Formulas FROM LECTURE 01 TO 22 W X. d n. fx f. Arslan Latif (mt ) & Mohsin Ali (mc ) Mean: Weighted Mean: Mean Deviation: Ungroup Data

 Anabel Nelson
 11 months ago
 Views:
Transcription
1 1 Formulas FROM LECTURE 01 TO Mea: fx f Weighted Mea: X w W X i i Wi Mea Deviatio: Ugroup Data d M. D Group Data fi di M. D f d ( X X ) Coefficiet of Mea Deviatio: M. D Coefficiet of M. D(for mea) Mea M. D Coefficiet of M. D(for media) Media Stadard Deviatio: Ugroup Data ( X X ) S. D Group Data f ( X X ) S. D f
2 Shortcut Formula for Ugroup data x x S. D Shortcut Formula for Group Data fx fx S. D f f CoEfficiet of Stadard Deviatio: S. D X Variace: Ugroup Data Variace ( X X ) Group Data Variace ( ) f X X f CoEfficiet of Variatio:. C. V S D 100 X Media: + 1 Media item h Media l + c f Harmoic Mea: Ugroup Data X 1 x Group Data f X f x th
3 3 Average Speed: Total Distace Travelled Average Speed Total Time Take Distace Time Speed Distace Speed Time Distace Speed Mode: Time ( fm f 1) X l + h ( fm f 1) + ( fm f ) Empirical Relatio betwee Mea, Media ad Mode: Mode 3Media Mea Rage: m 0 Rage X X MidRage: Mid Rage X 0 + X m Mid Quatile Rage: Mid Rage Q1 + Q3 Coefficiet of Dispersio: Coefficiet of Dispersio X X m m X + X 0 0 Geometric Mea: Ugroup Data log X G. M ati log Group Data f log X G. M ati log f
4 4 Quartile Deviatio: Q3 Q1 Q. D Q Q Co efficiet of Q. D Q + Q th + 1 Q1 item 4 h Q1 l + c f 4 h 3 Q3 l + c f 4 Skewess: SK Mea Mode S. D Pearso s Coefficiet of Skewess 3( Mea Media) SK S. D Bowley s coefficiet of Skewess Q1 + Q3 ( Media) SK Q Q Kurtosis: K Q. D P P Momets: Ugroup Data ( X i X ) m1 0 ( X i X ) m 3 ( X i X ) m3 Group Data m 3 f ( X X ) Momet Ratios: i f ( m3 ) m4 ad b ( m ) ( m ) b 1 3 3
5 5 Equatio of a Straight Lie Y a + bx Stadard Error of Estimate: s yx Y a Y b XY Normal Equatios: Y a + b X XY a X + b X Correlatio: r ( X X )( Y Y ) ( X X ) ( Y Y ) PEARSON S COEFFICIENT OF CORRELATION r XY ( X )( Y ) / [ X ( X ) / ][ Y ( Y ) / ] Liear Correlatio of Coefficiet: r xy ( x)( y) x ( x) y ( y) Regressio Lie of Y o X: xy ( x)( y) byx x ( x) Divisio of Circle: Cell Frequecy 360 Total Frequecy Biomial Coefficiet:( ( C r ) ( a + b) a b r 0 r r r
6 6 1. Cumulative Law: A B B A A B B A. Associative Law: ( A B) C A ( B C) ( A B) C A ( B C) 3. Distributive Law: A (B C) (A B) (A C) A (B C) (A B) (A C) 4. Idempotet Law: A A A A A A 5. Idetity Law: A S S A S A A φ A A φ φ0 6. Complemetatio Law: A A S A A φ ( A) A S φ φ S 7. De Morga s Law: ( ) ( ) A B A B A B A B Rule of Permutatio:! r! ( ) Rule of Combiatio:! r! r! ( )
7 7 Probability Formula: m P(A) No. of Favourable Outcomes P(A) Total No. of Possible Outcomes Mutually Exclusive Evets: If A ad B mutually exclusive evets, P(A B) P(A) + P(B) Complemetatio Law of Probability: P(A) 1 P( A) Additio Law: P(A B) P( A) + P( B) P( A B) Coditioal Probability: P(A/B) P(B/A) P(A B) P(B) P(A B) P(A) Multiplicatio Law: P(A B) P(A)P(B/A) Baye s Theorem: P(A B) 1 P( A ). P( B A ) 1 1 P( A ). P( B A ) + P( A ). P( B A ) 1 1 Computatio of Stadard Deviatio Probability Distributio: X P( X ) XP( X ) S. D.( X ) P( X ) P( X ) S. D.( X ) X P( X ) XP( X ) As we kow, P( X ) 1 Coefficiet of Variatio: 100 µ
8 8 Total Cards 5 Diamod 13 Heart 13 Club 13 Spades 13 Face Cards 1 Quee 4 Jack 4 Kig 4 Jocker 4 FROM LECTURE 3 TO 45 Mathematical Expectatios: E( X ) x f ( x ) i 1 i i E(X) is also called the mea of X ad is usually deoted by the letter µ. If the values are equally likely: 1 E( X ) X i If X is a discrete radom variable ad if a ad b are costats, the: E ( ax + b) ae( X ) + b If µ is a Populatio Mea the: E ( X µ ) E( X ) [ E( X )] Shortcut Formula for the Variace i Mathematical Expectatios: E( X ) [ E( X )] Skewess of Probability: µ 3 β1 3 µ Kurtosis of Probability: 4 µ β1 µ Mea o Probability: E( X ) XP( X )
