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1. P a g e | 1
CHAPTER 1
Descriptive Statistics
1.1 Introduction
1.2 Basic concepts
1.3 Sampling schemes
1.4 Graphical representation of data
1.5 Numerical description of data
1.6 Computers and statistics
1.7 Chapter summary
1.8 Computer examples
Projects for Chapter 1
Statistical software R is used for this book. All outputs and codes given are in R. R is a free
statistical software, and it can be downloaded from the website: http://www.r-project.org
Mathematical Statistics with Applications in R 2nd Edition Ramachandran Solutions Manual
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2. P a g e | 2
Exercises 1.2
1.2.1
The suggested solutions:
For qualitative data we can have color, sex, race, Zip code and so on. For quantitative data we
can have age, temperature, time, height, weight and so on. For cross section data we can have
school funding for each department in 2000. For time series data we can have the crude oil
price from 1995 to 2008.
1.2.2
The suggested solutions:
For qualitative data we collect the frequency information of the data and we want to see the
comparison by either bar chart or pie chart.
For quantitative data we collect the numerical information of the data and we want to see the
comparison by histogram distribution.
For cross section data we collect different section data on the same time and we want to make
comparison between them.
For time series data we collect same type of data on different time spot and we want to see if
there is any trend or pattern of this data with time shifting.
1.2.3
The suggested questions can be:
1. What types of data the amounts are?
2. Do these Federal Agency receive the same amount of funding? If not, why?
3. Which Federal Agency should receive more funding? Why?
The suggested inferences we can make are:
1. These Federal Agency get different amount of money.
2. There are big differences between funding the Agencies receive.
1.2.4
The suggested questions can be
1. How does the funding changes for each agency through time?
2. Should we change the proportion between the Agencies or not?
3. Should we increase the total amount or not?
The suggested inferences we can make is
1. The total money tends to be the same.
2. The proportion between the Agencies tends to be the same.

3. P a g e | 3
Exercises 1.3
1.3.1
Simple Random Sample:
Say we have a population of 1,000 students, and we want a sample of 100 students.
Using software or a random table, we randomly select 100 out of the 1,000 students. We want
the selection probability for all the students to be equal. That is no student is more likely to be
selected than any other student.
Systematic Sample:
Again, we have a population of 1,000 students, and we want a sample of 100 students.
We need the sampling interval k = N/n = 10. Now, we need a random starting point between 1
and k. Let say, we randomly select 4. This gives us the sample: 4, 14, 24, ..., 74, 84, 94. This
sample of numbers will correspond to ordered list of students.
Stratified Sample:
Suppose we decide to sample 100 college students from the population of 1000 ( that is
10% of the population). We know these 1000 students come from three different major, Math,
Computer Science and Social Science. We have Math 200, CS 400 and SS 400 students. Then
we choose 10% of each of them Math 20, CS 40 and SS 40 by using simple random sample
within each major.
Cluster Sample:
Presume we have a population of 1,000 students clustered into 10 departments. For our
sample of students, we will randomly select a subset from the 10 departments. Let say we
randomly select 3 out 10 departments. Now, all the students on those 3 department become
the sample from the population of students.
Exercises 1.4
1.4.1
(a) Bar graph
Very goodGoodFairMediocrePoor
35.00%
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
C1
C2
Bar graph for the percent of road mileage

4. P a g e | 4
(b) Pie chart
Poor
Very good
Good
Fair
Mediocre
Category
Pie chart of the percent of road mileage
1.4.2
(a) Bar graph
Other
Lepidoptera
Thysanoptera
O
donata
Collem
bola
O
rthoptera
Hem
iptera
Diptera
Coleoptera
40.00%
30.00%
20.00%
10.00%
0.00%
C1
C2
Bar graph of species
(b) Pareto graph
Percentage 0.35 0.26 0.15 0.06 0.06 0.05 0.03 0.04
Percent 35.0 26.0 15.0 6.0 6.0 5.0 3.0 4.0
Cum % 35.0 61.0 76.0 82.0 88.0 93.0 96.0 100.0
Species
O
thers
O
donata
Collem
bola
O
ther
O
rthoptera
Hem
iptera
Coleoptera
Diptera
1.0
0.8
0.6
0.4
0.2
0.0
100
80
60
40
20
0
Percentage
Percent
Pareto graph of species

