CURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:


 Chrystal Townsend
 1 years ago
 Views:
Transcription
1 CURRICULUM INSPIRATIONS: wwwmaaorg/ci MATH FOR AMERICA_DC: wwwmathforamericaorg/dc INNOVATIVE CURRICULUM ONLINE EXPERIENCES: wwwgdaymathcom TANTON TIDBITS: wwwjamestatocom TANTON S TAKE ON MEAN ad VARIATION JULY 01 I our early grades we lear that the average of a collectio of data measuremets represets, i some way, a typical or middle value for the data For example, the average of the umbers 1,, 5, is: = 5 Geometrically, the average is the level of a sadbox after we smooth out colums of sad of heights give by the data: I a statistics class the average value of a collectio of data values is called the mea of the data (The word still meas average ) Oe deotes the mea of the data by puttig a bar over whichever letter is beig uses to deote the data For example, the mea of a1, a, ad a is: a + a + a a= 1 ad the mea of x1, x, K, x is: x= x + x + L+ x 1 If the data set is extraordiarily large ad oe does t have ay hope of determiig the mea of the full data set, the that true, but ukow, mea is usually deoted with the Greek letter µ For example, we have o hope of kowig the average height of wwwjamestatocom ad wwwgdaymathcom
2 all humas o this plaet at this very momet But we ca measure the height of 100 humas ad collect 100 data values h, h, K, h We the hope that the sample mea h approximates the true mea µ to some reasoable degree Exercise: Data values x 1, x, L, x have mea x Prove that the sum of the differece of each data value from the mea is sure to be zero: ( x x) ( x x) ( x x) = 0 L Exercise: Some texts might give the followig formula for mea: x= f x + f x + L+ f x 1 1 f + f + L+ f 1 Ca you iterpret what the symbols i this formula mea ad why the formula is correct? SIMPSON S PARADOX Two studets Albert ad Bilbert each took a sample of math questios over a series of two days There were 100questios i total ad Albert scored 65% ad Bilbert 6% overall Thus Albert proved himself a better test taker But here are the scores daybyday: FIRST DAY: Albert = 71% Bilbert = 80% SECOND DAY: Albert = 50% Bilbert = 57% So each day Bilbert did a better job tha Albert, but did ot prove to be the better testtaker after the two days combied! How is this possible? The followig table shows raw data of their test results This paradox arises because Albert ad Bilbert did ot complete the same umber of questios each day ad the averages computed are ot equally weighted This curious pheomeo is kow as Simpso s paradox ad was discovered by the Statisticia Simpso i the 1960s after examiig graduate school admissio rates for me ad wome ito UC Berkeley ASIDE: There are several other measures of a typical or cetral value of a data set The mode of a set of data values is the value i the set that occurs most ofte (if there is oe) For the te data values, 6, 5,, 1, 6, 5,, 8, the mode is The data set 5, 5, 6, 6, 9, 9,,,, has o mode The data set 1, 1, 1, 1, 5, 5, 7, 7, 7, 7, 8, 8, 9, 9, 9 is bimodal (Is the secod example quitimodal?) For oumerical data, such as colours, or letters of the alphabet, the mode is the oly measure of cetral tedecy available If we arrage the data set i icreasig order of values, the the media of the data is the middle value of the ordered sequece or the average of the two middle values if there are a eve umber of terms wwwjamestatocom ad wwwgdaymathcom