9 9 If the Parameters i distributio are a ad b the µ a + b Mea of the Populatio Distributio E( X ) xf ( x) µ Mea of the SAMPLIN DISTRIBUTION of X µ E( X ) x f ( x) x µ µ x Variace of Probability: Var( X ) X P( X ) [ XP( X )] Expected value of Variace Var( X ) or E( X µ ) ( x µ ) f ( x) If the Parameters i distributio are a ad b the ( b a) 1 Variace of the Populatio Distributio x f ( x) x f ( x) Variace of the SAMPLIN DISTRIBUTION of X Ugroup Data X X ( X ) Group Data X X f ( X ) X f ( X ) I case of samplig with replacemet: x I case of samplig without replacemet from a fiite populatio: x N N 1 i Fiite Populatio Correctio (fpc) N N 1
10 10 Biomial Distributio: x P( X x) p q x x IN CASE OF NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION f ( x) x x x p q Joit Probability Fuctio: f ( x, y ) kth Momet: m ' r X r i Origi of Radom Variable µ k E( Xk) X k f ( x) Mea of Radom Variable µ k i E( X µ ) k ( X µ ) k f ( x) Chebychev s Iequality: i I the case of a Discrete Probability Distributio 1 P( µ k < X < µ + k ) 1 k E µ µ [( X ) ] E( X ) MARGINAL PROBABILITY: Margial Probability fuctio of X g( x ) f ( x, y ) i i j j 1
11 11 Margial Probability fuctio of Y m h( y ) f ( x, y ) P( Y y ) i i j j i 1 Coditioal Probability Fuctio: Coditioal Probability fuctio for X f ( x y ) P( X x Y y ) i j i j Coditioal Probability fuctio for Y f ( y x ) P( Y y X x ) j i j i HYPER GEOMETRIC PROBABILITY DISTRIBUTION: P( X x) k N k ( x )( x ) N ( ) Where N umber of uits i the populatio, umber of uits i the sample, ad k umber of successes i the populatio. HYPER GEOMETRIC DISTRIBUTION: Mea of Hyper Geometric Distributios k µ N Variace of Hyper Geometric Distributio k N k N N N N 1 Biomial Probability Distributio: Lim p 0 b(x;,p) x 0,1,,..., e.1788 µ p e x! µ x µ UNIFORM DISTRIBUTION: A radom variable X is said to be uiformly distributed if its desity fuctio is defied as
12 1 1 f ( x), a x b b a Poisso Process Formula: e P( X x) ( t) x! λ t x λ P( X x) e µ x µ x! Where λ average umber of occurreces of the outcome of iterest per uit of time, t umber of timeuits uder cosideratio, ad x umber of occurreces of the outcome of iterest i t uits of time. NORMAL DISTRIBUTION: 1 f ( x) e π Where : π or 7 e x µ Momet Ratios i Probability: The momet ratios of the ormal distributio come out to be 0 ad 3 respectively: µ 0 β µ ( ) 4 4 µ 3 β 3 µ ( ) Stadardizatio Formula: Z X µ I case of X X µ x Z x
13 13 Proportio of Successes i the Populatio: X p or p N X p Proportio of Success X Number of successes i the populatio Samplig distributio of p SAMPLING DISTRIBUTION OF DIFFERENCES BETWEEN MEANS: µ µ µ x x 1 ad µ x1 x 1 1 x x 1 ad x x 1 ( x1 x) f ( x1 x) d f ( d) df ( d) x1 + x Z ( X1 X ) ( µ 1 µ ) SAMPLING DISTRIBUTION OF p µ p p f ( p ) Stadard Deviatio of the Samplig Distributio: p q p1 + p p q Modified Formula for the Sample Variace: S ( x x) 1
14 14 Relative Efficiecy: E f Var( T ) Var( T ) 1 Differece betwee the Meas of Two Populatios: s s ( x1 x ) ± zα / CONFIDENCE INTERVAL FOR A POPULATION PROPORTION (P): p ± z α / p (1 p ) Where p Proportio of "success" i hte sample Sample size z α / 1.96 for 95% cofidece.58 for 99% cofidece CONFIDENCE INTERVAL FOR P1P: p1(1 p1) p(1 p ) ( p1 p) ± zα / + 1 SAMPLE SIZE FOR ESTIMATING POPULATION PROPORTION: p z α / pq ɵ tdistribution: 1 x f ( x) 1 1 v + v vβ, Variace of the tdistributio v v for v > ( v+ 1)/
15 15 FDistributio: [ ] Γ ( v + v ) / ( v / v ) x < x < 1 Γ( v / ) Γ ( v / ) 1 + v x / v v1 / ( v1 /) ( ), 0 ( v + v )/ f x [ ] 1 1 Mea of the Fdistributio: For v > v v Variace of the Fdistributio: For v > 4 v ( v + v ) ( ) ( 4) 1 v1 v v Mea Square: Mea Square Sum of Squares Degrees of Freedom Least Sigificat Differece: LSD t α /,( v) ( MSE) r
