考满分数学群四章

GRE数学主讲：张巍考满分GRE微信公众号：考满分GRE张巍老师全球G友群：190117008目标只有一个：170第四章

资料分析第一节

描述资料的图形方法4.11变量和频率

variable变量frequency

(count)频率frequencydistribution频率分布relativefrequency相对频率relativefrequencydistribution相对频率分布例：1

2

0

4

1

3

3

1

2

0

4

5

2

3

2

3

2

4

1

2

3

0

2

3

14.12条形图和柱状图bargraph条形图图histogram柱状图

4.13分段柱状图segmentedbargraph分段柱状图4.14饼状图circlegraph饼状图

4.15散点图scatterplot散点图univariate单变量bivariate二变量4.16时间曲线图timeplot(timeseries):时间曲线图第二节

描述资料的数量方法4.21集中趋势的计量centraltendency集中趋势arithmeticmean算术平均数average平均数mean平均数median中数mode众数练习：124599

求

arithmeticmean,median,modeweightedmean加权平均数练习：下面是一个同学的某一科的考试成绩：平时测验80，期中90，期末95学校规定的科目成绩的计算方式是：1)

平时测验占20%2)

期中成绩占30%3)

期末成绩占50%那么加权平均成绩是：例1：b-3,b-1,b+2,b+3,b+4Themedianofthefivetermslistedaboveis5,wherebisaconstant.Whatistheaverage(arithmeticmean)ofthefiveterms?A.3B.4C.5D.6E.7例2：If1+x+x2+x3=60,thentheaverage(arithmeticmean)ofx,x2,x3,x4isequaltowhichofthefollowing?A.12xB.15xC.20xD.30xE.60x例3：Inasetof24positiveintegers,12oftheintegersarelessthan50.Therestaregreaterthan50.

QuantityAQuantityBThemedianof

50the24integersA.

Quantity

A

is

greater.B.

Quantity

B

is

greater.C.

The

two

quantities

are

equal.D.

The

relationship

cannot

be

determined

from

the

information

given.

4.22位置的计量quartiles四分位数firstquartile第一四分位数secondquartile第二四分位数thirdquartile第三四分位数interquartilerange四分位距percentiles百分位数

例1：ThemedianeofagroupofCollegeCgraduatessixmonthsaftergraduationwas$3,000higherthanthemedianeofagroupofCollegeDgraduatessixmonthsaftergraduation.QuantityAQuantityBThe75thpercentileofThe75thpercentileofthetheesofthegroupesofthegroupofofCollegeCgraduatesCollegeDgraduatessixsixmonthsaftergraduationmonthsaftergraduationA.QuantityAisgreater.B.QuantityBisgreater.C.Thetwoquantitiesareequal.D.Therelationshipcannotbedeterminedfromtheinformationgiven.4.23分散度的计量measuresofdispersion分布的计量range极差outlier异常值boxplotsorbox-and-whiskerplots箱型图练习：variance方差standarddeviation=populationstandarddeviation标准方差

samplestandarddeviation样本方差练习：求下列数据的方差和标准方差

1,

2,

5,

8,

14例1：Alistofnumbershasameanof8andastandarddeviationof2.5.Ifxisanumberinthelistthatis2standarddeviationsabovethemean,whatisthevalueofx?例2：Therangeoftheheightsofthefemalestudentsinacertainclassis13.2inches,andtherangeoftheheightsofthemalestudentsintheclassis15.4inches.Whichofthefollowingstatementsindividuallyprovide(s)sufficientadditionalinformationtodeterminetherangeoftheheightsofallthestudentsintheclass?Indicateallsuchstatements.A.

The

tallest

male

student

in

the

class

is

5.8

inches

taller

than

the

tallest

female

student

in

the

class.B.

The

median

height

of

the

male

students

in

the

class

is

1.1

inches

greater

than

the

median

height

of

the

female

students

in

the

class.C.

The

average

(arithmetic

mean)

height

of

the

male

students

in

the

class

is

4.6

inches

greater

than

the

average

height

of

the

female

students

in

the

class例3：ThenumbersindatasetShaveastandarddeviationof5.Ifanewdatasetisformedbyadding3toeachnumberinS,whatisthestandarddeviationofthenumbersinthenewdataset?A.2B.3C.5D.8E.15例4：ListKconsistsofthenumbers-10,-5,0,5and10.WhichofthefollowinglistsofnumbershavethesamerangeasthenumbersinlistK?Indicateallsuchlists.A.

-15,

-1,

0,1,

15B.

-7,

-4,

-2,

1,

13C.

0,

1,

2,

5,

8,

10D.

2,

3,

5,

15,

19,

22E.

4,

5,

6,

24第三节

计数方法4.31集合set集合membersorelements元素finiteset有限集infiniteset无限集nonempty非空集Intersection

交集union并集disjointormutuallyexclusive不相交或互斥例1：SetAconsistsof40integers,andsetBconsistsof150integers.ThenumberofintegersthatareinbothsetAandsetBis20.

QuantityAQuantityBThe

total

number

of

integersthat

are

in

set

A

or

set

B,

170

or

bothA.

Quantity

A

is

greater.B.

Quantity

B

is

greater.C.

The

two

quantities

are

equal.D.

The

relationship

cannot

be

determined

from

the

information

given.例2：SetSconsistsof5objects.

QuantityAQuantityBThenumberofsubsets

The

number

of

subsetsofsetSthatconsistof

of

Set

S

that

consist

of

1object

4

objectsA.

Quantity

A

is

greater.B.

Quantity

B

is

greater.C.

The

two

quantities

are

equal.D.

