Mathematical Statistics with Applications in R 2nd Edition Ramachandran Solutions Manual

Mathematical Statistics with Applications in R 2nd Edition Ramachandran Solutions Manual Full Download: http://alibabadownload.com/product/mathematical-statistics-with-applications-in-r-2nd-edition-ramachandran-s P a g e |1

CHAPTER 1 Descriptive Statistics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction Basic concepts Sampling schemes Graphical representation of data Numerical description of data Computers and statistics Chapter summary 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

This sample only, Download all chapters at: alibabadownload.com

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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.

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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 Bar graph for the percent of road mileage 35.00% 30.00% 25.00%

C2

20.00% 15.00% 10.00% 5.00% 0.00%

Poor

Mediocre

Fair C1

Good

Very good

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Pie chart of the percent of road mileage C ategory Poor Very good Good Fair Mediocre

(b) Pie chart 1.4.2 (a) Bar graph Bar graph of species 40.00%

C2

30.00%

20.00%

10.00%

0.00%

C

eo ol

er pt

a

a er pt Di

m He

ra te ip

la ra te bo op em h l t l r O Co

a a ta er er na pt pt do no do i O a p ys Le Th

r he Ot

C1

(b) Pareto graph 100

0.8

80

0.6

60

0.4

40

0.2

20

0.0 Species

Percentage Percent Cum %

s la ra er ta er te th bo na th O m op do O e m h l t lle O e r o o H C O C 0.35 0.26 0.15 0.06 0.06 0.05 0.03 0.04 35.0 26.0 15.0 6.0 6.0 5.0 3.0 4.0 35.0 61.0 76.0 82.0 88.0 93.0 96.0 100.0

a er pt Di

r te op

a

ra te ip

0

Percent

Percentage

Pareto graph of species 1.0

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Pie chart of species species

Category Coleoptera Diptera Hemiptera Orthoptera Collembola Odonata Thy sanoptera Lepidoptera other

(c) Pie chart 1.4.3 (a) Bar graph Bar graph 40.00%

C2

30.00%

20.00%

10.00%

0.00%

Coal

Natural Gas Nyclear Electric Power

Petrolium

C1

Renewable Energy

(b) Pareto graph 100

0.8

80

0.6

60

0.4

40

0.2

20

0.0 C1

Pe

Percentage Percent Cum %

m li u tro

0.40 40.0 40.0

Na

as lG ra tu

0.23 23.0 63.0

Co

al

r ea cl Ny 0.22 22.0 85.0

ec El

ic tr

w Po

er

R

ew en

0.08 8.0 93.0

le ab

gy er En

0.07 7.0 100.0

0

Percent

Percentage

Pareto graph 1.0

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Pie chart of species species

C ategory C oal Natural Gas Ny clear Electric Power Petrolium Renewable Energy

(c) Pie chart 1.4.4 (a) Bar graph Bar graph 12 10

C2

8 6 4 2 0

Bat

Dolphin Chimpanzee

Lion

Hedgehog Red Fox

Rabbit

Black rat

C1

(b) Pareto graph Pareto graph 35

100

30

Percentage

60

20 15

40

Percent

80

25

10 20

5 0 C1

Percentage Percent Cum %

k ac Bl

t ra

it bb Ra

11 32.4 32.4

6 17.6 50.0

R

ed

Fo

x He

6 17.6 67.6

og eh dg 5 14.7 82.4

L io

t Ba

n

im Ch 1 2.9 94.1

3 8.8 91.2

ee nz pa 1 2.9 97.1

er th O

0

1 2.9 100.0

1.4.5 bar graph 6 5

Count

4 3 2 1 0

(a) Bar graph

A

B

C C1

D

F

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(b) Pie chart Pie chart species

Category A B C D F

1.4.6 (a) Pie chart Pie chart species

Category 16 to 19 y ears 20 to 24 y ears 25 to 34 y ears 35 to 44 y ears 45 to 54 y ears 55 to 64 y ears 65 y ears and ov er