16 16 Defiitios FROM LECTURE 01 TO Statistics: Statistics is that sciece which eables to draw coclusios about various pheomea oe the basis of real data collected o sample basis. OR Statistics is a sciece of facts ad figures. INFERENTIAL STATISTICS: That brach of Statistics which eables us to draw coclusios or ifereces about various pheomea o the basis of real data collected o sample basis. Data: A well defied collectio of objects is kow as data. Qualitative data: Data that are labels or ames used to idetify a attribute of each elemet. Qualitative data may be oumeric or umeric. Qualitative variable: A variable with qualitative data. Quatitative data: Data that idicate how much or how may of somethig. Quatitative data are always umeric. Variable: A measurable quatity which ca vary from oe idividual or object to aother is called a variable. Nomial Scale: The classificatio or groupig of observatios ito mutually exclusive qualitative categories is said to costitute a omial scale e.g. studets are classified as male ad female. Ordial Scale: It icludes the characteristic of a omial scale ad i additio has the property of orderig or rakig of measuremets e.g. the performace of studets ca be rated as excellet, good or poor. Iterval Scale: A measuremet scale possessig a costat iterval size but ot true zero poit is called a Iterval Scale. Ratio Scale: It is a special kid of a iterval scale i which the scale of measuremet has a true zero poit as its origi.
17 17 Biased Errors: A error is said to be biased whe the observed value is cosistetly ad costatly higher or lower tha the true value. Ubiased Errors or Radom Errors: A error, o the other had, is said to be ubiased whe the deviatios, i.e. the excesses ad defects, from the true value ted to occur equally ofte. Primary Data: The data published or used by a orgaizatio which origially collected them are called primary data thus the primary data are the first had iformatio collected, complied, ad published by a orgaizatio for a certai purpose. Secodary Data: The data published or used by a orgaizatio other tha the oe which origially collected them are kow as secodary data. DIRECT PERSONAL INVESTIGATION: I this method, a ivestigator collects the iformatio persoally from the idividuals cocered. Sice he iterviews the iformats himself, the iformatio collected is geerally cosidered quite accurate ad complete. INDIRECT INVESTIGATION: Sometimes the direct sources do ot exist or the iformats hesitate to respod for some reaso or other. I such a case, third parties or witesses havig iformatio are iterviewed. Populatio: The collectio of all idividuals, items or data uder cosideratio i statistical study is called Populatio. Sample: A sample is a group of uits selected from a larger group (the populatio). By studyig the sample it is hoped to draw valid coclusios about the larger group. OR Sample is that part of the Populatio from which iformatio is collected. SAMPLING FRAME: A samplig frame is a complete list of all the elemets i the populatio. Samplig Error: The samplig error is the differece betwee the sample statistic ad the populatio parameter. NoSamplig Error: Such errors which are ot attributable to samplig but arise i the process of data collectio eve if a complete cout is carried out. Samplig with replacemet: Oce a elemet has bee icluded i the sample, it is retured to the populatio. A previously selected elemet ca be selected agai ad therefore may appear i the sample more tha oce. Samplig without replacemet: Oce a elemet has bee icluded i the sample, it is removed from the populatio ad caot be selected a secod time.