3 The media of the data set,, 5, 6, 7, 16, 16, 19, 7 is 7 The media of,,, 5, 8, 8, 10, is = 6 5 The media is a value that divides the data set ito two equally sized groups The midrage of a data set is the average of the smallest ad largest values The midrage of the data set 5, 6, 9, 9 is = 7 The midrage provides a quick estimate to a cetral value It is easy to compute, but is highly affected by extremely low or high values i the data set Exercise: a) Fid FIVE data values with: Media = 10 Mode = 10 Mea = 1000 b) Now fid five data values with media = 10, mode = 1000 ad mea = 10 c) Ca you fid five data values with media = 1000, mode = 10, mea = 10? COOL Exercise: Repeat the previous exercise but this time for SIX data values DEVIATION FROM THE MEAN The data set 1,,5, has mea 5 So too does the data set: 01, 0, 1, 110 These are two very differet data sets, with the secod beig much more spread out tha the first We ca measure the degree of spread by calculatig the average deviatio from the mea for each DATA SET 1,,5, : Deviatios: 1 5 = 15 5 = = 5 5 = 05 Average deviatio: = 15 DATA SET 01, 0, 1, 110: Deviatios: = = = = 1075 Average deviatio: = 865 The umbers 15 ad 865, the average deviatios from the mea, do give a quatitative measure of the amout of spread of each data set THE POINT OF THIS ESSAY Usig the absolute value, the distace of a particular data value from the mea value of the data, is the atural ad appropriate way to measure data variatio But statisticias DON T use absolute values i their work! This is very strage ad cofusig for studets (There is also a secod piece of cofusio, which we shall leave to later i this essay) Here are two ratioales for the switch away from absolute values: wwwjamestatocom ad wwwgdaymathcom
4 RATIONALE ONE: Workig with absolute values is hard Ca we avoid them? Ideed, workig with absolute values i mathematical equatios is really tough! Optioal Exercises: a) Sketch the curve x + y = b) Fid all values of w which satisfy: w w 5 w = 7 c) (From last moth s essay) Three data poits A= (, ), B= (5,8) ad C = (7,5) are plotted o a graph A horizotal lie y = k will be draw but a value k eeds to be chose so that the sum of the three vertical deviatios from the horizotal lie is at a miimum (NOTE: We ve draw the horizotal lie so that A lies below it ad B ad C above it This eed ot be the case) O a calculator, type i a fuctio that represets the sum of these three deviatios ad graph it Which value of k seems to give a miimum value for this sum of three deviatios? But we still eed a measure, a positive umber that represets the deviatio of each data value from the mea If we wat to avoid absolute value, how else ca we obtai positive values? Aswer: Square the values! Let s square all the deviatios ad take the average of those squared deviatios: DATA SET 1,,5, : Deviatios squared: 1 5 = 5 5 = = 65 5 = 05 Average squared deviatio: = 5 DATA SET 01, 0, 1, 110: Deviatios squared: = = = = Average squared deviatio: = 5575 These average squared deviatios still give a good sese of the differet spreads the two data sets possess Oe subtle poit: Data ofte comes from physical measuremets the height of a perso, the speed of a car o a highway, ad so o ad so has uits associated with them If x 1, x, K, x are i uits of iches, say, x1 + x + L+ x the the mea x= also has uits of iches, but the average squared deviatio: wwwjamestatocom ad wwwgdaymathcom
5 + ( x + K+ ( x has uits of iches squared To brig all quatities ad comparisos betwee quatities back to the same uits, statisticias will take the square root of the average squared deviatio: + ( x + K+ ( x This quatity ow has uits of iches ad is called the stadard deviatio of the data WARNING: Statisticias might raise a eyebrow or two over at what I just said They might prefer to call the quatity: + ( x + K+ ( x 1 the stadard deviatio of the data set (Note i the deomiator, rather tha ) This chage  the secod cofusio for studets studyig statistics  is discussed at the ed of this essay RATIONALE TWO: Abstract mathematics tells us it is atural to work with quatities squared Suppose we ru a experimet or poll some people ad gai from the exercise data values: x1, x, K, x