5. P a g e | 5
(c) Pie chart
Coleoptera
Diptera
Hemiptera
Orthoptera
Collembola
Odonata
Thysanoptera
Lepidoptera
other
Category
Pie chart of species
species
1.4.3
(a) Bar graph
Renewable EnergyPetroliumNyclear Electric PowerNatural GasCoal
40.00%
30.00%
20.00%
10.00%
0.00%
C1
C2
Bar graph
(b) Pareto graph
Percentage 0.40 0.23 0.22 0.08 0.07
Percent 40.0 23.0 22.0 8.0 7.0
Cum % 40.0 63.0 85.0 93.0 100.0
C1
Renew
able
Energy
Nyclear Electric
Pow
er
Coal
Natural Gas
Petrolium
1.0
0.8
0.6
0.4
0.2
0.0
100
80
60
40
20
0
Percentage
Percent
Pareto graph

6. P a g e | 6
(c) Pie chart
Coal
Natural Gas
Nyclear Electric Power
Petrolium
Renewable Energy
Category
Pie chart of species
species
1.4.4
(a) Bar graph
Black ratRabbitRed FoxHedgehogLionChimpanzeeDolphinBat
12
10
8
6
4
2
0
C1
C2
Bar graph
(b) Pareto graph
Percentage 11 6 6 5 3 1 1 1
Percent 32.4 17.6 17.6 14.7 8.8 2.9 2.9 2.9
Cum % 32.4 50.0 67.6 82.4 91.2 94.1 97.1 100.0
C1
O
ther
Chim
panzee
Bat
Lion
Hedgehog
Red
Fox
Rabbit
Black
rat
35
30
25
20
15
10
5
0
100
80
60
40
20
0
Percentage
Percent
Pareto graph
1.4.5
(a) Bar graph
FDCBA
6
5
4
3
2
1
0
C1
Count
bar graph

7. P a g e | 7
(b) Pie chart
A
B
C
D
F
Category
Pie chart
species
1.4.6
(a) Pie chart
16 to 19 years
20 to 24 years
25 to 34 years
35 to 44 years
45 to 54 years
55 to 64 years
65 years and over
Category
Pie chart
species
(b) Bar graph
65
years
and
over
55
to
64
years
45
to
54
years
35
to
44
years
25
to
34
years
20
to
24
years
16
to
19
years
700
600
500
400
300
200
100
0
C1
C3
Bar graph

8. P a g e | 8
(c) Pareto graph
C2 628 605 600 498 393 334 260
Percent 18.9 18.2 18.1 15.0 11.8 10.1 7.8
Cum % 18.9 37.2 55.2 70.3 82.1 92.2 100.0
C1
16
to
19
years
20
to
24
years
65
years
and
over
25
to
34
years
35
to
44
years
55
to
64
years
45
to
54
years
3500
3000
2500
2000
1500
1000
500
0
100
80
60
40
20
0C2
Percent
Pareto Graph
1.4.7
(a) Pie chart
Mining
Construction
Manufacturing
Transportation
Wholesale
Retail
Finance
Services
Category
Pie chart
species
(b) Bar graph
Services
Finance
Retail
W
holesale
Transportation
M
anufacturing
Construction
M
ining
8000
7000
6000
5000
4000
3000
2000
1000
0
C1
C2
Bar graph

9. P a g e | 9
1.4.8
(a) Bar graph
AustraliaWesternEasternCaribbeanLatinEastSouthNorthSub-Saharan
25
20
15
10
5
0
C1
C2
Bar graph
(b) Pareto graph
Percentage 25.30 5.80 1.40 1.32 0.70 1.72
Percent 69.8 16.0 3.9 3.6 1.9 4.7
Cum % 69.8 85.8 89.7 93.3 95.3 100.0
C1 OtherEasternNorthLatinSouthSub-Saharan
40
30
20
10
0
100
80
60
40
20
0
Percentage
Percent
Pareto graph
1.4.9
Bar graph
20001990198019601900
80
70
60
50
40
30
20
10
0
C1
C2
Bar graph

