Exe

rcis

esQ 6

For E

xerc

ises

1 to

A, i

dent

ify

the

popu

lati

on, t

he p

aram

eter

,th

e sa

mpl

e, a

nd t

he st

atis

tic

in e

ach s

etti

ng.

) 1

j S

top

smok

ingl

A ra

ndom

sam

ple

of 1

000

peop

lew

ho s

igne

d a

cart

! say

ing

they

inte

nded

to

quit

smok

-in

g w

ere

cont

acte

d .n

ine

mon

ths

late

r It

turn

ed o

utth

at 2

.10

(21%

) of

the

sam

pled

ind

ivid

uals

had

not

smok

ed o

ver .

the

past

six

mon

ths.

.

A

<yv

|_3.

| Hot

tur

key

Top

i is

cook

ing

a la

rge

turk

ey b

reas

t for

a ho

liday

mea

l. H

e w

ants

to b

e su

re th

at th

e tu

rkey

is s

afe

to e

at, w

hich

req

uire

s a

min

imum

inte

rnal

tem

pera

ture

of

165°

F. T

om u

ses

a th

erm

omet

er to

mea

sure

the

tem

pera

ture

of t

he tu

rkey

mea

t at f

our

rand

omly

-cho

sen

poin

ts. T

he m

inim

um r

eadi

ng in

• th

e sa

mpl

e is

170

°F.

Trt

£

4-

pore

ach

bold

face

nu

mbe

r in

Exe

rcis

es5t

o& (

1) st

ate

whe

ther

it is

a p

aram

eter

or a

sta

tistic

and

(2)

use

app

ropr

i-a

te n

otat

ion

to d

escr

ibe

each

num

ber,

for

exam

ple,

p =

0.6

5.

Get

you

r be

arin

gs A

larg

e co

ntai

ner

of b

all

bear

ings

has

mea

n di

amet

er 2

.500

3 ce

ntim

eter

s (c

m).

Thi

sis

with

in th

e sp

ecif

icat

ions

for a

ccep

tanc

e of

the

cont

aine

r by

the

purc

hase

r. B

y ch

ance

, an

insp

ecto

rch

oose

s 10

0 be

arin

gs f

rom

the

cont

aine

r th

at h

ave

rnga

njia

met

er 2

.500

9 cm

. Bec

ause

this

is o

utsi

de th

esp

ecifi

ed l

imits

, the

con

tain

er i

s mis

take

nly

reje

cted

.

Flo

rida

vot

ers

Flor

ida

has

play

ed a

key

role

in r

ecen

tpr

esid

entia

l el

ectio

ns. V

oter

regi

stra

tion

reco

rds

show

that

41%

of F

lori

da v

oter

s ar

e re

gist

ered

asT

jem

o-^r

atsj

jb te

st a

rand

om d

igit

dial

ing

devi

ceT

you

use

itto

cal

l 25

0 ra

ndom

ly c

hose

n re

side

ntia

l te

leph

ones

inFl

oTid

a. O

f the

reg

iste

red

vote

rs c

onta

cted

, 33%

are

regi

ster

ed D

emoc

rats

.

y — ? A F

?c.

-

I I

VW

e

^\,<

all g

irls

Acc

ordi

ng to

the

Nat

iona

l C

ente

r fo

r[e

alth

Sta

tistic

s, th

e di

stri

butio

n of

hei

ghts

for

16-y

ear-

old

fem

ales

is

mod

eled

wel

l by

a N

orm

alde

nsity

cur

ve w

ith m

ean

fi =

64

inch

es a

nd s

tand

ard

Dev

iatio

n a ~

2-5

inch

es. T

o se

e if

this

dis

trib

utio

nap

plie

s at

thei

r hi

gh s

choo

l, an

AP

Sta

tistic

s cl

ass

take

s an

SR

S of

20

of th

e 30

0 16

-yea

r-ol

d fe

mal

es a

t/-

the

scho

ol a

nd m

easu

res

thei

r he

ight

s. W

hat v

alue

s . •

f of

the

sam

ple

mea

n x

wou

ld b

e co

nsis

tent

with

the

\pop

ulat

ion

dist

ribu

tion

bein

g N

(64,

2.5

)? T

o fi

nd o

ut,

~jw

e us£

d F

atho

m s

oftw

are

to s

imul

ate

choo

sing

250

-1 SR

Ss o

f siz

e n

= 2

0 st

uden

ts fr

om a

pop

ulat

ion

that

is/

N(6

4, 2

.5).

The

fig

ure

belo

w is

a d

otpl

ot o

f th

ej

sam

ple

mea

n he

ight

x o

f the

stu

dent

s in

the

sam

ple.

