The document discusses the concepts of sampling distributions, including: 1. The sampling distribution of the mean, which follows a normal distribution with mean μ and standard deviation σ/√n when observations are independent and identically distributed. 2. The sampling distribution of the standard deviation, which follows a chi-squared distribution with n-1 degrees of freedom when the population variance is σ2. 3. When the population variance is unknown, the distribution of the sample mean is a t-distribution rather than normal, with heavier tails and degrees of freedom equal to n-1.

1. Stat310 Sampling distributions
Hadley Wickham
Tuesday, 23 March 2010

2. 1. About the test
2. Sampling distribution of the mean
3. Sampling distribution of the standard
deviation
Tuesday, 23 March 2010

3. 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

4. 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

5. Sampling distribution
of the mean
Tuesday, 23 March 2010

6. Means
X1, X2, ... are iid N(μ, σ2)
n
¯ Sn
Sn = Xi Xn =
n
1
Then
2
σ
¯ n ∼ N(µ, )
X
n
Tuesday, 23 March 2010

7. Means
X1, X2, ... are iid N(μ, σ2)
n
¯ Sn
Sn = Xi Xn =
n
1
Then
2
σ
¯ n ∼ N(µ, )
X
n
Tuesday, 23 March 2010

8. Means
X1, X2, ... are iid E(X) = μ, Var(X) = σ2
n
¯ Sn
Sn = Xi Xn =
n
1
Then
2
σ
¯ n ∼ N(µ, )
X ˙
n
Tuesday, 23 March 2010

9. Means
X1, X2, ... are iid E(X) = μ, Var(X) = σ2
n
¯ Sn
Sn = Xi Xn =
n
1
Then
2
σ
¯ n ∼ N(µ, )
X ˙
n
Tuesday, 23 March 2010

10. Means
X¯n − µ
Zn = 2 √
σ / n
Zn ∼ N(0, 1)
˙
Tuesday, 23 March 2010

11. 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

12. Continuity correction
When 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

13. Your turn
What’s the probability the average score
for the Lakers is less than 100?
Tuesday, 23 March 2010

14. Steps
Write as probability statement.
Transform each side to get to known
distribution.
Apply continuity correction, if necessary.
Compute.
Tuesday, 23 March 2010

15. Multiplication
X ~ Poisson(λ)
Y = tX
Then Y ~ Poisson(λt)
Tuesday, 23 March 2010

16. Exactly
How could you use the Poisson
distribution to calculate the exact
probability that the average score is
100?
Tuesday, 23 March 2010

17. Sampling distribution of
the standard deviation
Tuesday, 23 March 2010

18. (n − 1)S 2
2
∼ χ (n − 1)
2
σ
¯ 2
(Xi − X)
If Xi ~ iid N(0, 1), S =
2
n−1
Tuesday, 23 March 2010

19. Five fun facts about
2
χ
Tuesday, 23 March 2010

20. Proof
(n − 1)S 2
2
∼ χ (n − 1)
2
σ
Tuesday, 23 March 2010

21. Sampling distribution of
mean if variance unknown
Tuesday, 23 March 2010

22. Your turn
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?
Tuesday, 23 March 2010

23. 0.3
df
1
dens
0.2 2
15
Inf
0.1
−3 −2 −1 0 1 2 3
x
Tuesday, 23 March 2010

24. t-distribution
Xi ∼ Normal(µ, σ )2
¯n − µ
X ¯n − µ
X
√ ∼Z √ ∼ tn−1
σ/ n s/ n
Parameter called
degrees of freedom
Tuesday, 23 March 2010

25. Properties of the t-dist
Heavier tails compared to the normal
distribution.
lim tn = Z
n→∞
Practically, if n 30, the t distribution is
practically equivalent to the normal.
Tuesday, 23 March 2010

26. t-tables
Basically 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