Stat 180
/
C236
Homework 5
J. Sanchez
UCLA Department of Statistics
Instructions
(1) Homework must be typed and answered in the order given (problem 1(a)(b)(c)(d) ﬁrst, problem 2(a)(b).
.. second,
etc.
..)
(2) Undergrads and grads will answer all questions.
(3) Include in each part of the homework only the answer. R code and R output (without mistakes), must be included
in the appendix to the question. For example, for question 1.a, write only the answer and your comments. The
code and output for that part of the question will be in the appendix (the last part of question 1).
(4) No late homework under any circumstances.
(5) Write your name and ID this way: Last name, ﬁrst name, UCLA ID, date, Homework number.
(6) Do not just give a number as an answer. For example, if asked for probability that posterior proportion is larger
than 0.7, write
Prob
(
p
>
0
.
7)
=
0
.
3, say and write comments or explanations if needed.
(7) The homework must be turned in in lecture (no mail box, no email).
Homework 5: Do problems 1, 2, 3 of Hopf’s Chapter 5. Can be found in the back matter.
Problem 1.
5.1 Studying: The ﬁles
school1.dat, school2.dat
and
school3.dat
contain data on the amount of
time students from three high schools spent on studying or homework during an exam period. Analyze data from each
of these schools separately, using the normal model with a conjugate prior distribution, in which
{
μ
0
=
5
, σ
2
0
=
4
, κ
0
=
1
, ν
0
=
2
}
and compute or approximate the following:
(a) posterior means and 95% conﬁdence intervals for the mean
θ
and standard deviation
σ
from each school;
Answer:
The joint posterior distribution for
θ
and
σ
is (from our notes and Hopf Chapter 5)
p
(
θ, σ
2

y
1
, ....
,
y
n
)
=
p
(
θ

σ
2
,
y
1
, ...,
y
n
)
P
(
σ
2

y
1
, ....
y
n
)
=
Normal
μ
n
,
σ
2
κ
n
!
IG
ν
n
2
, ν
n
σ
2
n
2
!
where
κ
n
=
κ
0
+
n
μ
n
=
κ
0
σ
2
μ
0
+
n
σ
2
¯
y
κ
0
σ
2
+
n
σ
2
=
κ
0
μ
0
+
n
¯
y
κ
n
ν
n
=
ν
0
+
n
σ
2
n
=
1
ν
n
"
ν
0
σ
2
0
+
(
n

1)
s
2
+
κ
0
n
κ
n
(¯
y

μ
0
)
2
#
Posterior inference for
θ
and
σ
School
E(
θ
)
95% int
θ
E(
σ
)
95% int
σ
n
School 1
9.281
(7.756, 10.801)
3.906
(3.011 ,5.166)
25
School 2
6.946
(5.197,8.763)
4.392
(3.333,5.888)
23
School 3
7.821
(6.191, 9.431)
3.748
(2.802, 5.086)
20
As we can see in the table, kids in school 2 study less on average, but there is more variability on how many hours
each kid there put than in the other two schools. See the R code at the end of this section.
January 14, 2010 (Revised November 7, 2010)
1
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View Full DocumentStat 180
/
C236
Homework 5
J. Sanchez
UCLA Department of Statistics
### For school1
####
Read the data
school1.dat=read.table("http://www.stat.washington.edu/hoff/Book/Data/hwdata/school1.dat")
attach(school1.dat)
#### summarize the data
hours1=school1.dat$V1
y1bar=mean(hours1) ; y1s2=var(hours1)
y1s=sd(hours1) ; n1=length(hours1)
####
Prior distribution parameters
mu0=5 ; sigma02=4; k0=1; nu0=2
####
posterior distribution parameters
kn.1=k0+n1
mun.1=(k0*mu0 + n1*y1bar)/kn.1
nun.1=nu0+n1
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 Spring '10
 wu
 Statistics, posterior distribution, J. Sanchez, J. Sanchez UCLA, Sanchez UCLA Department

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