IB Diploma Program Mathematics Course Companion Higher Level Option: Statistics [1 ed.] 0198304854, 9780198304852

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O X

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HI GH E R

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S TAT I S T I C S C O U R S E

C O M PA N I O N

Josip Harcet

Lorraine Heinrichs

Palmira Mariz Seiler

Marlene Torres Skoumal

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Great

Clarendon

Oxford

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Acknowledgements

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v

Contents

Chapter

1

Exploring

Introduction

1.1

Probability

Cumulative

Discrete

1.2

Other

Expected

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Review

Chapter

2

2.1

of

Limit

Biased

and

32

and

Central

Limi t

probability

Case

I:

Case

II:

of

of

single

two

or

variable

more

independent

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42

variables

normal

47

random

variables

mean

64

68

statistical

analysis

how

can

methods

we

make

78

sense

of

the

mean

and

variance

of

a

normal

81

denition?

for

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82

mean

inter val

inter val

for

inter val

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84

μ

for

when

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testing

for

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when

σ

is

unknown

testing

for

test

exercise

matched

pairs

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101

105

109

errors

distribution

Two-tail

90

100

testing

II

unknown

85

97

testing

Type

79

80

for

Hypothesis

and

data?

estimates

Condence

Normal

58

74

inter vals

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40

41

Theorem

Condence

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Review

Theorem

models

42

estimators

Condence

vi

25

variable

Condence

18

20

and

a

information:

well-dened

I

variable

algebra

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Estimators

Type

random

exercise

random

3.4

algebra

distribution

Unbiased

3.3

geometric

variables

distributions

combination

Central

Review

A

a

36

of

3

of

functions

transformation

linear

Introduction

12

variance

Linear

Chapter

5

5

distribution

of

The

3.2

and

4

12

transformation

Sampling

decisions

quantities

Linear

2.3

informed

function

independent

Normal

2.2

make

exercise

Expectation

A

3.1

value

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Introduction

to

2

distributions

generating

sum

tool

distri butions

distribution

binomial

Probability

a

probabi li ty

continuous

probability

Negative

as

distribution

and

Geometric

1.3

further

112

for

a

one-tail

test

115

116

117

Chapter

4

Introduction

4.1

Statistical

Bivariate

modeling

122

distributions

123

Correlation

124

Correlation

Sampling

4.2

Covariance

4.3

Hypothesis

and

causation

128

distributions

130

135

Proper ties

of

covariance

136

testing

138

Introduction

t-Statistic

4.4

Linear

for

138

dependence

regression

Review

exercise

of

X

and

Y

139

141

152

Answers

156

Index

161

vii

Exploring

further

1 probability

distributions

CHAPTER

OBJECTIVES:

Cumulative

7.1

distribution

Geometric

distribution.

Probability

generating

Using

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Find

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i

−

0



2

3

4

0.3

0.

0.3

0.

0.05

0.5

i

P(X

=

x ) i

⎧5

Mode

(X )

=

3,

because

P(X

=

3)

=

b

which

is

the

highest

probability

of

the

P( X

=

x )

=

⎨

⎪ ⎩

four

random

Median,

P( X

≤

m

1)

=

variables.

=

2

since

0 .3

and

P( X

≤

2)

=

0 .55

4

=

E (X )

=

∑

x

p i

i

i =1

= 1 × 0. 3

+

2 × 0. 25

+ 3 × 0. 35

+

4 × 0. 1 =

2. 25

4

2

σ

=

Var ( X

)

=

∑

2

x

p i

−

μ

i

i =1

2

=

1

2

(0.3)

+

2

2

(0.25)

+

3

2

(0.35)

+

4

2

(0.1)

2

=

2

6.05 − 2.25

Exploring

further

=

0.994

probability

distributions

−

x

,

⎪

0.35,

0.225

x

=

1 ,

2,

3,

10

0,

otherwise

4

2

Find

the

mode,

standard

random

median,

deviation

variable,

function

of

given

the

by

a

of

a

e.g.

mean

the

discrete

and

2

continuous

probability

random

Find

the

mode,

standard

density

variable X

random

is

median,

deviation

variables

probability

=

(x )

dened

density

⎧ 3

f

x,

0

≤

x

≤

a

2

f

(x )