The

relationship

cannot

be

determined

from

the

information

given.4.32韦恩图Venndiagram韦恩图universalset全集考点：inclusion-exclusionprinciple容斥原理4.33乘法原理multiplicationprinciple乘法原理

4.34排列和阶乘permutationorarrangement排列

factorial阶乘

考点：1)阶乘的换算公式阶乘换算公式n!=n(n-1)(n-2)…(2)(1)=n(n-1)!=n(n-1)(n-2)!一般规律：

阶乘补充：0!=1考点：2)排列的运算排列公式：练习：Howmanydifferentfive-digitpositiveintegerscanbeformedusingthedigits1,2,3,4,5,6,and7ifnoneofthedigitscanoccurmorethanonceintheinteger?4.35组合combination组合考点：1)组合的公式：练习：Supposeyouwanttoselecta3-personcommitteefromagroupof9students.Howmanywaysaretheretodothis?例1：A

manager

is

forming

a

6-person

team

to

work

on

a

certain

project.

From

the

11

candidates

available

for

the

team,

the

manager

has

already

chosen

3

to

be

on

the

team.

In

selecting

the

other

3

team

members,

how

many

different

combinations

of

3

of

the

remaining

candidates

does

the

manager

have

to

choose

from?A.

6B.

24C.

56D.

120E.

462例2：Forthebiologicalsciencesandhealthsciencesfacultycombined,1/3ofthefemaleand2/9ofthemalefacultymembersaretenuredprofessors.Whatfractionofallthefacultymembersinthosetwofieldscombinedaretenuredprofessors?例3：Areadinglistforahumanitiescourseconsistsof10books,ofwhich4arebiographiesandtherestarenovels.Eachstudentisrequiredtoreadaselectionof4booksformthelist,including2ormorebiographies.Howmanyselectionsof4bookssatisfytherequirements?A.90B.115C.130D.144E.195例4：Inagraduatingclassof236students,142tookalgebraand121tookchemistry.Whatisthegreatestpossiblenumberofstudentsthatcouldhavetakenbothalgebraandchemistry?

students第四节

概率4.41等可能事件（古典概率）probabilityexperiment概率实验randomexperiment

随机试验samplespace样本空间event事件probability概率randomselection随机挑选考点：1)等可能事件（古典概率）的计算公式：练习：盒子里面共有3个黑球，4个白球，6个红球，从盒子里面随机取出一个球，此球是红球的概率是多少？例1：Eightpointsareequallyspacedonacircle.If4ofthe8pointsaretobechosenatrandom,whatistheprobabilitythataquadrilateralhavingthe4pointschosenasverticeswillbeasquare?A.1/70B.1/35C.1/7D.1/4E.1/24.42互斥事件和对立事件mutuallyexclusive互斥事件complementaryevents对立事件考点：1)互斥事件的概率公式：P(A+B)=P(A)+P(B)2)对立事件的概率公式：P(A+B)=P(A)+P(B)=14.43相互独立事件independentevent相互独立事件考点：1)相互独立事件同时发生的概率：P(A·B)=P(A)·P(B)练习：连续扔两次色子，第一次1朝上且第二次5朝上的概率是多少？例1：Acertainjarcontains60jellybeans—22white,18green,11yellow,5red,and4purple.Ifajellybeanistobechosenatrandom,whatistheprobabilitythatthejellybeanwillbeneitherrednorpurple?A.0.09B.0.15C.0.54D.0.85E.0.91例2：Ofthe700membersofacertainorganization,120arelawyers.Twomembersoftheorganizationwillbeselectedatrandom.Whichofthefollowingisclosesttotheprobabilitythatneitherofthemembersselectedwillbealawyer?A.0.5B.0.6C.0.7D.0.8E.0.9例3：pistheprobabilitythateventEwilloccur,andsistheprobabilitythateventEwillnotoccur.

QuantityAQuantityB

p+s

psA.

Quantity

A

is

greater.B.

Quantity

B

is

greater.C.

The

two

quantities

are

equal.D.

The

relationship

cannot

be

determined

from

the

information

given.例4：Ofthe20lightbulbsinabox,2aredefective.Aninspectorwillselect2lightbulbssimultaneouslyandatrandomfromthebox.Whatistheprobabilitythatneitherofthelightbulbsselectedwillbedefective?Giveyouranswerasafraction.第五节资料分布，随机变量，概率分布4.51资料分布distributionsofdata资料的分布distributioncurve（densitycurveandfrequencycurve）密度曲线4.52随机变量randomvariable随机变量4.53概率分布probabilitydistribution概率分布discreterandomvariables离散型随机变量expectedvalue=meanoftherandomvariableX数学期望考点：1)数学期望的计算公式：数学期望=每个变量的值乘以每个变量概率，再求和。练习：如图，计算随机变量X的数学期望4.54正态分布approximatelynormallydistributed近似正态分布continuousprobabilitydistribution连续型概率分布normaldistribution正态分布standard

normal

distribution标准正态分布考点1：正态分布的特点1)Themean,median,andmodeareallnearlyequal.2)Thedataaregroupedfairlysymmetricallyaboutthemean.3)Abouttwo-thirdsofthedataarewithin1standarddeviationofthemean.4)

Almostallofthedataarewithin2standarddeviationsofthemean.考点2：标准正态分布的基本性质1)

The

mean,

median,

and

mode

are

all

equal.2)

The

data

are

grouped

fairly

symmetrically

about

the

mean3)

About

68%

of

the

data

are

within

1

standard

deviation

of

the

mean.4)

About

96%

of

the

data

are

within

2

standard

deviations

of

the

mean.练习：一个学校所有的学