10. P a g e | 10
1.4.10
84 LookalikeLusealMid Button FlagHammer
60
50
40
30
20
10
0
C1
C2 Bar graph
1.4.11
(a) Bar graph
SuicideStrokePneumoniaKidneyHeartDiabetesCancerCChronicAccidents
300
250
200
150
100
50
0
C1
C2
Bar graph
(b) Pareto graph
Percentage 268.0 199.4 58.5 42.3 35.1 34.5 23.9 30.2
Percent 38.7 28.8 8.5 6.1 5.1 5.0 3.5 4.4
Cum % 38.7 67.6 76.0 82.1 87.2 92.2 95.6 100.0
C1
Other
Diabetes
Accidents
Pneum
oniaC
Stroke
Cancer
Heart
700
600
500
400
300
200
100
0
100
80
60
40
20
0
Percentage
Percent
Pareto graph

11. P a g e | 11
1.4.12
(a) Expenditure
Bar graph
PersonalTransfersDebtOperatingCapitalReserves
35
30
25
20
15
10
5
0
C1
C2
Bar graph
Revenues
Bar graph
TransfersInterestFinesChargesInterLicensesUtilityProperty
40
30
20
10
0
C1
C2
Bar graph
(b) Expenditure
Pie chart
Reserves
Capital
Operating
Debt
Transfers
Personal
Category
Pie chart
species

12. P a g e | 12
Revenues
Pie chart
Property
Utility
Licenses
Inter
Charges
Fines
Interest
Transfers
Category
Pie chart
species
1.4.13
90807060
9
8
7
6
5
4
3
2
1
0
C1
Frequency
Histogram
1.4.14
(a) Stem and leaf
Stem-and-Leaf Display: C1
Stem-and-leaf of C1 N = 40
Leaf Unit = 1.0
2 0 00
12 0 2222223333
13 0 5
20 0 6666677
20 0 888899
14 1 111
11 1 223333
5 1 55
3 1 677

13. P a g e | 13
(b) Histogram
1612840
6
5
4
3
2
1
0
C1
Frequency
Histogram
(c) Pie chart
17-19
0-1
1-3
3-5
5-7
7-9
9-11
11-13
13-15
15-17
Category
Pie Chart
1.4.15
( a ) Stem and leaf
Stem-and-leaf of SAT Mathematics scores N = 20
Leaf Unit = 10
1 4 7
3 4 99
8 5 00011
10 5 22
10 5 4455
6 5 6667
2 5 9
1 6 0

14. P a g e | 14
(b) Histogram
600580560540520500480
5
4
3
2
1
0
C1
Frequency
Histogram
(c) Pie chart
470-490
490-510
510-530
530-550
550-570
570-590
590-610
Category
Pie Chart
1.4.16
Frequency table
Interval Frequency Relative Freq Percentage
5-9 1 .04 4
10-14 3 .12 12
15-19 5 .2 20
20-24 10 .4 40
25-29 5 .2 20
30-35 1 .04 4

15. P a g e | 15
Histogram
1.4.17
Non-Hispanic Black or African American
Non-Hispanic Asian
Non-Hispanic American Indian or Alaska Native
Non-Hispanic Native Hawaiian or other Pacific Islander
Non-Hispanic Some Other Race
Non-Hispanic Two or more races
Hispanic or Latino
White or European American Hispanic
Black or African American Hispanic
American Indian or Alaska Native Hispanic
White or European American
Asian Hispanic
Some Other Race Hispanic
Two or more races Hispanic
Black or African American
Asian American
American Indian or Alaska Native
Native Hawaiian or other Pacific Islander
Some other race
Two or more races
Not Hispanic nor Latino
Non-Hispanic White or European American
Category
Pie Chart
Exercises 1.5
1.5.1
       
2 2 2 2
2
2
176105... 7896
165.67
12
176165.67 105165.67 ... 78165.67 96165.67
121
3988.42
3988.4263.15
x
s
s
s
 
 
      



 

16. P a g e | 16
1.5.2
(a)
       
2 2 2 2
2
2
7.6257.5... 5.3757.5
7.013
10
7.6257.013 7.57.013 ... 5.3757.013 7.57.013
101
.548
.548.0738
x
s
s
s
  
 
      



 
(b)
 
