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Hadley Wickham
Stat310Sampling distributions
Tuesday, 23 March 2010
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1. About the test
2. Sampling distribution of the mean
3. Sampling distribution of the standard deviation
Tuesday, 23 March 2010
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Test
Next Tuesday.
Covers bivariate random variables and inference up to Thursday.
Same format as last time: 4 questions, 80 minutes. 2 sides of notes. Half applied and half theoretical.
Hopefully a little easier than last time.
Tuesday, 23 March 2010
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Test tips
Work through the learning objectives online, looking them up in your notes if you’re not sure.
Work through the practice problems.
Go back over previous quizzes and homeworks and make sure you know how to answer each question.
Tuesday, 23 March 2010
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Sampling distribution of the mean
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MeansX1, X2, ... are iid N(μ, σ2)
Then
Sn =n�
1
Xi X̄n =Sn
n
X̄n ∼ N(µ,σ2
n)
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MeansX1, X2, ... are iid N(μ, σ2)
Then
Sn =n�
1
Xi X̄n =Sn
n
X̄n ∼ N(µ,σ2
n)
Tuesday, 23 March 2010
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MeansX1, X2, ... are iid E(X) = μ, Var(X) = σ2
Then
Sn =n�
1
Xi X̄n =Sn
n
X̄n ∼̇ N(µ,σ2
n)
Tuesday, 23 March 2010
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MeansX1, X2, ... are iid E(X) = μ, Var(X) = σ2
Then
Sn =n�
1
Xi X̄n =Sn
n
X̄n ∼̇ N(µ,σ2
n)
Tuesday, 23 March 2010
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Means
Zn =X̄n − µ
σ2/√
n
Zn ∼̇ N(0, 1)
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Your turn
Back to the Lakers. Let Oi ~ Poisson(λ = 103.9) - their offensive score for a single game.
What is the distribution of their average score for the entire season? (There are 82 games in a season)
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Continuity correctionWhen using the normal distribution to approximate a discrete distribution we need to make a small correction
P(X = 1) = P(0.5 < Z < 1.5)
P(X < 1) = P(Z < 0.5)
P(X ≤ 1) = P(Z < 1.5)
P(X > 1) = P(Z > 1.5)
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Your turn
What’s the probability the average score for the Lakers is less than 100?
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Steps
Write as probability statement.
Transform each side to get to known distribution.
Apply continuity correction, if necessary.
Compute.
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Multiplication
X ~ Poisson(λ)
Y = tX
Then Y ~ Poisson(λt)
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Exactly
How could you use the Poisson distribution to calculate the exact probability that the average score is < 100?
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Sampling distribution of the standard deviation
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(n− 1)S2
σ2∼ χ2(n− 1)
If Xi ~ iid N(0, 1), S2 =�
(Xi − X̄)2
n− 1Tuesday, 23 March 2010
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χ2Five fun facts about
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(n− 1)S2
σ2∼ χ2(n− 1)
Proof
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Sampling distribution of mean if variance unknown
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When we have to estimate the sd, what do you think happens to the distribution of our estimate of the mean? (Would it get more or less accurate?)
What about as n gets bigger?
Your turn
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x
dens
0.1
0.2
0.3
−3 −2 −1 0 1 2 3
df1215Inf
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X̄n − µ
σ/√
n∼ Z
X̄n − µ
s/√
n∼ tn−1
t-distribution
Xi ∼ Normal(µ, σ2)
Parameter called degrees of freedom
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Properties of the t-dist
Heavier tails compared to the normal distribution.
Practically, if n > 30, the t distribution is practically equivalent to the normal.
limn→∞
tn = Z
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t-tablesBasically the same as the standard normal. But one table for each value of degrees of freedom.
Easiest to use calculator or computer: http://www.stat.tamu.edu/~west/applets/tdemo.html
(For homework, use this applet, for exams, I’ll give you a small table if necessary)
Tuesday, 23 March 2010