Measures of Spread and Boxplots Discrete Math, Section 9.4


 Blanche O’Brien’
 3 years ago
 Views:
Transcription
1 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9, 10, 11, 13, 16} Now plot both data sets o these umber lies: S 1 S What do you observe? We'll look at a few differet measures of spread (also called measures of dispersio): Rage Iterquartile rage Five Number Summary Variace Stadard Deviatio I. The Rage As we defied the other day: The rage of a set of umbers is the differece betwee the largest ad smallest umbers i the set, i.e. rage = (largest value)  (smallest value) Questio: What importat thigs does the rage tell us? What problems ca occur if the rage is used to measure the spread of a set of data? Page 1
2 II. The Iterquartile Rage (IQR) Before we defie the iterquartile rage, we eed to discuss the topics of percetiles ad quartiles. Questio: Suppose you score a 760 o the SAT Math ad you are told that this placed you i the 95 th percetile. What does this mea? Quartile defiitios: The lower quartile (or first quartile) of a set is the 5 th percetile. Notatio: Q 1. The upper quartile (or third quartile) of a set is the 75 th percetile. Notatio: Q 3. The secod quartile is the media. Notatio: Q or M. To compute quartiles: 1. Sort the data from smallest to largest.. Fid the media. This is the secod quartile, Q. 3. Look at the first half of the data (ot icludig Q ). Fid the media of the first half of the data. This is the first quartile, Q Look at the secod half of the data (ot icludig Q ). Fid the media of the secod half of the data. This is the third quartile, Q 3. Example : Quartiles Calculate Q 1, the media, ad Q 3 for the followig data set: Defiitio: The iterquartile rage (IQR) is defied to be the differece betwee the third quartile ad the first quartile. Thus, we ca express the IQR usig this formula: IQR = Q 3  Q 1 Example 3: IQR Fid the IQR for Example. Questio: Why would IQR be a "good" measure of spread? We ca also use the IQR to determie whether a umber is a outlier of a data set: A Test for Outliers: A data poit is cosidered to be a outlier if it lies more tha 1.5 iterquartile rages below Q 1 (i.e., the umber is less tha Q 11.5IQR) or 1.5 iterquartile rages above Q 3 (i.e., the umber is greater tha Q IQR). Page
3 III. FiveNumber Summary ad Boxplots Aother way to describe both the ceter ad spread of a set of umbers is to use its fiveumber summary. The fiveumber summary cosists of: miimum value Q 1 media Q 3 maximum value Example 4 (Yates, et. al.): Bods' Home Rus The followig data are the umbers of home rus Barry Bods hit i his first 16 seasos, sorted: a. Create a fiveumber summary of this data. b. We suspect that Bods' 73homeru seaso is a outlier. Is it? c. For good measure, is the 16homeru seaso a outlier? Give specific calculatios. We ca represet the fiveumber summary graphically. A boxplot or boxadwhisker plot is a graphical represetatio of the fiveumber summary: The box exteds from Q 1 to Q 3. The box is divided at the media. The whiskers exted from Q 1 to the mi ad from Q 3 to the max. Example 5 Make a boxadwhisker plot for the last example. Notes of importace: 1. The fiveumber summary is a excellet way to measure the spread of a skewed data set.. Two sidebyside boxplots ca be a good way of comparig two related data sets. 3. It is importat to label the umbers whe makig boxplots to compare data. 4. Boxplots ca be draw either horizotally or vertically. Page 3
4 That was fu, but While they're ice, boxplots coceal outliers. As a result, we adopt a modified boxplot. The modified boxplot is similar, except that outliers are plotted as idividual poits. Modified Boxplot: A modified boxplot is draw as follows: A cetral box exteds from Q 1 to Q 3. The box is divided at the media. Observatios more tha 1.5IQR outside the cetral box (the outliers) are plotted idividually The whiskers exted from Q 1 to the smallest value that is ot a outlier ad from Q 3 to the largest value that is ot a outlier. Example 6 Now draw a modified boxplot for the previous example. Usig the Calculator to Your Advatage The TI graphig calculators ca do fiveumber summaries, boxplots, ad modified boxplots: The fiveumber summary is foud uder 1Var Stats. Scroll dow i the list for mix, Q 1, Med, Q 3, ad maxx. Boxplots ad modified boxplots are foud uder the Stat Plot meu. Look for them i the secod row of stat plot optios. Be sure to use ZoomStat! (You otherwise may miss the outliers!) Aother Test for Outliers: Eter the data i a list. Use the calculator to make a modified boxplot. The modified boxplot will show whether or ot there are outliers. Example 49 (Uderstadig Statistics) I a hurry? O the ru? Hugry as well? How about a ice cream bar as a sack. Ice cream bars are popular amog all age groups. Cosumer Reports did a study of ice cream bars. Twetyseve bars with taste ratigs of at least "fair" were listed, ad the cost per bar was icluded i the report. Just how much does a icecream bar cost? The data, expressed i dollar, appear below. As you ca see, the cost varies quite a bit, partly because the bars are ot uiform i size a. Compute the fiveumber summary ad the iterquartile rage for this data set. (Who said example umbers had to be borig?) Page 4