=

,

0

=

2

because

it

has

the

maximum

0,

⎪ b

at

the

≤

x

≤

2

elsewhere

elsewhere

π

⎧

point

given

2

⎪

0,

(X )

the

16

⎩

Mode

by

function:

x

⎨ 4

⎨ 2

⎩

and

continuous

3

⎪

⎪

mean

the

formula

⎧ 1 ⎪

of

end

of

the

f

(x )

=

cos

(

)

2x

,

π ≤

x

≤

4

⎨

4

inter val. ⎪

m

0,

⎩

elsewhere

2

1

∫

Median,

f

(x )

dx

m

=

1

⇒

=

2

0

⇒

4

m

=

2 ⎧

2

6

, 3

⎪

x

≤

≤

6

2 2

c

2

1

μ

=

E (X )

=

∫

xf

(x)dx

(x )

=

⎨ x

4

2

x

=

f

dx

⎪

=

∫

0,

⎩ 2

0

elsewhere

3

0

2

2

=

Var

(

)

X

=

x

2

f

( x )

−

∫ 0

2

2

1

⎛

3

x

∫

dx

−

4

⎜

2

⎝

16

⎞

=

⎟ 3

2

−

2

= 9

⎠

3

0

3

Find

the

series

sum

by

of

using

an

the

innite

geometric

3

a

1

+

u

1

+

u

2

+

...

=

, 0

the

sum

geometric

formula

u

u

Find


2

⎛

5 ⎞

⎜

⎟

⎝

6

⎛ 1 ⎞

⇒

log

⎠

⎜ ⎝

⎟ 2

⎛

>

n

×

log

⎠

5

Logarithms

⎝

⎛

Method

f orget

to

the

inequality

symbol.

3

80

⇒

n

=

4

5 ⎞

⎜ ⎝

don’t

⎠

= ⎛

so

⎟ 2

>

log

negative

1 ⎞

⎜ ⎝

n

are

6

reverse

log

here

⎜

⎟ 6

⎠

II

Use

the

cumulative

distribution

function

on

the

Scratchpad

GDC.

y

2

step

1 f1(x)=geomCdf

(

6

)

tracing 2

(4, 0.018)

x

0 2

3

4

5

6

–1

f1:

(4, 0.018)

–2

P( X

16

≤

n)

Exploring

>

0 .5

⇒

further

n

=

function

the

values

and

1 , 1, x

1

1

Since

4

probability

distributions

the

graph.

we

are

can

discrete

nd

the

we

obtain

solution

by

a

Exercise

1

2

3

Given

1B

that

X

a

p

=

0.6,

b

p

=

0.14,

c

p

=

0.5,

d

p

=

0.88,

Given

that

a

P (X

≤

b

P (X

>

c

P (5

d

P (1

Mario

k

k

=

k

4)

6)

p

=

0.3




k )

=

q

+

,

k ∈ 

.

Chapter

1

17

Investigation

Let

X

∼

Geo( p ) .

Calculate

the

a

p

=

0.4,

P( X

>

5 |X

>

3),

b

p

=

0.7,

P( X

>

6 |X

>

2 ),

c

p

=

0.12,

Make

and

a

tr y

this

to

need

for

prove

to

the

of

a

about

we

will

n

P( X

5),

the

lear n

>

4)

P( X

form

>

of

of

how

to

a

7)

nd

of

the

conditional

random

expected

In

order

sequences.

a

the

and

simple

conjecture.

geometric

variable.

terms

between

your

geometric

consecutive

probabilities:

2)

connection

random

innite

>

P( X

variance

geometric

of

>

general

and

manipulate

sum

12 | X

the

value

section

variance

>

conjecture

Expected

In

P( X

following

to

value

do

Recall

geometric

variable

so

and

you

the

the

will

formula

sequence:

n +1

2

1

+

q

+

3

q

+

1

n

q

+ ... +

q

q

=

, 1

When

n

becomes

between

−

Therefore

and

we

an

,

can

extremely

the

where

|q|




c

P(5

≤

d

P(8