shade
(D)Table A: Pr(Z < –2.26) = .0119
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Sampling Distributions
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Central Limit Theorem
No matter the shape of the population, the
distribution of x-bars tends toward Normality
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Sampling Distributions
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Central Limit Theorem
Time to Complete Activity Example
Let X ≡ time to perform an activity.
X has µ = 1 & σ = 1 but is NOT Normal:
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Sampling Distributions
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Central Limit Theorem
Time to Complete Activity Example
These figures
illustrate the
sampling
distributions
of x-bars
based on
(a) n = 1
(b) n = 10
(c) n = 20
(d) n = 70
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Sampling Distributions
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Central Limit Theorem
Time to Complete Activity Example
The variable X is NOT Normal, but the sampling distribution
of x-bar based on n = 70 is Normal with μx-bar = 1 and σx-bar =
1 / sqrt(70) = 0.12, i.e., xbar~N(1,0.12)
What proportion of x-bars will be less than 0.83 hours?
(A) State: Pr(xbar < 0.83)
(B) Standardize:
z = (0.83 – 1) / 0.12 = −1.42
(C) Sketch: right
(D) Pr(Z < −1.42) = 0.0778
xbar
z
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Sampling Distributions
-1.42
0
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