H o c

u f v1

E <c;

i r

s/h^

j«-j

/\?6-

• oo

«••

•• •

\ tx

>e<)

Da»

*«

« I

o

oaaa

oB <

K»«

»n

I•

«e

*s»*

M«M

OB

<w •

•

eo

omoo

* «

e>o

BOB

09 «

• a

OS

OS

SW ••••

•BTO

0S®

@>

O

•B«o

e»«M

MiiM

H)«

t»»«

>«aB

ca

6ZO

62

.5

63.0

63

.5

64.0

64

.5

65.0

65

.5

66.0

\>OT

(a)

IslH

is th

e sa

mpl

ing

dist

ribu

tion

of x

? Ju

stify

you

ran

swer

.

(b)

Des

crib

e th

e di

stri

butio

n. A

re th

ere

any

obvi

ous

outli

ers?

(c)

Supp

ose

that

the

aver

age

heig

ht o

f the

20

girl

s in

the

clas

s's a

ctua

l sam

ple

is x

—

64.

7. W

hat w

ould

you

conc

lude

abo

ut th

e po

pula

tion

mea

n he

ight

p f

orth

e 16

-yea

r-ol

d fe

mal

es a

t the

sch

ool?

Exp

lain

.

12.\l

girl

s R

efer

to E

xerc

ise

10.

(a)

Mak

e a

grap

h of

the

popu

latio

n di

stri

butio

n.

(b)

Sket

ch a

pos

sibl

e gr

aph

of th

e di

stri

butio

n of

sam

ple

data

for t

he S

RS

of si

ze 2

0 ta

ken

by th

eAP

Stat

istic

s cl

ass.

.

Z.&

!

u-'D

oU

V»^

\

1 17

JIR

S a

udit

s T

he I

nter

nal

Rev

enue

Ser

vice

pla

ns to

ex-

amin

e an

SR

S o

f ind

ivid

ual f

eder

al i

ncom

e ta

x re

turn

sfr

om e

ach

stat

e. O

ne v

aria

ble

of in

tere

st is

the

prop

or-

tion

of r

etur

ns c

laim

ing

item

ized

ded

ucti

ons.

The

tot

alnu

mbe

r of

tax

retu

rns

in e

ach

stat

e va

ries

fro

m o

ver

15m

illio

n in

Cal

ifor

nia

to a

bout

240

,000

in W

yom

ing.

(a)

Will

the

sam

plin

g va

riab

ility

' of t

he s

ampl

e pr

opor

-tio

n ch

ange

fro

m s

tate

to s

tate

if a

n SR

S o

f 200

0 ta

xre

turn

s is

sele

cted

in

each

sta

te?

Exp

lain

you

r ans

wer

.

(b)

Will

the

sam

plin

g va

riab

ilit

y of

the

sam

ple

pro-

port

ion

chan

ge f

rom

sta

te t

o st

ate

if a

n SR

S o

f 1%

of

all t

ax re

turn

s is

sele

cted

in

each

sta

te?

Exp

lain

you

ran

swer

.F

or

Re

-K^n

^ /v

««&

.

Pre

dict

the

elec

tion

Jus

t bef

ore

a pr

esid

enti

al e

lec-

tion,

a n

atio

nal

opin

ion

poll

incr

ease

s th

e si

ze_o

f its

_w

eekl

y ra

ndom

sam

ple

from

the

usu

al 1

50Q

peo

ple

to40

00 p

eopl

e.

(a)

Doe

s th

e la

rger

ran

dom

sam

ple

redu

ce th

e bi

as o

fth

e po

ll r

esul

t? E

xpla

in.

(b)

Doe

s it

red

uce

the

vari

abili

ty o

f the

res

ult?

Exp

lain

.

re<i

oc<L

-H

-teo

fc S

A^

,OL

|-N

!G

si'z

/=/O

OT

<x

oll

^TH

o^e.

p

eop

lern

-ru

/-rn

cV\A

/He

.

\.

19.

Bia

s an

d va

riab

ility

T

he f

igur

e be

low

sho

ws

his-

togr

ams

of fo

ur s

ampl

ing

dist

ribu

tions

of

diff

eren

tst

atis

tics

inte

nded

to e

stim

ate

the

sam

e pa

ram

eter

.

^"f

t"

(a)

Whi

ch s

tatis

tics

are

unbi

ased

est

imat

ors?

Just

ifyyo

ur a

nsw

er.

(b)

Whi

ch s

tatis

tic d

oes

the

best

job

of e

stim

atin

g th

epa

ram

eter

? E

xpla

in.