We, ot beig omisciet, kow othig about the data values we shall obtai: we do t kow what to expect for the mea of the values (what is the true average height of all humas o this plaet?), what variatio from the mea to expect, what the frequecies of particular values should be, ad so o But if the experimet was ideal or the populatio we were pollig from is truly uiform, the the experimet or pollig would be absolutely ad utterly repeatable ad we d expect o variatio i data values at all That is, i the perfect ideal, all measuremets would adopt exactly the same value q, say, over ad over agai Let s ask: How close is our data, x, K, x) from some ideal set of repeatable data ( q, q,, q) K? Now we leared last moth that, i twodimesioal geometry, the distace A= a, a ad betwee two poits ( 1 ) B= ( b, b ) is give by: 1 (, ) d A B = a b + a b 1 1 Ad the distace betwee two poits A a, a, a B= b, b, b i = ad 1 threedimesioal space is: 1 (, ) = ( ) + ( ) + ( ) d A B a b a b a b 1 1 Ad so o, for ay dimesio of space So to aswer this questio we seek a value M = q, q, K, q is as q so that the poit close as possible to our poit P=, x, K, x) i dimesioal geometry We wat to choose a value q that miimizes the distace: = ( ) + ( ) + L ( ), 1 d P M x q x q x q It is easier to just to miimize the quatity uder the square root sig Notice that we are ow led to study a sum of quatities squared Expad the sum uder the root ad collect terms: wwwjamestatocom ad wwwgdaymathcom
6 ( x q) + ( x q) + L( x q) 1 ( 1 L ) ( 1 L ) = q x + + x q+ x + + x We see that the sum we wish to miimize is just a quadratic i q It has miimum value for: + L+ x) x1+ L+ x q= = = x  the data s mea! We have: The mea of a data set is the value of closest ideal, repeatable, experimet to the give data From this perspective we see that it is atural to thik about sums of deviatios squared Dividig through by, we call: + L+ ( x the variace of the data Ad to match uits, we take the square root ad call this the stadard deviatio of the data: + L+ ( x Commet: We have ow see that the mea x of a set of data values x1, x, K, x has two properties: i) The sum ( x 1 x ) + ( x x ) + L + ( x x ) is zero ON VERSUS Some text authors will argue that it is better to divide by i the formulas for variace ad stadard deviatio rather tha by for the followig philosophical reaso: We have that ( x 1 x ) + ( x x ) + L + ( x x ) is sure to equal zero This meas that if oe kows the first values x1 x, x x, K, x 1 x, the the value of the th oe, x So amog the values x, is forced, ( x, K, ( x there are oly 1real pieces of iformatio To reflect this, let s divide by rather tha ad set the variace as: + L+ ( x 1 ad the stadard deviatio as: + L+ ( x 1 But this seems usatisfactory a explaatio Text authors will ofte add: If the data sets are large, that is, if is a large umber, the there will be little differece i dividig through by over dividig through by ii) Of all the sums of the form: ( x q) + ( x q) + ( x q) the sum 1 L + ( x + L ( x has the smallest value The correct studet respose to this add o is: So, really, why bother makig this chage? To uderstad why statisticias prefer to divide by,ot, let s go back to a previous example wwwjamestatocom ad wwwgdaymathcom
7 Because the data set is so large, we have o hope of kowig the true average height µ of all humas o this plaet right at this momet All we ca do is measure the heights of a sample of humas, compute the data mea h of that sample, ad hope that h offers a good approximatio forµ We would expect there to be some uiformity amog all the possible samples we could work with Certiaily, if we select a sample of 100 humas ad measure their heights we would obtai a sample mea h If we chose a differet collectio of 100 people we would probably obtai a slightly differet mea h I fact, if we