18 18 Stadard error: The stadard deviatio of a poit estimator. AND The degree of scatter of the observed values about the regressio lie measured by what is called stadard deviatio of regressio or stadard error of estimate. Samplig Uit: The uits selected for samplig. A samplig uit may iclude several elemets. NONRANDOM SAMPLING: Noradom samplig implies that kid of samplig i which the populatio uits are draw ito the sample by usig oe s persoal judgmet. This type of samplig is also kow as purposive samplig. Quota Samplig: Quota samplig is a method of samplig widely used i opiio pollig ad market research. Iterviewers are each give a quota of subjects of specified type to attempt to recruit for example, a iterviewer might be told to go out ad select 0 adult me ad 0 adult wome, 10 teeage girls ad 10 teeage boys so that they could iterview them about their televisio viewig. RANDOM SAMPLING: The theory of statistical samplig rests o the assumptio that the selectio of the sample uits has bee carried out i a radom maer. By radom samplig we mea samplig that has bee doe by adoptig the lottery method. Simple radom samplig: Fiite populatio: a sample selected such that each possible sample of size has the same probability of beig selected. Ifiite populatio: a sample selected such that each elemet comes from the same populatio ad the elemets are selected idepedetly. Pie Chart: Pie Chart cosists of a circle which is divided ito two or more mars i accordace with the umber of distict classes that we have i our data. SIMPLE BAR CHART: A simple bar chart cosists of horizotal or vertical bars of equal width ad legths proportioal to values they represet. MULTIPLE BAR CHARTS: This kid of a chart cosists of a set of grouped bars, the legths of which are proportioate to the values of our variables, ad each of which is shaded or colored differetly i order to aid idetificatio. CLASS BOUNDARIES: The true class limits of a class are kow as its class boudaries. HISTOGRAM: A histogram cosists of a set of adjacet rectagles whose bases are marked off by class boudaries alog the Xaxis, ad whose heights are proportioal to the frequecies associated with the respective classes. FREQUENCY POLYGON: A frequecy polygo is obtaied by plottig the class frequecies agaist the midpoits of the classes, ad coectig the poits so obtaied by straight lie segmets.
19 19 FREQUENCY CURVE: Whe the frequecy polygo is smoothed, we obtai what may be called the frequecy curve. CUMULATIVE FREQUENCY DISTRIBUTION: As i the case of the frequecy distributio of a discrete variable, if we start addig the frequecies of our frequecy table columwise, we obtai the colum of cumulative frequecies. AVERAGES (I.E. MEASURES OF CENTRAL TENDENCY): A sigle value which iteded to represet a distributio or a set of data as a whole is called a average. It is more or less a cetral value aroud which the observatios ted to cluster so it is called measure of cetral tedecy. Sice measure of cetral tedecy idicate the locatio of the distributio o X axis so it is also called measure of locatio. The Arithmetic, Geometric ad Harmoic meas Are averages that are mathematical i character, ad give a idicatio of the magitude of the observed values. The Media Idicates the middle positio while the mode provides iformatio about the most frequet value i the distributio or the set of data. THE MODE: The Mode is defied as that value which occurs most frequetly i a set of data i.e. it idicates the most commo result. DOT PLOT: The horizotal axis of a dot plot cotais a scale for the quatitative variable that we wat to represet. The umerical value of each measuremet i the data set is located o the horizotal scale by a dot. GROUPING ERROR: Groupig error refers to the error that is itroduced by the assumptio that all the values fallig i a class are equal to the midpoit of the class iterval. Ogive: A graph of a cumulative distributio. Dispersio: The variability that exists betwee data set. Rage: The rage is defied as the differece betwee the maximum ad miimum values of a data set. The coefficiet of variatio: The coefficiet of variatio expresses the stadard deviatio as the percetage of the arithmetic mea. Quartiles: Quartiles are those three quatities that divide the distributio ito four equal parts. Quartile Deviatio: The quartile deviatio is defied as half of the differece betwee the first ad third quartiles.
20 0 Quatiles: Collectively the quartiles, the deciles, percetiles ad other values obtaied by equall subdivisio of the data are called quatiles. Percetiles: Percetiles are those iety ie quatities that divide the distributio ito hudred equall parts Absolute measure of dispersio: A absolute measure of dispersio is oe that measures the dispersio i terms of the same uits or i the square of uits, as the uits of the data. Relative measure of dispersio: Relative measure of dispersio is oe that is expressed i the form of a ratio, coefficiet of percetage ad is idepedet of the uits of measuremet. COEFFICIENT OF QUARTILE DEVIATION: The Coefficiet of Quartile Deviatio is a pure umber ad is used for COMPARING the variatio i two or more sets of data. Mea Deviatio: The mea deviatio is defied as the arithmetic mea of the deviatios measured either from the mea or from the media, all deviatios beig couted as positive. Variace: Variace is defied as the square of the stadard deviatio. Stadard Deviatio: Stadard Deviatio is defied as the positive square root of the mea of the squared deviatios of the values from their mea. Fiveumber summary: A exploratory data aalysis techique that uses the followig five umbers to summarize the data set: smallest value, first quartile, media, third quartile, ad largest value. Momets: Momets are the arithmetic meas of the powers to which the deviatios are raised. Correlatio: Correlatio is a measure of the stregth or the degree of relatioship betwee two radom variables. OR Iterdepedece of two variables is called correlatio. OR Correlatio is a techique which measures the stregth of associatio betwee two variables. RANDOM EXPERIMENT: A experimet which produces differet results eve though it is repeated a large umber of times uder essetially similar coditios is called a Radom Experimet. SAMPLE SPACE: A set cosistig of all possible outcomes that ca result from a radom experimet (real or coceptual), ca be defied as the sample space for the experimet ad is deoted by the letter S. Each possible outcome is a member of the sample space, ad is called a sample poit i that space.