5 b. Use the TI graphig calculator to create a boxplot ad a modified boxplot. Draw these graphs below, labelig the fiveumber summary o each graph. c. Did this data set cotai ay outliers? Example 8: Comparig Sog Legths usig SideBySide Boxplots Below are the legths of the tracks o three differet CDs, listed i miutes ad secods: U's How to Dismatle a Atomic Bomb: 3:14 3:59 5:08 4:50 5:47 3:39 4:30 5:03 3:51 4:41 4:1 Dave Matthews Bad's Crash: 4:07 6:7 5:16 4:1 6:39 6:11 5:54 4:07 5:4 5:53 5:00 9:11 Somethig Corporate's North: 3:08 :57 3:7 3:7 4:07 3:03 3:4 3:38 3:16 3:49 3:18 3:51 Use the TI graphig calculator to make 3 sidebyside modified boxplots. [Hit: Begi by eterig this data ito lists. Before we ca make boxplots, we eed to covert miutes ad secods to all secods. You ca use lists o the calculator to do this too.] a. Sketch the boxplots here, labelig the legths i secods: b. Do ay of these albums cotai sogs whose track legths are outliers? Which oe? c. What does the modified boxplot say about how track legths vary o North? d. Use the modified boxplots to compare the legths of tracks o these three CDs. e. Suppose you are DJ at a radio statio ad you have these three albums at your disposal. You have approximately 4 miutes of time to fill before a program begis ad you eed to fid a sog. Lookig at the boxplots, which CD would be the best place to look? (Keep i mid that ot all sogs o all CDs are radio sigles.) Page 5
6 IV. Sample Variace ad Sample Stadard Deviatio Before we begi, we must make a importat distictio: A umber that describes a Populatio is called a Parameter. A umber that describes a Sample is called a Statistic. Recall: What is the differece betwee a populatio ad a sample? Note that, i practice, whe a parameter is ukow, we use the correspodig statistic. Whe we talked about the mea, we talked about the sample mea, a statistic. We used the otatio x for the sample mea. If we are talkig about the populatio mea, a parameter, we use the otatio µ. Just as the mea is the most commolyused measure of ceter, the stadard deviatio is the most commolyused measure of spread. I order to defie stadard deviatio, we first defie variace. Defiitio: Sample Variace The sample variace, deoted s, of a set of observatios { x 1, x,..., x}, is give by s = "( xi! x) i= 1! 1 Defiitio: Sample Stadard Deviatio The sample stadard deviatio, deoted s, of a set of observatios { x 1, x,..., x}, is the square root of the sample variace, ad is give by s = "( xi! x) i= 1! 1 Note that variace is't used all that ofte, but stadard deviatio is defied i terms of variace, so we iclude it. Your calculator ca easily compute both the sample stadard deviatio ad the sample variace, but we will first work with them a bit to get a uderstadig of how the formulas work. Example 9 Compute the variace ad stadard deviatio of the followig sample: Page 6
7 Notes of Importace: 1. The parameters populatio variace ad populatio stadard deviatio are deoted by σ ad σ, respectively.. If you kow all of the data i a populatio, you ca also compute the populatio variace ad stadard deviatio by very similar formulas:! i= 1 ( x " µ ) i! i= 1 ( x " µ ) i # = ad # = Note that the distictio betwee these formulas ad the sample formulas is that we use the populatio mea here ad we divide by istead of by 1. Your book presets these formulas i a somewhat misleadig way. Be careful to use sample formulas whe you are workig with a sample. 3. The TI calculators ca compute stadard deviatio. It's uder 1Var Stats. The fourth etry, Sx, is the sample stadard deviatio, ad the fifth etry, σx, is the populatio stadard deviatio. You usually wat the sample stadard deviatio. 4. The stadard deviatio of a set of umbers measures how umbers are spread out from the mea. 5. As we saw o a recet worksheet, the quatity x i! x is called the deviatio from the mea ad the sum of all the deviatios for ay data set always equals The stadard deviatio is oresistat to outliers. 7. The quatity 1, which appears i the deomiator of the formulas for sample variace ad sample stadard deviatio, is called the umber of degrees of freedom. 8. (Yates et. al.) The mea ad stadard deviatio are excellet measures of spread for data sets which a symmetric. For skewed data sets, the media ad fiveumber summary may be more helpful. Example 10 (Freedma): Spread Each of the followig lists has a average of 50. For which oe is the spread of the umbers aroud the average biggest? Smallest? a b c Example 11 (Freedma): Spread Repeat the directios show i Example 10 for the followig lists: a b c Page 7
8 Example 1 (Freedma): Stadard Deviatios Each of the followig lists has a average of 50. For each oe, guess whether the stadard deviatio is aroud 1,, or 10. (This example does't require ay arithmetic.) a b c d e Example 13 (Ima): Fial Exams A graduate studet i biology has bee asked to grade 40 fial exams, selected at radom from several large sectios of a itroductory course. There are the grades: a. Calculate the sample mea ad sample stadard deviatio: sample mea: sample stadard deviatio: b. What percet of the raw data lies withi oe stadard deviatio from the mea? c. What percet lies withi two stadard deviatios? d. What percet lies withi three stadard deviatios? Homework: 9.4: #3, 5, 7, 1 Page 8