looked at every possible collectio of 100, we d have a whole spread of values for h, all approximatig the true mea value µ Sice the set of all samples of 100humas well ad truly covers the etire huma populatio, it would be a shock if, o average, the set of all possible values of h tured out to be differet from µ The same should be true for variace We ca t possibly kow the true variace of the etire set of huma populatio heights, but we ca take a sample of 100heights ad fid the value of the variace for that sample Ad it would be a shock if agai, o average, the variaces over all possible samples of 100 people tured out to be a value differet from the true variace of the etire populatio Let s see what ca happe with some actual umbers EXAMPLE: Cosider the data set 1,,, This is a set of = data values with true mea µ = ad true variace, whe dividig by = : ( 1 0) + ( ) + ( ) + ( ) V = 1 = ad true variace, whe dividig by = : V = = ( 1 0) + ( ) + ( ) + ( ) But suppose we do t kow these values a data set of four values is too large for us to maage so we decide to look at samples of size three istead ad work out their sample meas ad sample variaces Here is a table of all possible subsets of size three (hadlig the repeated s) ad the sample meas ad variaces we would see: V V x {1,, } / 9+ 1 / 9+ 1 / 9 5 / = / 9 = 1/ {1,,} / 9 1 {1,,} / 9 1 {,,} 7 / / 9 1/ / 9+ 1/ 9+ 1 / 9 Average 1/ / We see that the meas ad the variaces do deped o which sample of three you happe to choose We also see, i this example, that our first dream is true: the average of all the sample meas matches µ = o the ose wwwjamestatocom ad wwwgdaymathcom
8 Ad our secod dream is true too if we divide by istead of whe computig variaces: the average of the values of V over all samples matches the value of V for the overall data These two claims are ot a coicidece for our particular example: they are true i geeral It is for this reaso that statisticias prefer to work with the formula: + L+ ( x 1 for variace ad the square root of this for stadard deviatio Exercise: There are six twoelemet subsets of the data set 1,,, (if you hadle the repeated s appropriately) List all six subsets, compute the mea ad variace V of each, ad take the average value of 1 these six meas ad variaces Show these average values match µ = ad V = / of the origial data set MATHEMATICAL PROOFS: The mathematics here is tedious algebra ad is hard to read Oe ca phrase the algebra i terms of expected values ad variaces of radom variables ( E( X ) ad Var( X )) ad make matters less complicated visually, but oe does this at the price of obscurig the coceptual straightforwardess If you are game, here s how these proofs proceed Suppose a populatio possesses a total of N data poits ad has mea: y1 + y + L+ yn µ = N Our job is to look at a subset of data poits, x1, x, K, x, compute their data mea x, ad take the average of all possible values for x over all possible subsets ad show this average equals µ We must also compute the variaces + L+ ( x 1 over all subsets ad show that their average equals: Now there are ( µ ) + L+ ( µ ) y1 y N N 1 = subsets! N! ( ) wwwjamestatocom ad wwwgdaymathcom N C of size amog N data poits, so i each case, our average is a sum divided by this umber For the sample meas we eed to show: x1 + x + L+ x x ' 1+ x ' + L+ x ' + + L! N! ( ) equals µ where the umerator is the sum of sample meas over all possible subsets (There is a similar, but more complicated formula, for the average of the variaces) This expressio is equivalet to: ( ) 1!( N )! ( + x + L+ x) + ( x ' 1+ x ' + L+ x ' ) + L) Now a particular data poit x appears i ( N 1)! N 1C 1 = subsets of size 1!( N )! So i the sum we have each data poit metioed this may times Our expressio is thus equivalet to: 1!( N )! N 1! N 1! x+ y+ L 1! N! 1! N!