21 1 SIMPLE & COMPOUND EVENTS: A evet that cotais exactly oe sample poit is defied as a simple evet. A compoud evet cotais more tha oe sample poit, ad is produced by the uio of simple evets. MUTUALLY EXCLUSIVE EVENTS: Two evets A ad B of a sigle experimet are said to be mutually exclusive or disjoit if ad oly if they caot both occur at the same time i.e. they have o poits i commo. EXHAUSTIVE EVENTS: Evets are said to be collectively exhaustive, whe the uio of mutually exclusive evets is equal to the etire sample space S. EQUALLY LIKELY EVENTS: Two evets A ad B are said to be equally likely, whe oe evet is as likely to occur as the other. I other words, each evet should occur i equal umber i repeated trials. Skewess: Skewess is the lack of symmetry i a distributio aroud some cetral value (mea, media or mode).it is thus the degree of a symmetry. Kurtosis: Kurtosis is the degree of peakess of a distributio usually take relative to a ormal distributio. MOMENTS: A momet desigates the power to which deviatios are raised before averagig them. Probability: Probability is defied as the ratio of favorable cases over equally likely cases. SUBJECTIVE OR PERSONALISTIC PROBABILITY: Probability i this sese is purely subjective, ad is based o whatever evidece is available to the idividual. It has a disadvatage that two or more persos faced with the same evidece may arrive at differet probabilities. Coditioal Probability: The probability of the occurrece of a evet A whe it is kow that some other evet B has already occurred is called the coditioal probability. MULTIPLICATION LAW: The probability that two evets A ad B will both occur is equal to the probability that oe of the evets will occur multiplied by the coditioal probability that the other evet will occur give that the first evet has already occurred. INDEPENDENT EVENTS: Two evets A ad B i the same sample space S, are defied to be idepedet (or statistically idepedet) if the probability that oe evet occurs, is ot affected by whether the other evet has or has ot occurred, Baye s theorem: A method used to compute posterior probabilities. RANDOM VARIABLE: Such a umerical quatity whose value is determied by the outcome of a radom experimet is called a radom variable.
22 Discrete radom variable: A radom variable that may assume either a fiite umber of values or a ifiite sequece of values. Set: A set is ay welldefied collectio or list of distict objects, e.g. a group of studets, the books i a library, the itegers betwee 1 ad 100, all huma beigs o the earth, etc. SUBSET: A set that cosists of some elemets of aother set, is called a subset of that set. For example, if B is a subset of A, the every member of set B is also a member of set A. If B is a subset of A, we write: (B A) PROPER SUBSET: If a set B cotais some but ot all of the elemets of aother set A, while A cotais each elemet of B, the the set B is defied to be a proper subset of A. Uiversal Set: The origial set of which all the sets we talk about, are subsets, is called the uiversal set (or the space) ad is geerally deoted by S or Ω. VENN DIAGRAM: A diagram that is uderstood to represet sets by circular regios, parts of circular regios or their complemets with respect to a rectagle represetig the space S is called a Ve diagram, amed after the Eglish logicia Joh Ve ( ). The Ve diagrams are used to represet sets ad subsets i a pictorial way ad to verify the relatioship amog sets ad subsets. UNION OF SETS: The uio or sum of two sets A ad B, deoted by (A B), ad read as A uio B, meas the set of all elemets that belog to at least oe of the sets A ad B, that is (A B) INTERSECTION OF SETS: The itersectio of two sets A ad B, deoted by A B, ad read as A itersectio B, meas that the set of all elemets that belog to both A ad B. OPERATIONS ON SETS: The basic operatios are uio, itersectio, differece ad complemetatio. SET DIFFERENCE: The differece of two sets A ad B, deoted by A B or by A (A B), is the set of all elemets of A which do ot belog to B. DISJOINT SETS: Two sets A ad B are defied to be disjoit or mutually exclusive or ooverlappig whe they have o elemets i commo, i.e. whe their itersectio is a empty set i.e. A B φ Cojoit Sets: Two sets A ad B are said to be cojoit whe they have at least oe elemet i commo. COMPLEMENTATION: The particular differece S A, that is, the set of all those elemets of S which do ot belog to A, is called the complemet of A ad is deoted by A or by Ac.