(b) Bar graph Bar graph 700 600

C3

500 400 300 200 100 0

16

to

19

ye

s ar 20

to

24

a ye

rs 25

to

34

ye

s ar 35

to

44

a ye

rs 45

C1

to

54

ye

s ar 55

to

64

a ye

rs

65

a ye

rs

d an

ov

er

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Pareto Graph 3500

100

3000

C2

2000

60

1500

40

1000 20

500 0

C1

45

(c) Pareto graph

to

54

C2 Percent Cum %

a ye

rs 55

to

64

628 18.9 18.9

ye

s ar 35

to

44

a ye

rs 25

605 18.2 37.2

to

600 18.1 55.2

34

a ye 65

rs a ye

rs

498 15.0 70.3

d an

er ov 20

to

24

393 11.8 82.1

1.4.7 (a) Pie chart Pie chart species

C ategory Mining C onstruction Manufacturing Transportation W holesale Retail Finance Serv ices

(b) Bar graph Bar graph 8000 7000 6000

C2

5000 4000 3000 2000 1000 0

g in in M

n g n io in tio ct ur rta ru ct t o a ns uf sp Co an an M Tr

W

le sa le o h

C1

Re

il ta

e nc na Fi

ic e rv Se

s

a ye

rs 16

to

334 10.1 92.2

19

a ye

rs

260 7.8 100.0

0

Percent

80

2500

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1.4.8 Bar graph 25

20

C2

15

10

5

0

Sub-Saharan North

South

East

Latin

Caribbean Eastern

C1

(a) Bar graph (b) Pareto graph Pareto graph 40

100 80 60

20

40 10 20 0 C1 Percentage Percent Cum %

Sub-Saharan 25.30 69.8 69.8

South 5.80 16.0 85.8

Latin 1.40 3.9 89.7

North 1.32 3.6 93.3

Eastern 0.70 1.9 95.3

0

Other 1.72 4.7 100.0

1.4.9 Bar graph Bar graph 80 70 60

C2

50 40 30 20 10 0

1900

1960

1980 C1

1990

2000

Percent

Percentage

30

Western

Australia

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1.4.10 Bar graph 60 50

C2

40 30 20 10 0

Hammer

Mid Button Flag

Luseal

84 Lookalike

C1

1.4.11 (a) Bar graph Bar graph 300 250

C2

200 150 100 50 0

A ccidents C hronic

C

Cancer

Diabetes

Heart

Kidney Pneumonia Strok e

Suicide

C1

(b) Pareto graph Pareto graph 700

100 80

500 60

400 300

40

200 20

100 0 C1

Percentage Percent Cum %

a He

rt

268.0 38.7 38.7

er nc Ca 199.4 28.8 67.6

ro St

ke

58.5 8.5 76.0

C

ia on

s ts te en be id a c i D Ac

m eu Pn 42.3 35.1 6.1 5.1 82.1 87.2

34.5 5.0 92.2

23.9 3.5 95.6

r he Ot 30.2 4.4 100.0

0

Percent

Percentage

600

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1.4.12 (a) Expenditure Bar graph Bar graph 35 30 25

C2

20 15 10 5 0

Reserves

Capital

Operating

Debt

Transfers

Personal

C1

Revenues Bar graph Bar graph 40

C2

30

20

10

0

Property

Utility

Licenses

Inter

Charges C1

Fines

Interest Transfers

(b) Expenditure Pie chart Pie chart species

C ategory Reserv es C apital Operating Debt Transfers Personal

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Revenues Pie chart Pie chart species

C ategory Property Utility Licenses Inter C harges Fines Interest Transfers

1.4.13 Histogram 9 8 7

Frequency

6 5 4 3 2 1 0

60

70

80 C1

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

90

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Histogram 6

Frequency

5 4 3 2 1 0

0

4

8

12 C1

(b) Histogram (c) Pie chart Pie Chart

Category 0-1 1-3 3-5 5-7 7-9 9-11 11-13 13-15 15-17 17-19

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

16

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Histogram 5

Frequency

4

3

2

1

0

480

500

(b) Histogram (c) Pie chart

520

540 C1

560

580

600

Pie Chart Category 470-490 490-510 510-530 530-550 550-570 570-590 590-610

1.4.16 Frequency table Interval 5-9 10-14 15-19 20-24 25-29 30-35

Frequency 1 3 5 10 5 1

Relative Freq .04 .12 .2 .4 .2 .04

Percentage 4 12 20 40 20 4

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Histogram 1.4.17 Pie Chart Category White or European American 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 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 Asian Hispanic Some Other Race Hispanic Two or more races Hispanic