 
1
3
6.625
7.5 7.625
7.5625
2
7.375
7.5625 6.625 .9375
6.625 1.5 .9375 5.21875
7.625 1.5 .9375 9.0312
.
5
Q
Q
M
IQ
Ther
R
e ar
LL
e no outli
L
ers
L


 

  
  
  
1.5.3
Given information: mean=6 , median = 4 , mode = 3
We know that the value 3 can only be in the data twice. If not the median would be different
than 4. This give us the following: 3, 3, x, y. Where x and y are the missing values. We
introduce a system of equation to solve for x and y.
3 9 3
6 4
4 2
24 6 8 3
18 5
18 5
13,x=5
x y x
x y x
x y x
y
y
   
 
    
  
 

Data: 3, 3, 5, 13
       
 
2 2 221
3 6 3 6 5 6 13 6
3
1
= 68
3
=22.667
Sd=
= 22.667
=4.76
Var
Var
        
 

17. P a g e | 17
1.5.4
(a)
       
2 2 2 2
2
2
11881050... 1578261
1243.5
14
11881243.5 10501243.5 ... 15781243.5 2611243.5
141
792365.81
792365.81890.15
28822612621
x
s
s
s
Range
  
 
      



 
  
(b)
 
 
1
3
537
1578
1117 1050
1083.5
2
1578 537 1041
537 1.5 1041 1024.5
1578 1.5 1041 3139.5
.
Q
Q
M
IQR
L
There are no outlier
L
LL
s



 
  
   
  
(c)
1.5.5
5001000150020002500
(a)
 
 
1
3
80
115
95
115 80 35
80 1.5 35 27.5
115 1.5 35 167.5
Q
Q
M
IQR
LL
LL



  
  
  

18. P a g e | 18
(b)
406080100120
(c) There are no outliers.
1.5.6
    
         
2 2 2 2 2
2
5214715121017622
11.8
50
5211.814711.8151211.8101711.862211.8
34.653
501
34.6535.887
x
s
s
   
 
       
 

 
1.5.7
(a)
i1 i1 i1
( ) () () 0
l l l
i i i i ifmx fmfxnxnx
  
  
(b)
         
     
5
1
5211.814711.8151211.8101711.862211.8
59.8144.815.2105.2610.2
4967.235261.20
i i
i
fmx


 



19. P a g e | 19
1.5.8
(a)
 2 2 2 2 2
i 1 1 i 1 i 1 i 1
2
2 2 2 2 2 2 1
i 1 i 1 i 1
2
i 12
i 1
( ) 2 2
2
n n n n n
i i i i i
i
n
in n n
i
i i i
n
in
i
x x x xx x x x x x
x
x nx nx x nx x n
n
x
x
n
    

  


      
 
 
       
 
 
 
 
 
  
    

  


(b)
     
 
2 2 2 22
i1
2
2
i12
i1
( ) 10592.4678092.467...11592.4679592.4679737.7333
1387
137989 9737.7333
15
n
i
n
in
i
xx
x
x
n



    
 
 
    



1.5.9
(a)
 
32
1
32
22
1
1059.36
33.105
32 32
1
33.
5488.332
177.043
31
53.50 5.31 48.
105
1
9
3
1
i
i
i
i
x
r
x
s x
ange


  
 
  
 


(b)
 
 
1
3
24.75 25.44
25.095
2
42.19 43.25
42.72
2
32 32
32
2
42.72 25.095 17.625
25.095 1.5 17.625 1.3425
42.72 1.5 17.625 69.1575
.
Q
Q
M
IQR
LL
LL
There are no outliers

 

 

 
  
   
  

20. P a g e | 20
(c)
1020304050
(d)
Histogram of y
y
Frequency
0 10 20 30 40 50 60
02468
(e)
33.105x
 19.80,46.41x s 21 data point (65.625%) fall within 1 SD, empirical rule = 68%
 2 6.49,59.72x s  31 data point (96.875%) fall within 2 SD, empirical rule = 95%
 3 6.81,73.02x s  32 data point (100%) fall within 3 SD, empirical rule = 99.7%

21. P a g e | 21
1.5.10
(a)
 
40
1
22
0
1
4
333.6
8.34
40 40
1
8
944.376
24.215
39
17.2 .5
.3
167
4
3
.
9
i
i
i
i
x
x
s x
range