23 3 POWER SET: The class of ALL subsets of a set A is called the Power Set of A ad is deoted by P(A). For example, if A {H, T}, the P(A) {φ, {H}, {T}, {H, T}}. CARTESIAN PRODUCT OF SETS: The Cartesia product of sets A ad B, deoted by A B, (read as A cross B ), is a set that cotais all ordered pairs (x, y), where x belogs to A ad y belogs to B. This set is also called the Cartesia set of A ad B set of A ad B, amed after the Frech mathematicia Ree Descartes ( ). TREE DIAGRAM: The tree is costructed from the left to the right. A tree diagram is a useful device for eumeratig all the possible outcomes of two or more sequetial evets. PERMUTATION: A permutatio is ay ordered subset from a set of distict objects. OR A arragemet of all or some of a set of objects i a defiite order is called permutatio. COMBINATION: A combiatio is ay subset of r objects, selected without regard to their order, from a set of distict objects. OCCURRENCE OF AN EVENT: A evet A is said to occur if ad oly if the outcome of the experimet correspods to some elemet of A. PARTITION OF SETS: A partitio of a set S is a subdivisio of the set ito oempty subsets that are disjoit ad exhaustive, i.e. their uio is the set S itself. CLASS OF SETS: A set of sets is called a class. For example, i a set of lies, each lie is a set of poits. DISTRIBUTION FUNCTION: The distributio fuctio of a radom variable X, deoted by F(x), is defied by F(x) P(X < x). FROM LECTURE 3 TO 45 Chebychev s Iequality: If X is a radom variable havig mea µ ad variace > 0, ad k is ay positive costat, the the probability that a value of X falls withi k stadard deviatios of the mea is at least. CONTINUOUS RANDOM VARIABLE: A radom variable X is defied to be cotiuous if it ca assume every possible value i a iterval [a, b], a < b, where a ad b may be ad + respectively. JOINT DISTRIBUTIONS: The distributio of two or more radom variables which are observed simultaeously whe a experimet is performed is called their JOINT distributio.
24 4 BIVARIATE PROBABILITY FUNCTION: The joit or bivariate probability distributio cosistig of all pairs of values (xi, yj). MARGINAL PROBABILITY FUNCTIONS: The poit to be uderstood here is that, from the joit probability fuctio for (X, Y), we ca obtai the INDIVIDUAL probability fuctio of X ad Y. Such idividual probability fuctios are called MARGINAL probability fuctios. CONDITIONAL PROBABILITY FUNCTION: Let X ad Y be two discrete r.v. s with joit probability fuctio f(x, y). The the coditioal probability fuctio for X give Y y, deoted as f(x y). INDEPENDENCE: Two discrete r.v. s X ad Y are said to be statistically idepedet, if ad oly if, for all possible pairs of values (xi, yj) the joit probability fuctio f(x, y) ca be expressed as the product of the two margial probability fuctios. COVARIANCE OF TWO RANDOM VARIABLES: The covariace of two r.v. s X ad Y is a umerical measure of the extet to which their values ted to icrease or decrease together. It is deoted by XY or Cov (X, Y), ad is defied as the expected value of the product. BINOMIAL DISTRIBUTION: The biomial distributio is a very importat discrete probability distributio. It was discovered by James Beroulli about the year PROPERTIES OF A BINOMIAL EXPERIMENT: Every trial results i a success or a failure. The successive trials are idepedet. The probability of success, p, remais costat from trial to trial. The umber of trials,, is fixed i advaced. Biomial probability distributio: A probability distributio showig the probability of x successes i trials of a biomial experimet. Biomial probability fuctio: The fuctio used to compute probabilities i a biomial experimet. Biomial experimet: A probability experimet havig the followig four properties: cosists of idetical trials, two outcomes (success ad failure) are possible o each trial, probability of success does ot chage from trial to trail, ad the trials are idepedet. PROPERTIES OF A BINOMIAL EXPERIMENT: Every item selected will either be defective (i.e. success) or ot defective (i.e. failure) Every item draw is idepedet of every other item The probability of obtaiig a defective item i.e. 7% is the same (costat) for all items. (This probability figure is accordig to relative frequecy defiitio of probability. Hyper Geometric Probability fuctio: The fuctio used to compute the probability of x successes i trials whe the trials are depedet.
25 5 PROPERTIES OF HYPERGEOMETRIC EXPERIMENT: The outcomes of each trial may be classified ito oe of two categories, success ad failure. The probability of success chages o each trial. The successive trials are ot idepedet. The experimet is repeated a fixed umber of times. Hyper Geometric Distributio: There are may experimets i which the coditio of idepedece is violated ad the probability of success does ot remai costat for all trials. Such experimets are called hyper geometric experimets. PROPERTIES OF THE HYPERGEOMETRIC DISTRIBUTION: If N becomes idefiitely large, the hyper geometric probability distributio teds to the BINOMIAL probability distributio. The above property will be best uderstood with referece to the followig importat poits: There are two ways of drawig a sample from a populatio, samplig with replacemet, ad samplig without replacemet. Also, a sample ca be draw from either a fiite populatio or a ifiite populatio. Poisso Probability Distributio: A probability distributio showig the probability of x occurreces of a evet over a specified iterval of time or space. Poisso Probability Fuctio: The fuctio used to compute Poisso probabilities. PROPERTIES OF POISSON DISTRIBUTION: It is a limitig approximatio to the biomial distributio, whe p, the probability of success is very small but, the umber of trials is so large that the product p µ is of a moderate size; a distributio i its ow right by cosiderig a POISSON PROCESS where evets occur radomly over a specified iterval of time or space or legth. POISSON PROCESS: It may be defied as a physical process govered at least i part by some radom mechaism. NORMAL DISTRIBUTION: A cotiuous radom variable is said to be ormally distributed with mea μ ad stadard deviatio if its probability desity fuctio is give by (Formula of Normal Distributio) THE STANDARD NORMAL DISTRIBUTION: A ormal distributio whose mea is zero ad whose stadard deviatio is 1 is kow as the stadard ormal distributio. THE PROCESS OF STANDARDIZATION: I other words, the stadardizatio formula coverts our ormal distributio to the oe whose mea is 0 ad whose stadard deviatio is equal to 1. SAMPLING DISTRIBUTION: The probability distributio of ay statistic (such as the mea, the stadard deviatio, the proportio of successes i a sample, etc.) is kow as its samplig distributio.
