Will Landau
August 18, 2011
STAT 231 Problem Set 1 Solutions
Exercise 1.1 (Devore 1.42).
a. Suppose our sample of xi ’s is {x1 , . . . , xn } for some n. The sample has
e. Then:
mean x and median x
Pn
y=
i=1
yi
n
Pn
=
xi + c
=
n
i=1
Pn
i=1
xi
n
+
nc
= x+c
n
If n is odd, then:
ye = y n+1 = x n+1 + c = x
e+c
2
2
If n is even, then the answer is the same:
ye =
y n2 + y( n2 +1)
x n + c + x( n2 +1) + c
x n + x( n2 +1)
c+c
= 2
= 2
+
=x
e+c
2
2
2
2
If our sample of xi ’s is {2, 3, 3, 4, 5}, then x = 3.4 and x
e = 3. If c = 3,
then y = 6.4 and ye = 6. If c = 7, then y = 10.4 and ye = 10.
b.
Pn
y=
i=1
yi
n
Pn
i=1
=
n
cxi
Pn
i=1
=c
xi
n
= cx
If n is odd, then:
ye = y n+1 = c · x n+1 = ce
x
2
2
If n is even, then the answer is the same:
y n + y( n2 +1)
c · x n2 + c · x( n2 +1)
=
=c·
ye = 2
2
2
x n2 + x( n2 +1)
2
= ce
x
e = 3. If c = 3,
If our sample of xi ’s is {2, 3, 3, 4, 5}, then x = 3.4 and x
then y = 10.2 and ye = 9. If c = 7, then y = 23.8 and ye = 21.
Exercise 1.2 (Devore 1.51).
1
P19
a. Use the formula on page 38 of Devore. i=1 xi = 2563 and
P19 2
368,501−(25632 /19)
2
= 1264.766 and
i=1 xi = 368, 501, so s =
19−1
s=35.564
b. Let the xi ’s be our original sample and yi be the i’th time in hours.
hour
Then, yi = cxi , where c = 601minutes
. Use the formula on page 38 to get:
s2y =
X
yi2 −
(
P
P
X
X
yi )2
( cxi )2
( xi )2
=
(cxi )2 −
= c2
x2i −
= c2 s2x
19
19
19
P
So that
sy = csx
Plugging in the numbers, we get:
s2y =
1264.766
= 0.351
3600
And:
sy = csx =
35.564
= 0.593
60
Exercise 1.3.
a. Type 1: mean = 370.8, sample standard deviation = 14.408
b. Type 2: mean = 371, sample standard deviation = 11.892
c. Type 3: mean = 371, sample standard deviation = 10.542
Exercise 1.4.
2
exp1-10: Distribution of C1
Page 1 of 3
Distributions
C1
18
16
14
12
18
10
16
8
14
6
12
4
10
2
8
6
Quantiles
exp1-10: Distribution of C1
Page 2 of 3
4
Distributions
C12
Quantiles
100.0% maximum
99.5%
97.5%
90.0%
75.0%
quartile
50.0%
median
25.0%
quartile
10.0%
2.5%
0.5%
0.0%
minimum
18.26
18.26
16.669
13.587
12.1675
9.835
7.915
6.726
4.33
2.97
2.97
Moments
Mean
Std Dev
Std Err Mean
Upper 95% Mean
Lower 95% Mean
N
10.038444
2.8679903
0.3023127
10.639133
9.4377555
90
Stem and Leaf
3
exp1-10: Distribution of C1
Page 3 of 3
Distributions
C1
Stem and Leaf
Stem
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
Leaf
3
Count
1
19
12
024
144456
223367799
011234677
002333444556
134456688888
03345566778
2223667799
04678899
269
0
0
2
2
3
6
9
9
12
12
11
10
8
3
1
1
3|0 represents 3.0
Exercise 1.5.
a. Before you begin, you should have 2 columns of data: one called Data,
the fatigue limit values, and Label (Type 1, Type 2, or Type 3).
Generate the reports by following the clues in the screenshot:
4
Turn the report for each label horizontal by selecting the red triangle to
the left of Data, and then selecting Display Options ¿ Horizontal Layout
in the drop-down menu:
Make sure to include QUANTILE box plots and not outlier box plots:
5
The result should look something like this:
b. Again, make sure to have all the data in one column and all the labels
(Type 1, Type 2, or Type 3) in the other column of your spreadsheet.
Open Graph Builder:
6
Click and drag Data into the box labeled ”y” and Label into the box
labeled ”x”.
You should get something like this:
7
The data is displayed by label, but in a scatterplot. Change the display
to three side-by-side box plots:
And make sure they are QUANTILE box plots and not outlier box plots:
8
You should get something like this:
9