9 where the sum is over each ad every data poit i the set This simplifies to: 1 x y N ( + +L ) which is ideed µ! For the average value of the variaces, we eed to work with: + L+ ( x 1 +! ( N )! 1 + L This is equivalet to: ( ) ( N ) ( x ' 1 x ') + L+ ( x ' x ') + L+ ( x + ( x ' 1 x ') + L+ ( x ' x ') 1!! + L = 1!! ( ) ( N ) 1 1 x1 + L+ x) + L+ x + L+ x) x' 1 ( x ' 1+ L+ x ' ) + L+ x' ( x ' 1+ L+ x' ) + L = 1! N! (( 1) x1 x L + ( x1 + ( 1) x L + L (( 1 ) x ' 1 x ' x ' ) ( x ' 1 ( 1 ) x ' x' ) L + + L + L + L By expadig terms ad coutig how may times a particular data poit squared x appears ad how may times the pair 1 x x appears (ad these couts are the 1 same for all data poits), oe ca show that this expressio does ideed equal: ( µ ) + L+ ( µ ) y1 y N N 1 the variace over all the data poits We ll leave the details to the truly gugho reader! Exercise: To get a (maageable) feel for the algebra, do work through the details for the case of N = data poits: x1, x, x, x Write dow ad simplify the formulas for the variaces of each of the subsets,,,, x, x, x, { x1 x x },{ x1 x x },{ 1 } {,, } x x x ad a expressio for the average of these four values Show this average equals: x1 + x + x + x x1 x + x + x + x + x 1 x + x + x + x + x x + x + x + x + x 1 1 1, 01 James Tato wwwjamestatocom ad wwwgdaymathcom
Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More information1 Lesson 6: Measure of Variation
1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages
More informationMedian and IQR The median is the value which divides the ordered data values in half.
STA 666 Fall 2007 Webbased Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5umber summary mea ad stadard deviatio Media
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso productmomet correlatio
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the zaxis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationMeasures of Spread: Variance and Standard Deviation
Lesso 16 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample
More informationMA131  Analysis 1. Workbook 2 Sequences I
MA3  Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationActivity 3: Length Measurements with the FourSided Meter Stick
Activity 3: Legth Measuremets with the FourSided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a foursided meter
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationMA131  Analysis 1. Workbook 9 Series III
MA3  Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationP1 Chapter 8 :: Binomial Expansion
P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationSequences III. Chapter Roots
Chapter 4 Sequeces III 4. Roots We ca use the results we ve established i the last workbook to fid some iterestig limits for sequeces ivolvig roots. We will eed more techical expertise ad low cuig tha
More informationHow to Maximize a Function without Really Trying
How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet
More informationTable 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab
Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationOnce we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1
. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More informationSolutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationStudents will calculate quantities that involve positive and negative rational exponents.
: Ratioal Expoets What are ad? Studet Outcomes Studets will calculate quatities that ivolve positive ad egative ratioal expoets. Lesso Notes Studets exted their uderstadig of iteger expoets to ratioal
More informationLinear Regression Models
Liear Regressio Models Dr. Joh MellorCrummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More information11.6 Absolute Convergence and the Ratio and Root Tests
.6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationShannon s noiseless coding theorem
18.310 lecture otes May 4, 2015 Shao s oiseless codig theorem Lecturer: Michel Goemas I these otes we discuss Shao s oiseless codig theorem, which is oe of the foudig results of the field of iformatio
More informationPostedPrice, SealedBid Auctions
PostedPrice, SealedBid Auctios Professors Greewald ad Oyakawa 2070208 We itroduce the postedprice, sealedbid auctio. This auctio format itroduces the idea of approximatios. We describe how well this
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationCHAPTER III RESEARCH METHODOLOGY
CHAPTER III RESEARCH METHODOLOGY A. Method of the Research I this research the writer used the experimetal method. The experimetal research was aimed to kow if there were effect or ot for the populatio
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationAcademic. Grade 9 Assessment of Mathematics. Released assessment Questions
Academic Grade 9 Assessmet of Mathematics 2014 Released assessmet Questios Record your aswers to the multiplechoice questios o the Studet Aswer Sheet (2014, Academic). Please ote: The format of this booklet
More informationFourier Series and the Wave Equation