9 This example illustrates the followig rule of thumb: Rule of Thumb o Spread: Rule: For may samples of data, 1. About /3 of the sample observatios fall withi oe sample stadard deviatio of the mea.. About 95% of the observatios fall withi two sample deviatios of the mea. 3. About 99.7% of the observatios fall withi three sample deviatios of the mea. Note: This rule is particularly accurate for data which have a bellshaped graph. Example 14: Fidig Stadard Deviatio a. Fid the sample stadard deviatio for the followig sample: b. Suppose the above data were all of the data for a populatio. Compute the populatio stadard deviatio. Chebyshev's Theorem Form 1: The probability that ay radom variable X will assume a value withi k stadard deviatios of the mea is at least 11/k. I terms of a formula, 1 P ( µ! k# < X < µ + k# ) " 1! k Form : The probability that ay radomlychose outcome lies betwee µk ad µ+k is at least! 1" k Note: Chebyshev's Theorem is amed for Russia Mathematicia Pafuty L. Chebyshev ( ). As this is a Aglicizatio of his last ame, it's oe of several ways you'll see it spelled i books. Please do ot cofuse his first ame with ay rappers. Page 9
10 Example 15 (Mizrahi/Sulliva): Usig Chebyshev Suppose tha a experimet with umerical outcomes has mea 4 ad stadard deviatio 1. Use Chebyshev's Theorem to estimate the probability that a outcomes lies betwee ad 6. Example 16 (Walpole/Myers/Myers): More from Pafuty A radom variable X has a mea µ = 8, a variace σ =9, ad a ukow probability distributio. Fid: a. P (! 4 < X < 0) b. P ( X! 8 > 6) Homework: 9.4: #19, 0 Page 10
Descriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationMannWhitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)
NoParametric ivariate Statistics: WilcoxoMaWhitey 2 Sample Test 1 MaWhitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo) MaWhitey (WMW) test is the oparametric equivalet of a pooled
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
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 informationNow here is the important step
LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationChapter XIV: Fundamentals of Probability and Statistics *
Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive
More informationExploratory Data Analysis
1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationMEP Pupil Text 9. The mean, median and mode are three different ways of describing the average.
9 Data Aalysis 9. Mea, Media, Mode ad Rage I Uit 8, you were lookig at ways of collectig ad represetig data. I this uit, you will go oe step further ad fid out how to calculate statistical quatities which
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More information, a Wishart distribution with n 1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematiskstatistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 00409 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More informationConfidence Intervals for Linear Regression Slope
Chapter 856 Cofidece Iterval for Liear Regreio Slope Itroductio Thi routie calculate the ample ize eceary to achieve a pecified ditace from the lope to the cofidece limit at a tated cofidece level for
More informationOnesample test of proportions
Oesample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationTopic 5: Confidence Intervals (Chapter 9)
Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationG r a d e. 2 M a t h e M a t i c s. statistics and Probability
G r a d e 2 M a t h e M a t i c s statistics ad Probability Grade 2: Statistics (Data Aalysis) (2.SP.1, 2.SP.2) edurig uderstadigs: data ca be collected ad orgaized i a variety of ways. data ca be used
More informationTI83, TI83 Plus or TI84 for NonBusiness Statistics
TI83, TI83 Plu or TI84 for NoBuie Statitic Chapter 3 Eterig Data Pre [STAT] the firt optio i already highlighted (:Edit) o you ca either pre [ENTER] or. Make ure the curor i i the lit, ot o the lit
More information15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011
15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes highdefiitio
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT  Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationQuadrat Sampling in Population Ecology
Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may
More informationUnit 8: Inference for Proportions. Chapters 8 & 9 in IPS
Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater
More informationProject Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments
Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 612 pages of text (ca be loger with appedix) 612 figures (please
More informationPractice Problems for Test 3
Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a page formula sheet. Please tur over Mathematics/P DoE/Exemplar
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 informationCentral Limit Theorem and Its Applications to Baseball
Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead
More informationConfidence Intervals
Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more
More informationCOMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationMultiserver Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu
Multiserver Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio coectio
More informationResearch Method (I) Knowledge on Sampling (Simple Random Sampling)
Research Method (I) Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationSTA 2023 Practice Questions Exam 2 Chapter 7 sec 9.2. Case parameter estimator standard error Estimate of standard error
STA 2023 Practice Questios Exam 2 Chapter 7 sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (1) oe p ( 1 p) CI: prop.
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More information