18 Sampling Mean Sd

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


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.


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.


Sampling distribution of the mean


MeansX1, X2, ... are iid N(μ, σ2)

Then

Sn =n�

1

Xi X̄n =Sn

n

X̄n ∼ N(µ,σ2

n)


MeansX1, X2, ... are iid E(X) = μ, Var(X) = σ2

Then

Sn =n�

1

Xi X̄n =Sn

n

X̄n ∼̇ N(µ,σ2

n)


Means

Zn =X̄n − µ

σ2/√

n

Zn ∼̇ N(0, 1)


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)


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)


Your turn

What’s the probability the average score for the Lakers is less than 100?


Steps

Write as probability statement.

Transform each side to get to known distribution.

Apply continuity correction, if necessary.

Compute.


Multiplication

X ~ Poisson(λ)

Y = tX

Then Y ~ Poisson(λt)


Exactly

How could you use the Poisson distribution to calculate the exact probability that the average score is < 100?


Sampling distribution of the standard deviation


(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


(n− 1)S2

σ2∼ χ2(n− 1)

Proof


Sampling distribution of mean if variance unknown


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


x

dens

0.1

0.2

0.3

−3 −2 −1 0 1 2 3

df1215Inf


X̄n − µ

σ/√

n∼ Z

X̄n − µ

s/√

n∼ tn−1

t-distribution

Xi ∼ Normal(µ, σ2)

Parameter called degrees of freedom


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


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)


http://www.stat.tamu.edu/~west/applets/tdemo.html




18 Sampling Mean Sd

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