Exercises 1.5 1.5.1

1 7 6  1 0 5  . . . 7 89 6 x   1 6 5 .

6 7 1 2 1 7 6  1 6 5 .

6 7 1 0 5  1 6 5 .

6 7 . . . 7 8  1 6 5 .

6 7 9 6  1 6 5 .

6 7         s 2

2

2

1 2  1

s 3 9 8 8 .

4 2 2

s 3 9 8 8 .

4 26  3 .

1 5

2

2

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1.5.2 (a)

7 .

6 2 5  7 .

5  . . .  5 .

3 7 5  7 .

5 x   7 .

0 1 3 1 0 7 .

6 2 5  7 .

0 1 3 7 .

5  7 .

0 1 3 . . .  5 .

3 7 5  7 .

0 1 3 7 .

5  7 .

0 1 3         s 2

2

2

2

2

1 0  1

s .

5 4 8 2

s .

5 4 8  .

0 7 3 8 (b)

Q1  6.625 7.5  7.625  7.5625 2 M  7.375

Q3 

IQR  7.5625  6.625  .9375 LL  6.625 1.5.9375  5.21875 LL  7.625 1.5.9375  9.03125 There are no outliers.

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xy 6 4 xy2 46

x3 4 2 x83

xy1 8

x5

y1 85 y1 3 ,x = 5 Data: 3, 3, 5, 13

1 2 2 2 2 Var 36 36 56 136   3 1 = 68 3 =22.667 Sd= Var = 22.667 =4.76

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1.5.4

1 1 8 8  1 0 5 0  . . .1 5 7 82 6 1 x   1 2 4 3 .

5 1 4 1 1 8 8  1 2 4 3 .

5 1 0 5 0  1 2 4 3 .

5 . . . 1 5 7 8  1 2 4 3 .

5 6 1  1 2 4 3 .

5      2  s 2

2

2

(a)

1 4  1

s 7 9 2 3 6 5 .

8 1 2

s 7 9 2 3 6 5 .

8 18 9 0 .

1 5 R a n g e  2 8 8 22 6 12 6 2 1 Q1  537 Q3  1578 1117 1050  1083.5 2 (b) IQR  1578  537  1041 M

LL  537 1.5 1041  1024.5 LL  1578 1.5 1041  3139.5 There are no outliers.

500

1000

1500

2000

2500

(c)

1.5.5 Q1  80 Q3  115

(a)

M  95 IQR  115  80  35 LL  80 1.535  27.5 LL  115 1.535  167.5

2

2

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40

60

80

100

120

(b)

(c) There are no outliers.

1.5.6

5 2  1 4 7  1 5 1 2  1 0 1 7  6 2 2        x   1 1 .

8 5 0 5 2  1 1 .

8  1 4 7  1 1 .

8  1 5 1 2  1 1 .

8  1 0 1 7  1 1 .

8  6 2 2  1 1 .

8           s  3 4 .

6 5 3 5 0  1 2

2

2

2

2

2

s  3 4 .

6 5 3  5 .

8 8 7 1.5.7 l

l

l

f ( m  x )  f ( m )  f ( x )  n x  n x  0 (a)    i i i i i i  1

i  1

i  1

5

f m  x  5 2  1 1 .

8  1 4 7  1 1 .

8  1 5 1 2 1 1 .

8  1 0 1 7 1 1 .

8  6 2 2  1 1 .

8           

(b)

i i i  1

 5  9 .

8  1 4  4 .

8  1 5 .

2 1 0 5 .

