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2011
Subject
Mathematics
Date
Jan 9, 2024
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Prac week 3 - Solutions
Dr Clara Grazian
1.
Create a vector
urn1=rep(c("white","black"),c(100,100))
representing an urn with 100 white and
100 black chips.
set.seed(
1234
)
urn1
=
rep(c(
"white"
,
"black"
),c(
100
,
100
))
urn1
##
[1] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[10] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[19] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[28] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[37] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[46] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[55] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[64] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[73] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[82] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[91] "white" "white" "white" "white" "white" "white" "white" "white" "white"
## [100] "white" "black" "black" "black" "black" "black" "black" "black" "black"
## [109] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [118] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [127] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [136] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [145] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [154] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [163] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [172] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [181] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [190] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [199] "black" "black"
2. Create a 100-by-100 matrix of zeroes called
with1
.
with1
=
matrix(
0
,
100
,
100
)
3.
Execute the following
for
loop, which draws 100 samples of size 100 with replacement, storing each in
a column of
with1
:
for
(i
in
1
:
100
){
with1[,i]
<-
sample(
x=
urn1,
size=
100
,
replace=
TRUE)
}
4.
Obtain a vector
Xwith1
consisting of the numbers of
white
in each column. There are various ways to
do this.
Xwith1
=
apply(with1==
"white"
,
2
,sum)
Xwith1_b
=
c()
for
(i
in
1
:ncol(with1)){
1
Xwith1_b[i]
=
sum(with1[,i]==
"white"
)
}
Xwith1
##
[1] 47 47 42 49 49 43 53 49 52 46 54 53 48 47 52 48 48 45 47 51 52 54 50 48 57
##
[26] 49 46 49 46 50 50 44 49 57 49 56 52 53 55 55 52 48 46 51 48 46 36 43 52 55
##
[51] 50 54 46 46 49 52 51 46 41 41 45 55 57 56 56 50 54 62 51 39 41 54 47 53 50
##
[76] 51 54 57 50 47 44 48 49 36 55 43 43 59 54 48 50 58 42 48 50 44 48 55 39 48
Xwith1_b
##
[1] 47 47 42 49 49 43 53 49 52 46 54 53 48 47 52 48 48 45 47 51 52 54 50 48 57
##
[26] 49 46 49 46 50 50 44 49 57 49 56 52 53 55 55 52 48 46 51 48 46 36 43 52 55
##
[51] 50 54 46 46 49 52 51 46 41 41 45 55 57 56 56 50 54 62 51 39 41 54 47 53 50
##
[76] 51 54 57 50 47 44 48 49 36 55 43 43 59 54 48 50 58 42 48 50 44 48 55 39 48
5.
Repeat questions 2 to 4, but this time sample without replacement. Use object names
without1
and
Xwithout1
.
without1
=
matrix(
0
,
100
,
100
)
for
(i
in
1
:
100
){
without1[,i]
<-
sample(
x=
urn1,
size=
100
,
replace=
FALSE)
}
Xwithout1
=
apply(without1==
"white"
,
2
,sum)
Xwithout1
##
[1] 49 45 42 50 46 52 54 49 49 48 56 48 53 51 52 53 54 54 49 51 50 47 55 49 53
##
[26] 46 49 51 52 52 45 47 51 50 53 48 47 56 54 52 50 55 47 53 49 52 45 48 49 46
##
[51] 53 49 57 50 48 53 50 51 48 52 50 49 43 51 47 49 50 58 50 53 47 51 52 46 44
##
[76] 48 46 54 51 52 50 46 52 46 44 54 48 55 50 50 49 50 48 51 47 42 52 48 52 51
6.
Repeat questions 1 to 5, but this time use an urn with 20 white and 180 black chips; use object names
urn2
,
with2
,
without2
, etc.
urn2
=
rep(c(
"white"
,
"black"
),c(
20
,
180
))
urn2
##
[1] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[10] "white" "white" "white" "white" "white" "white" "white" "white" "white"
##
[19] "white" "white" "black" "black" "black" "black" "black" "black" "black"
##
[28] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[37] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[46] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[55] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[64] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[73] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[82] "black" "black" "black" "black" "black" "black" "black" "black" "black"
##
[91] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [100] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [109] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [118] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [127] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [136] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [145] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [154] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [163] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [172] "black" "black" "black" "black" "black" "black" "black" "black" "black"
2
## [181] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [190] "black" "black" "black" "black" "black" "black" "black" "black" "black"
## [199] "black" "black"
with2
=
matrix(
0
,
100
,
100
)
for
(i
in
1
:
100
){
with2[,i]
<-
sample(
x=
urn2,
size=
100
,
replace=
TRUE)
}
Xwith2
=
apply(with2==
"white"
,
2
,sum)
without2
=
matrix(
0
,
100
,
100
)
for
(i
in
1
:
100
){
without2[,i]
<-
sample(
x=
urn2,
size=
100
,
replace=
FALSE)
}
Xwithout2
=
apply(without2==
"white"
,
2
,sum)
7.Prepare the graph window for a 2-by-2 array using
par(mfrow=c(2,2))
.
8. Create 4 histograms all on the same scale using of the four objects defined before.
par(
mfrow=
c(
2
,
2
))
hist(Xwith1,
breaks=
0
:
100
,
ylim=
c(
0
,
0.25
),
prob=
TRUE)
hist(Xwith2,
breaks=
0
:
100
,
ylim=
c(
0
,
0.25
),
prob=
TRUE)
hist(Xwithout1,
breaks=
0
:
100
,
ylim=
c(
0
,
0.25
),
prob=
TRUE)
hist(Xwithout2,
breaks=
0
:
100
,
ylim=
c(
0
,
0.25
),
prob=
TRUE)
Histogram of Xwith1
Xwith1
Density
0
20
40
60
80
100
0.00
0.15
Histogram of Xwith2
Xwith2
Density
0
20
40
60
80
100
0.00
0.15
Histogram of Xwithout1
Xwithout1
Density
0
20
40
60
80
100
0.00
0.15
Histogram of Xwithout2
Xwithout2
Density
0
20
40
60
80
100
0.00
0.15
9. Compute the means (with
mean()
) and standard deviations (with
sd()
) of the four
Xwith*
vectors.
3
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mean(Xwith1)
## [1] 49.34
mean(Xwith2)
## [1] 10.01
mean(Xwithout1)
## [1] 49.93
mean(Xwithout2)
## [1] 9.76
sd(Xwith1)
## [1] 4.999434
sd(Xwith2)
## [1] 3.301347
sd(Xwithout1)
## [1] 3.235707
sd(Xwithout2)
## [1] 2.225291
10.
Comment on similarities, differences and any other interesting features of these vectors; in particular
consider the questions below.
•
What do you expect the means to be roughly equal to in each case?
•
Do you expect some standard deviations to be bigger or smaller than others? Explain clearly.
We expect the sds to be smaller for urn 2 in both cases (with and without replacement), since for a true p
near 0 or 1 the observed proportion is less variable. Also, we expect without replacement to be less variable
than with replacement, since the very smallest and largest conceivable samples in the case with replacement
are not possible when sampling without replacement. The observed sd’s agree with these comments.
4