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Hadley Wickham
Stat310Sampling distributions
Tuesday, 23 March 2010
1. About the test
2. Sampling distribution of the mean
3. Sampling distribution of the standard deviation
Tuesday, 23 March 2010
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
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
Sampling distribution of the mean
Tuesday, 23 March 2010
MeansX1, X2, ... are iid N(μ, σ2)
Then
Sn =n�
1
Xi X̄n =Sn
n
X̄n ∼ N(µ,σ2
n)
Tuesday, 23 March 2010
MeansX1, X2, ... are iid N(μ, σ2)
Then
Sn =n�
1
Xi X̄n =Sn
n
X̄n ∼ N(µ,σ2
n)
Tuesday, 23 March 2010
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
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
Means
Zn =X̄n − µ
σ2/√
n
Zn ∼̇ N(0, 1)
Tuesday, 23 March 2010
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)
Tuesday, 23 March 2010
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)
Tuesday, 23 March 2010
Your turn
What’s the probability the average score for the Lakers is less than 100?
Tuesday, 23 March 2010
Steps
Write as probability statement.
Transform each side to get to known distribution.
Apply continuity correction, if necessary.
Compute.
Tuesday, 23 March 2010
Multiplication
X ~ Poisson(λ)
Y = tX
Then Y ~ Poisson(λt)
Tuesday, 23 March 2010
Exactly
How could you use the Poisson distribution to calculate the exact probability that the average score is < 100?
Tuesday, 23 March 2010
Sampling distribution of the standard deviation
Tuesday, 23 March 2010
(n− 1)S2
σ2∼ χ2(n− 1)
If Xi ~ iid N(0, 1), S2 =�
(Xi − X̄)2
n− 1Tuesday, 23 March 2010
χ2Five fun facts about
Tuesday, 23 March 2010
(n− 1)S2
σ2∼ χ2(n− 1)
Proof
Tuesday, 23 March 2010
Sampling distribution of mean if variance unknown
Tuesday, 23 March 2010
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
Tuesday, 23 March 2010
x
dens
0.1
0.2
0.3
−3 −2 −1 0 1 2 3
df1215Inf
Tuesday, 23 March 2010
X̄n − µ
σ/√
n∼ Z
X̄n − µ
s/√
n∼ tn−1
t-distribution
Xi ∼ Normal(µ, σ2)
Parameter called degrees of freedom
Tuesday, 23 March 2010
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
Tuesday, 23 March 2010
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
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