  
 
  
 


(b)
 
 
1
3
3.7 3.6
3.65
2
12.8 12.3
12.55
2
8.3 7.9
8.1
2
12.55 3.65 8.9
3.65 1.5 8.9 9.7
12.55 1.5 8.9 25.9
.
Q
Q
M
IQR
LL
LL
There are no outliers

 

 

 
  
   
  
(c)
051015
(d)
Histogram of y
y
Frequency
0 5 10 15
0246810

22. P a g e | 22
(e) 8.34x 
 3.42,13.26x s 24 data point (60%) fall within 1 SD, empirical rule = 68%
 2 1.5,18.18x s  40 data point (100%) fall within 2 SD, empirical rule = 95%
 3 6.42,23.1x s  40 data point (100%) fall within 3 SD, empirical rule = 99.7%
1.5.11
(a)
2
211,000 1 11,000
110, 1,900,000 6969697
100 1001 100
6969.69783.4847
x s
s
 
     
 
 
(b)
.68(400,000) 272,000
2 .95(400,000) 380,000
3 .997(400,000) 398,800
xs
x s
x s
 
  
 
1.5.12
(a)
 
10
1
22
1
1
3
10
418
41.
39
46
42 40
41
2
46 39 7
1149.6
127.733
9
127.733 11.
8
10 10
1
8.34
9
302
i
i
i
i
Q
Q
M
IQR
s
x
x
s x






 

  
 
 

 


(b)  
10
i1
( )4181041.8418-4180ixx

  
(c)
2030405060

23. P a g e | 23
(d)
 
 
7
391.57 28.5
461.57 56.5
18 60.
IQR
LL
LL
Therearetwooutliers and

  
  
1.5.13
(a)
 
30
1
30
22
1
112.3
3.7433
30 30
1
3.7433 3.502
29
3.502 1.871
ii
i
i
x
x
s x
s


  

 



(b) Frequency table
Class Interval Frequency Mi Mi∙fi
1 0-1.6 4 .8 3.2
2 1.7-3.3 10 2.5 25
3 3.4-5 9 4.2 37.8
4 5.1-6.7 5 5.9 29.5
5 6.8-8.4 2 7.6 15.2
(c) Grouped data:
         
2 2 2 2 22
4(.8)10(2.5)9(4.2)5(5.9)2(7.6)
3.69
30
1
40.83.69 2.53.69 4.23.69 5.93.69 7.63.10 9 5 2 693.62
29
3.621.90
x
s
s
  
 
         
 
 
The results from the grouped data are similar to the actual data.
1.5.14
(a)
 
30
1
30
22
1
1814
60.467
30 30
1
60.467 685.085
29
685. 26.1708 45
ii
i
i
x
x
s x
s


  
 
 



(b) Frequency table
Class Interval Frequency Mi Mi∙fi
1 0-20 1 10 10
2 20-40 8 30 240
3 40-60 6 50 300
4 60-80 5 70 350
5 80-100 10 90 900

24. P a g e | 24
(c)
Grouped data:
         
2 2 2 2 22
10240300350900
60
30
1
1060 3060 508 6 560 7060 9060 682.7592
29
682.75926.13
x
s
s

 
          
 
 
The results from the grouped data are similar to the actual data.
1.5.15
25L 139615mf 4w
178859bF 514661n
24822.27)5(.M  b
m
Fn
f
w
L
1.5.16
(a)
       
         
2 2 2 2 22
8159.511169.518179.59189.54199.5
177.5
50
1
8159.5177.5 169.5177.5 179.5177.5 189.5177.5 199.5177.5
49
134.6
11 1
94
134.69411.6
8
06
9 4
x
s
s
   
 
       
 

 
 
(b) 517L , 81mf , 9w
91bF , 05n
178)5(.M  b
m
Fn
f
w
L
1.5.17
       
         
2 2 2 2 22
38103129.55949.54569.5789.5
44.272
180
1
381044.272 29.544.272 49.544.272 69.544.272 89.544.272
49
536.146
536.14623
31 59 45 7
.155
x
s
s
   
 
     
 


   


(b) 40L , 59mf , 19w , 69bF
Mathematical Statistics with Applications in R 2nd Edition Ramachandran Solutions Manual
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