I Pop

ulat

ion

para

met

er

(d)

v, 0

20.

A s

ampl

e of

teen

s A

stud

y of

the

heal

th o

f te

enag

ers

plan

s to

mea

sure

the

blo

od c

hole

ster

ol l

evel

s of

an

SRS

of 1

3- to

16-

year

-old

s. T

he r

esea

rche

rs w

ill r

epor

tth

e m

ean

x fr

om t

heir

sam

ple

as a

n es

timat

e of

the

mea

n ch

oles

tero

l le

vel f

j, in

thi

s po

pula

tion.

(a)

Exp

lain

to

som

eone

who

kno

ws n

o st

atis

tics

wha

tit

mea

ns to

say

that

x

is a

n un

bias

ed e

stim

ator

of p

.

(b)

The

sam

ple

resu

lt X

is

an u

nbia

sed

estim

ator

of th

e po

pula

tion

mea

n p

no m

atte

r w

hat s

ize S

RS

the

stud

y ch

oose

s. E

xpla

in to

som

eone

who

kno

ws

no s

tatis

tics

why

a la

rge

rand

om s

ampl

e gi

ves

mor

etr

ustw

orth

y re

sults

than

a s

mal

l ra

ndom

sam

ple.

"t V

kO

F

t>

Exe

rcis

es

ie ca

ndy

mac

hine

Sup

pose

a la

rge

cand

y m

achi

neha

s 15

% o

rang

e ca

ndie

s. U

se F

igur

e 7.

13 '

to h

elp

answ

er th

e fo

llow

ing

ques

tions

.

(a)

Wou

ld y

ou b

e su

rpri

sed

if a

sam

ple

of 2

5 ca

ndie

sR

om th

e m

achi

ne c

onta

ined

-8-e

rang

e ca

ndie

s (t

hat's

32%

ora

nge)

? H

ow a

bout

5 o

rang

e ca

ndie

s (20

%or

ange

)? E

xpla

in.

Lo

o*

,

F~.iL

.vr

t'l

*0 (

32.^

))(3

(b)

Whi

ch i

s mor

e su

rpri

sing

: get

ting

a sa

mpl

e of

25

cand

ies

in w

hich

32%

are

ora

nge

or g

ettin

g a

sam

ple

of 5

0 ca

ndie

s in

whi

ch 3

2% a

re o

rang

e? E

xpla

in.

TO

ITB

e

cV

C

I

<"

I -0

.1

0 0.

10.

3.

0.4

TTM

ea

n =

0.1

54

S

td D

ev =

0.0

71

0.1

't'

0.2

TTM

ea

n =

0.1

49

S

td D

ev

= 0

.048

p =

0.2

4

sa

mp

le s

ize:

25

Cu

rren

t S

amp

le:

40

0

| I"

"] C

ount

Sam

ple

s...

nu

m s

amp

les:

|4

0Q

[o!l5

p =

0.1

8

sam

ple

siz

e:

nu

m s

amp

les:

|

-4OQ

(b)

Cu

nen

t S

amp

le:

40

0

f~1 C

ount S

am

ple

s...

f" P

lot

Norm

al C

uiv

e

FIG

UR

E 7

.13

The

resu

lt of

taki

ng 4

00 S

RSs

of (

a) si

ze n =

25 a

nd (b

) size

n =

50

cand

ies

from

a la

rge

cand

y m

achin

e in

whic

h 15

%of

the

cand

ies

are

oran

ge.T

he d

otpl

ots

show

the

appr

oxim

ate

sam

plin

g di

strib

utio

n of

pin

eac

h ca

se.

cand

y m

achi

ne S

uppo

se a

larg

e ca

ndy

mac

hine

has

45%

ora

nge

cand

ies.

Im

agin

e ta

king

an

SRS

of 2

5ca

ndie

s fr

om t

he m

achi

ne a

nd o

bser

ving

the

sam

ple

prop

ortio

n p

of o

rang

e ca

ndie

s.

(a)

Wha

t is

the

mea

n of

the

sam

plin

g di

stri

butio

n of

p? W

hy?

:- hL

<^

— m

ck

c so

re

~rc

i'se

d.i

(b)

Fin

d th

e st

anda

rd d

evia

tion

of th

e sa

mpl

ing

dist

ribu

tion

of p

. C

heck

to

see

if th

e 10

% c

ondi

tion

is

p =

0H

^

So

met

.U

tk ,

id

(c)

Is th

e sa

mpl

ing

dist

ribu

tion

of p

app

roxi

mat

ely

Nor

mal

? C

heck

to s

ee if

the

Nor

mal

con

ditio

n is

met

(d)

If th

e sa

mpl

e si

ze w

ere

50 ra

ther

than

25,

how

wou

ld th

is c

hang

e th

e sa

mpl

ing

dist

ribu

tion

of p

?*?

lU

>o

n

BOT

J 5

0

'9

In E

xerc

ises

33

and

34,

expl

ain

why

you

im

etho

ds o

f th

is s

ectio

n to

find

the

desi

red

prob

abili

ty.