Fourier Series ad the Wave Equatio We start with the oedimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationSEQUENCES AND SERIES
9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationAddition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c
Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity
More informationBHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13
BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationLecture 10 October Minimaxity and least favorable prior sequences
STATS 300A: Theory of Statistics Fall 205 Lecture 0 October 22 Lecturer: Lester Mackey Scribe: Brya He, Rahul Makhijai Warig: These otes may cotai factual ad/or typographic errors. 0. Miimaxity ad least
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationChapter 1 (Definitions)
FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 00900 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationMATH1035: Workbook Four M. Daws, 2009
MATH1035: Workbook Four M. Daws, 2009 Roots of uity A importat result which ca be proved by iductio is: De Moivre s theorem atural umber case: Let θ R ad N. The cosθ + i siθ = cosθ + i siθ. Proof: The
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σalgebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationPaired Data and Linear Correlation
Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0
More informationOutput Analysis and RunLength Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad RuLegth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More information5 3B Numerical Methods for estimating the area of an enclosed region. The Trapezoidal Rule for Approximating the Area Under a Closed Curve
5 3B Numerical Methods for estimatig the area of a eclosed regio The Trapezoidal Rule for Approximatig the Area Uder a Closed Curve The trapezoidal rule requires a closed o a iterval from x = a to x =
More informationSTAT 203 Chapter 18 Sampling Distribution Models
STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationIs mathematics discovered or
996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies
More information1 Section 2.2, Absolute value
.Math 0450 Hoors itro to aalysis Sprig, 2009 Notes #6 1 Sectio 2.2, Absolute value It is importat to uderstad iequalities ivolvig absolute value. I class we cosidered the iequality jx 1j < jxj ; ad discussed
More informationEECE 301 Signals & Systems
EECE 301 Sigals & Systems Prof. Mark Fowler Note Set #8 DT Covolutio: The Tool for Fidig the ZeroState Respose Readig Assigmet: Sectio 2.12.2 of Kame ad Heck 1/14 Course Flow Diagram The arrows here
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter
More informationSequences, Series, and All That
Chapter Te Sequeces, Series, ad All That. Itroductio Suppose we wat to compute a approximatio of the umber e by usig the Taylor polyomial p for f ( x) = e x at a =. This polyomial is easily see to be 3
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AMGM iequality The most basic arithmetic meageometric mea (AMGM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationUNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series
UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationMath 176 Calculus Sec. 5.1: Areas and Distances (Using Finite Sums)
Math 176 Calculus Sec. 5.1: Areas ad Distaces (Usig Fiite Sums) I. Area A. Cosider the problem of fidig the area uder the curve o the f y=x 2 +5 over the domai [0, 2]. We ca approximate this area by usig
More information11.1 Radical Expressions and Rational Exponents
Name Class Date 11.1 Radical Expressios ad Ratioal Expoets Essetial Questio: How are ratioal expoets related to radicals ad roots? Resource Locker Explore Defiig Ratioal Expoets i Terms of Roots Remember
More informationLecture 4 The Simple Random Walk
Lecture 4: The Simple Radom Walk 1 of 9 Course: M36K Itro to Stochastic Processes Term: Fall 014 Istructor: Gorda Zitkovic Lecture 4 The Simple Radom Walk We have defied ad costructed a radom walk {X }
More information1036: Probability & Statistics
036: Probability & Statistics Lecture 0 Oe ad TwoSample Tests of Hypotheses 0 Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso
More informationLast time, we talked about how Equation (1) can simulate Equation (2). We asserted that Equation (2) can also simulate Equation (1).
6896 Quatum Complexity Theory Sept 23, 2008 Lecturer: Scott Aaroso Lecture 6 Last Time: Quatum ErrorCorrectio Quatum Query Model DeutschJozsa Algorithm (Computes x y i oe query) Today: BersteiVazirii
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More information1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4
. Hilbert s Grad Hotel The Hilbert s Grad Hotel has ifiite may rooms umbered,,,.. Situatio. The Hotel is full ad a ew guest arrives. Ca the mager accommodate the ew guest?  Yes, he ca. There is a simple
More informationARITHMETIC PROGRESSION
CHAPTER 5 ARITHMETIC PROGRESSION Poits to Remember :. A sequece is a arragemet of umbers or objects i a defiite order.. A sequece a, a, a 3,..., a,... is called a Arithmetic Progressio (A.P) if there exists
More information