2  6 1 0 .

2           4 9  6 7 .

2  3  5 2  6 1 .

2  0

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1.5.8

(x x) x 2xxx x 2xx x n

n

n

2

i

i1

2 i

i1

2

i

i1

n

2 i

n

2

i1

i

i1

2

  xi   n n n  xi2 2nx2 nx2  xi2 nx2  xi2 n i1   n  i1 i1 (a) i1     n

2

n  xi   n  2 i1  xi  n i1 (b) n

( x  x )  1 0 5 9 2 .

4 6 78  0  9 2 .

4 6 7  . . .1  1 5  9 2 .

4 6 79  5  9 2 .

4 6 7  9 7 3 7 .

7 3 3 3        2

2

2

i i  1

2

n   x 2  i   n 1 3 8 7   i  1 2 x  1 3 7 9 8 9   9 7 3 7 .

7 3 3 3  i n 1 5 i  1

1.5.9 3 2

x 

i 1 0 5 9 .3 6 xi1  3 3 .1 0 5 3 2 3 2 (a) 132 4 8 8 .3 3 2 2 5 s2   1 7 7 .0 4 3 xi 33.105  3 1i1 3 1 ra n g e5 3 .5 0 5 .3 14 8 .1 9

24.75  25.44  25.095 2 42.19  43.25 Q3   42.72 2 32  32  32 (b) M  2 IQR  42.72  25.095  17.625 Q1 

LL  25.095 1.5 17.625  1.3425 LL  42.72  1.5 17.625  69.1575 There are no outliers.

2

2

10

20

30

40

50

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(c)

4 0

2

Frequency

6

8

Histogram of y

0

(d)

10

20

30

40

50

60

y

(e)

x 33.105

x s 1 9 .8 0 ,4 6 .4 1 21 data point (65.625%) fall within 1 SD, empirical rule = 68% x 2 s .4 9 ,5 9 .7 2 6 31 data point (96.875%) fall within 2 SD, empirical rule = 95% x  3 s 6 .8 1 ,7 3 .0 2  32 data point (100%) fall within 3 SD, empirical rule = 99.7%

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1.5.10 4 0

x 

i 3 3 3 .6 xi1  8 .3 4 4 0 4 0 (a) 1 40 4 4 .3 7 6 2 9 s2  2 4 .2 1 5 xi 8.34  3 9i1 3 9 ra n g e1 7 .2.51 6 .7

3.7  3.6  3.65 2 12.8  12.3 Q3   12.55 2 8.3  7.9 M   8.1 (b) 2 IQ R  12.55  3.65  8.9 Q1 

L L  3.65  1.5  8.9    9.7 L L  12.55  1.5  8.9   25.9

0

5

10

15

T here are no outliers .

(c)

6 4 0

2

Frequency

8

10

Histogram of y

0

(d)

5

10 y

15

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(e) x  8.34

x s 3 .4 2 ,1 3 .2 6 24 data point (60%) fall within 1 SD, empirical rule = 68% x 2 s 1 .5 ,1 8 .1 8  40 data point (100%) fall within 2 SD, empirical rule = 95% x 3 s 6 .4 2 ,2 3 .1  40 data point (100%) fall within 3 SD, empirical rule = 99.7% 1.5.11 2   1 1 , 0 0 0 2 1 1 1 , 0 0 0 x   1 1 0 , s   1 , 9 0 0 , 0 0 0    6 9 6 9  6 9 7 1 0 0 1 0 0  1 1 0 0 (a)  

s  6 9 6 9 .

6 9 7  8 3 .

4 8 4 7 xs  .

6 8 ( 4 0 0 ,0 0 0 ) 2 7 2 ,0 0 0  2 s  .

9 5 ( 4 0 0 ,0 0 0 ) 3 8 0 ,0 0 0 (b) x x  3 s  .