'—I

133\

c w

orke

rs A

fact

ory

empl

oys

3000

uni

oniz

ed•""

" w

orke

rs, o

f who

m 3

0% a

re H

ispa

nic.

The

15-

mem

ber

unio

n ex

ecut

ive

com

mitt

ee c

onta

ins

3 H

ispa

nics

.W

hat w

ould

be

the

prob

abili

ty o

f 3 o

r few

er H

ispa

n-ic

s if

the

exec

utiv

e co

mm

ittee

wer

e ch

osen

at r

ando

mfr

om a

ll th

e w

orke

rs?

~Th-

e

(V\V here

Stu

diou

s at

hlet

es A

uni

vers

ity is

con

cern

ed a

bout

the

acad

emic

sta

ndin

g of

its

inte

rcol

legi

ate

athl

etes

.A

stu

dy c

omm

ittee

cho

oses

an

SRS_

of50

of t

he_3

16at

hlet

es to

int

ervi

ewjn

deta

il S

uppo

se th

at 4

0% o

fth

^m

kS

^av

eTee

nld

id b

y co

ache

s to

neg

lect

thei

r st

udie

s on

at l

east

one

occ

asio

n. W

hat i

s th

epr

obab

ility

that

at l

east

15

in th

e sa

mpl

e ar

e am

ong

this

gro

up?

—*—

--^

_

(35]

Do

you

drin

k th

e ce

real

milk

?. A

USA

Tod

ay P

oll

aske

d a-

rand

om s

ampl

e of

1,0

12 tJ

:S. a

dults

wha

t the

ydo

with

the

milk

in th

e bo

wl-a

fter

they

hav

e ea

ten

the

cere

al. O

f th

e re

spon

dent

s, 6

7% sa

id t

hat t

hey

drin

kjt

Sup

pose

lhat

70%

of U

.S.

adul

ts a

ctua

lly d

rink

the

cere

al m

ilk. L

et p

be

the

prop

ortio

n of

peo

ple

in th

esa

mpl

e w

ho d

rink

the

cere

al m

ilk.

(a)

Wha

t is

the

mea

n of

the

sam

plin

g di

stri

butio

nof

p?

Why

?

(b)

Fin

d th

e st

anda

rd d

evia

tion

of th

e sa

mpl

ing

dist

ribu

tion

of p

. Che

ck to

see

if th

e 10

% c

ondi

tion

ism

et

(c)

Is th

e sa

mpl

ing

dist

ribu

tion

of p

app

roxi

mat

ely

Nor

mal

? C

heck

to s

ee if

the

Nor

mal

con

ditio

n is

met

.(d

) F

ind

the

prob

abili

ty o

f obt

aini

ng a

sam

ple

of10

12 a

dults

in w

hich

67%

or f

ewer

say

they

dri

nk th

ece

real

milk

. D

o yo

u ha

ve a

ny d

oubt

s ab

out t

he r

esul

tof

this

pol

l? h_

37.1

Do

you

drin

k th

e ce

real

milk

? W

hat s

ampl

e si

zew

ould

be

requ

ired

to

redu

ce t

he s

tand

ard

devi

atio

nof

the

sam

plin

g di

stri

butio

n to

one

-hal

f th

e va

lue y

oufo

und

in E

xerc

ise

35(b

)? Ju

stify

you

r ans

wer

.

> \K

ice

.

2

s_

Hr

-s

e~}0

A

A'

r:

,t7

.7

. ."

'•H

)=i

Z O

4^

^U

4^

One

time

ship

ping

You

r mai

l-or

der

com

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ExercisesJ \«dr\bU VcV

49. Songs on an iPod David's iPod has about 10,000* songs. The distribution of the play times for these

songs is heavily skewed to the right with a meanof 225 seconds and a standard deviation of 60 sec-onds. Suppose we choose an SRS of 10 songs fromthis population and calculate the mean play time xof these songs. What are the mean and the standarddeviation of the sampling distribution of x? Explain.

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51.1 Songs on an iPod Refer to Exercise 49. How manysongs would you need to sample if you wanted the

standard deviation of the sampling distribution of x tobe 30 seconds? Justify your answer.