26 6 CENTRAL LIMIT THEOREM: The theorem states that: If a variable X from a populatio has mea μ ad fiite variace, the the samplig distributio of the sample mea X approaches a ormal distributio with mea μ ad variace / as the sample size approaches ifiity. Samplig Distributio: A probability distributio cosistig of all possible values of a sample statistic. SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTION: I this regard, the first poit to be oted is that, wheever the elemets of a populatio ca be classified ito two categories, techically called success ad failure, we may be iterested i the proportio of successes i the populatio. POINT ESTIMATION: Poit estimatio of a populatio parameter provides as a estimate a sigle value calculated from the sample that is likely to be close i magitude to the ukow parameter. UNBIASEDNESS: A estimator is defied to be ubiased if the statistic used as a estimator has its expected value equal to the true value of the populatio parameter beig estimated. CONSISTENCY: A estimator θˆ is said to be a cosistet estimator of the parameter θ if, for ay arbitrarily small positive quatity e. EFFICIENCY: A ubiased estimator is defied to be efficiet if the variace of its samplig distributio is smaller tha that of the samplig distributio of ay other ubiased estimator of the same parameter. METHODS OF POINTESTIMATION: The Method of Momets The Method of Least Squares The Method of Maximum Likelihood These methods give estimates which may differ as the methods are based o differet theories of estimatio. THE METHOD OF LEAST SQUARES: The method of Least Squares, which is due to Gauss ( ) ad Markov ( ), is based o the theory of liear estimatio. It is regarded as oe of the importat methods of poit estimatio. METHOD OF MAXIMUM LIKELIHOOD: This method was itroduced i 19 by Sir Roald A. Fisher ( ).The mathematical techique of fidig Maximum Likelihood Estimators is a bit advaced, ad ivolves the cocept of the Likelihood Fuctio. HYPOTHESISTESTING: It is a procedure which eables us to decide o the basis of iformatio obtaied from sample data whether to accept or reject a statemet or a assumptio about the value of a populatio parameter. NULL HYPOTHESIS: A ull hypothesis, geerally deoted by the symbol H0, is ay hypothesis which is to be tested for possible rejectio or ullificatio uder the assumptio that it is true.
27 7 ALTERNATIVE HYPOTHESIS: A alterative hypothesis is ay other hypothesis which we are willig to accept whe the ull hypothesis H0 is rejected. It is customarily deoted by H1 or HA. TYPEI AND TYPEII ERRORS: O the basis of sample iformatio, we may reject a ull hypothesis H0, whe it is, i fact, true or we may accept a ull hypothesis H0, whe it is actually false. The probability of makig a Type I error is covetioally deoted by α ad that of committig a Type II error is idicated by β. TESTSTATISTIC: A statistic (i.e. a fuctio of the sample data ot cotaiig ay parameters), which provides a basis for testig a ull hypothesis, is called a teststatistic. PROPERTIES OF STUDENT S tdistribution: i) The tdistributio is bellshaped ad symmetric about the value t 0, ragig from to. ii) The umber of degrees of freedom determies the shape of the tdistributio. PROPERTIES OF FDISTRIBUTION: 1. The Fdistributio is a cotiuous distributio ragig from zero to plus ifiity.. The curve of the Fdistributio is positively skewed. ANALYSIS OF VARIANCE (ANOVA): It is a procedure which eables us to test the hypothesis of equality of several populatio meas. EXPERIMENTAL DESIGN: By a experimetal desig, we mea a pla used to collect the data relevat to the problem uder study i such a way as to provide a basis for valid ad objective iferece about the stated problem. The pla usually icludes: The selectio of treatmets, whose effects are to be studied, The specificatio of the experimetal layout, ad The assigmet of treatmets to the experimetal uits. SYSTEMATIC AND RANDOMIZED DESIGNS: I this course, we will be discussig oly the radomized desigs, ad, i this regard, it should be oted that for the radomized desigs, the aalysis of the collected data is carried out through the techique kow as Aalysis of Variace. THE COMPLETELY RANDOMIZED DESIGN (CR DESIGN): A completely radomized (CR) desig, which is the simplest type of the basic desigs, may be defied as a desig i which the treatmets are assiged to experimetal uits completely at radom, i.e. the radomizatio is doe without ay restrictios. THE RANDOMIZED COMPLETE BLOCK DESIGN (RCB DESIGN): A radomized complete block (RCB) desig is the oe i which The experimetal material (which is ot homogeeous overall) is divided ito groups or blocks i such a maer that the experimetal uits withi a particular block are relatively homogeeous. Each block cotais a complete set of treatmets, i.e., it costitutes a replicatio of treatmets. The treatmets are allocated at radom to the experimetal uits withi each block, which meas the radomizatio is restricted. (A ew radomizatio is made for every block.)the object of this type of arragemet is to brig the variability of the experimetal material uder cotrol.