9 9 7 ( 4 0 0 ,0 0 0 ) 3 9 8 ,8 0 0 1.5.12

Q1  39 Q3  46 42  40  41 2 IQR  46  39  7

M (a)

10

x

x i 1

i



418  41.8 10

10 1 10 1149.6 2 s 2    xi  8.34   127.733 9 i 1 9 s  127.733  11.302 1 0

( x  x )  4 1 8  1 0 4 1 .

8  4 1 8 4 1 8  0    i i  1

20

30

40

50

60

(b)

(c)

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IQ R  7 L L 3 9  1 .57 8 .5  2 (d)

L L4 6  1 .57 6 .5  5 T h e r ea r etw oo u tlie r s1 8a n d6 0 .

1.5.13 (a)

x 112.33.7433 x 30

i1 i

30 30 30 1 2 s2   xi 3.7433 3.502 29i1 s 3.5021.871 (b) Frequency table Class Interval 1 0-1.6 2 1.7-3.3 3 3.4-5 4 5.1-6.7 5 6.8-8.4 (c) Grouped data:

Frequency 4 10 9 5 2

Mi .8 2.5 4.2 5.9 7.6

Mi∙fi 3.2 25 37.8 29.5 15.2

4 ( .

8 )  1 0 ( 2 .

5 )  9 ( 4 .

2 )  5 ( 5 .

9 )  2 ( 7 .

6 ) x   3 .

6 9 3 0 2 2 2 2 2 21   s  4 0 .

8  3 .

6 9  1 0 2 .

5  3 .

6 9  9 4 .

2  3 .

6 9  5 5 .

9  3 .

6 9  2 7 .

6  3 .

6 93  .

6 2             2 9 s  3 .

6 2  1 .

9 0 The results from the grouped data are similar to the actual data.

1.5.14 (a)

x 181460.467 x 30

i 1i

30 30 30 1 2 s2   xi 60.467 685.085 29i1 s 685.08526.174 (b) Frequency table Class 1 2 3 4 5

Interval 0-20 20-40 40-60 60-80 80-100

Frequency 1 8 6 5 10

Mi 10 30 50 70 90

Mi∙fi 10 240 300 350 900

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(c) Grouped data:

1 0  2 4 0  3 0 0  3 5 0  9 0 0 x   6 0 3 0 2 2 2 2 2 2 1   s  1 0  6 0 8 3 0  6 0 6 5 0  6 0 5 7 0  6 0 2 9 0  6 0  6 8 2 .

7 5 9             2 9 s  6 8 2 .

7 5 9  2 6 .

1 3 The results from the grouped data are similar to the actual data.

1.5.15

w4 L  25 fm 139615

Fb 178859 n514661 w M  L (.

5 n  F )  27 .

24822 b f m 1.5.16 (a)

8 1 5 9 .

5  1 1 1 6 9 .

5  1 8 1 7 9 .

5  9 1 8 9 .

5  4 1 9 9 .

5         x   1 7 7 .

5 5 0 2 2 2 2 2 21   s  8 1 5 9 .

5  1 7 7 .

5  1 1 1 6 9 .

5  1 7 7 .

5  1 8 1 7 9 .

5  1 7 7 .

5  9 1 8 9 .

5  1 7 7 .

5  4 1 9 9 .

5  1 7 7 .

5             4 9  1 3 4 .

6 9 4 s  1 3 4 .

6 9 4  1 1 .

6 0 6 (b) L  175 , f m  18 , w  9

Fb  19 , n  50 w M  L  (.

5 n  F )  178 b f m 1.5.17

3 8 1 0  3 1 2 9 .

5  5 9 4 9 .

5  4 5 6 9 .

5  7 8 9 .

5          x   4 4 .

2 7 2 1 8 0 2 2 2 2 2 21   s  3 8 1 0  4 4 .

2 7 2  3 1 2 9 .

5  4 4 .

2 7 2  5 9 4 9 .

5  4 4 .

2 7 2  4 5 6 9 .

5  4 4 .

2 7 2  7 8 9 .

5  4 4 .

2 7 2             4 9  5 3 6 .

1 4 6 s  5 3 6 .

1 4 6  2 3 .

1 5 5 (b) L  40 , f m  59 , w  19 , Fb  69

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