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pa

n

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(514 Larger sample Suppose that the blood cholesterol "*~llevel of all men aged' 20 to 34 follows the Normal dis- \h mean n = 188 milligrams peTdecIfiter I

(mg/dl) and standard deviation cr = 41 mg/dl. __J

(a) Choose an SRS of 100 men from this population.TT71 . .—-|—— -—=— •-=—-.. f t

. . / , ' u

" N ( / tf £, 1 1

} Find the probability that x estimates p withinmg/dl. (This is the probability that x takes a value

between 185 and 191 mg/dl.) Show your work.

(fj£ Choose an SRS of 1000 men from this popula-tion. Now what is the probability that x falls within±3 mg/dl of /j,? Show your work. In what sense is thelarger sample "better"?

£ X ) =

-The o4- X =

^ ft i.s-e he. c <io ̂ a~

/56J ACT scores The composite scores of individualstudents on the ACT college entrance examination in2009 followed a Normal distribution with mean 21.1and standard deviation 5.1.(a) What is the probability, that a single studentrandomly chosen from all those taking the test scores23 or higher? Show your work.(b) Now take an SRS of 50 students who took thetest What is the probability that the mean .score x ofthese students is 23 or higher? Show your work.

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!57. What does thjCLTJay? Asked what the centrallimit theorem says, a student replies, "As you"taEelarger and larger samples from a population,the histogram of the sample values looks more andmore Normal." Is the student right? Explain youranswer. :•

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4-0 be<.u

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a

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more c

59. Songs on an iPod Refer to Exercise 49. ^--' '(a) Explain why you cannot safely calculate theprobability that the mean play time_Zis_morethan4_rninutes (240 seconds) fojju£SRS of 10 songs.̂

( (b);Suppose we take a^SRS_of_36jl>ngs instead.Explain how the central limit theorem allows us tofind the probability that the mean play time is morethan 240 seconds. Then calculate this probability.Show your work.

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A

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to

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a.

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orm a x ,

30 or

60. Lightning strikes The number of lightning strikeson a square kilometer of open ground in a year hasmean 6 and standard deviation 2.4. (These valuesare typical of much of the United States.) TheNational Lightning Detection Network (NLDN) usesautomatic sensors to watch for lightning in a randomsample of 10 one-square-kilometer plots

c.4-

= 6

(a) What are the mean and standard deviation ofx, the sample mean number of strikes per squarekilometer?(b). Explain why you cannot safely calculate the prob-ability that x < 5 based on a sample of size 10.(c) Suppose the NLDN takes a random sample of" ̂ JO-Square kilometers instead. E)q^3alrrriOTvThecentrallimit theorem allows us to find the probabilitythat the mean number of lightning strikes per squarekilometer is less than 5. Then calculate this probabil-ity. ShowyouTwofE

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63 More on insurance An insurance company knowsthat in the entire population of homeowners, themean annual loss from fire is p = $250 and the

Wf?-^- Thely right-skewed: many

policies nave ju iu^but a few have large losses. Ifthe cornDanjLsells (ROOOioHci^ can it safely baseits rates on the assumption that its ayeragejpsswillbeno greater than $275? Follow the four-step process.

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3

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Exercises 69 to 72 refer to the following setting. In the lan-^guage of government statistics, you are "in the labor force"if you are available for work and either working or actively

^eeKing.^orJ^^^ &the labor force (not of the entire population) who are unem-ployed. Here are data from the Current PppuktionSuryeyfor the civilian populatiotfage dT5 years and over in a recentjyear. The table entries are counts in tkousandTof people.

Highest education. Total populatio l̂n labor force ^mployed."27,669 ' X^-T2TTO---^'l 1,408 ~

rt \ C o \ r-e_ O <^

Didn't finish high • ;••school . . . -

Highschpp I but no...,-. .;=;: .&ffiO,r -:...-.,- 37,834;: ..> 35,857 ;.College ; . • . - . • • .. ;. - - v., •-'•^'•..-..:••. .•:::;.>;-p-

Less than bachelor's

1,977

47,556 34,439 32,977

;(l . 1 ) Findffie ;uneniplb;yment ratefor people with each level of education. How does theunemployment rate change with education?

70; Unemployment (5.1) What is the probability that akfe randomly chosen person 25 years of age or older is in

f°rce? Show your work, ^j (.

•11. Unemployment (5.3) If you know that a randomlychosen person 25 years of .age or older is a collegegraduate, what is the probability that he or she is inthe labor force? Show your work. /U

72. Unemployment(5.3) -•. Areathe events "in *he laborforce" and "college graduate"..independent? Justify'youranswer. •/• " • • ' • ' ' . •;_"*; :.

" : ' 135-133