28 8 Least Squares Method: The method used to develop the estimated regressio equatio. It miimizes the sum of squared residuals (the deviatios betwee the observed values of the depedet variable, yi, ad the estimated values of the depedet variable, yi) Level of Sigificace: Level of sigificace of a test is the probability used as a stadard for rejectig ull hypothesis Ho whe Ho is assumed to be true. The level of sigificace acts as a basis for determiig the critical regio of the test. THE LEAST SIGNIFICANT DIFFERENCE (LSD) TEST: Accordig to this procedure, we compute the smallest differece that would be judged sigificat, ad compare the absolute values of all differeces of meas with it. This smallest differece is called the least sigificat differece or LSD. PROPERTIES OF THE CHISQUARE DISTRIBUTION: The ChiSquare (χ) distributio has the followig properties: 1. It is a cotiuous distributio ragig from 0 to +.The umber of degrees of freedom determies the shape of the chisquare distributio. (Thus, there is a differet chisquare distributio for each umber of degrees of freedom. As such, it is a whole family of distributios.). The curve of a chisquare distributio is positively skewed. The skewess decreases as ν icreases. ASSUMPTIONS OF THE CHISQUARE TEST OF GOODNESS OF FIT: While applyig the chisquare test of goodess of fit, certai requiremets must be satisfied, three of which are as follows: 1. The total umber of observatios (i.e. the sample size) should be at least 50.. The expected umber ei i ay of the categories should ot be less tha 5. (So, whe the expected frequecy ei i ay category is less tha 5, we may combie this category with oe or more of the other categories to get ei 5.) 3. The observatios i the sample or the frequecies of the categories should be idepedet. DEGREES OF FREEDOM: As you will recall, whe discussig the tdistributio, the chisquare distributio, ad the Fdistributio, it was coveyed to you that the parameters that exists i the equatios of those distributios are kow as degrees of freedom. P value: The pvalue is a property of the data, ad it idicates how improbable the obtaied result really is. LATEST STATISTICAL DEFINITION: Statistics is a sciece of decisio makig for goverig the state affairs. It collects aalyzes, maages, moitors, iterprets, evaluates ad validates iformatio. Statistics is Iformatio Sciece ad Iformatio Sciece is Statistics. It is a applicable sciece as its tools are applied to all scieces icludig humaities ad social scieces.
MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
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 informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Crosssectioal data. 2. Time series data.
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 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 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 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 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 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 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 informationLecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH email: za@agh.edu.pl http://home.agh.edu.pl/~za
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 informationGeneral IxJ Contingency Tables
page1 Geeral x Cotigecy Tables We ow geeralize our previous results from the prospective, retrospective ad crosssectioal studies ad the Poisso samplig case to x cotigecy tables. For such tables, the test
More informationSample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for
Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they
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 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 informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationSampling, Sampling Distribution and Normality
4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig
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 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 information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
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 informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hardcopy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationIIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
More informationGoodnessOfFit For The Generalized Exponential Distribution. Abstract
GoodessOfFit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationStat 200 Testing Summary Page 1
Stat 00 Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,
More informationA PROBABILITY PRIMER
CARLETON COLLEGE A ROBABILITY RIMER SCOTT BIERMAN (Do ot quote without permissio) A robability rimer INTRODUCTION The field of probability ad statistics provides a orgaizig framework for systematically
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 informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oedimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationProbability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationUniversity of California, Los Angeles Department of Statistics. Hypothesis testing
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Elemets of a hypothesis test: Hypothesis testig Istructor: Nicolas Christou 1. Null hypothesis, H 0 (claim about µ, p, σ 2, µ
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 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 information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
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 informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
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 informationImportant Concepts not on the AP Statistics Formula Sheet
Part I: IQR = Q 3 Q 1 Test for a outlier: 1.5(IQR) above Q 3 or below Q 1 The calculator will ru the test for you as log as you choose the boplot with the oulier o it i STATPLOT Importat Cocepts ot o the
More informationExam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.
Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material
More informationUnbiased Estimation. February 712, 2008
Ubiased Estimatio February 72, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
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 informationB Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets
B671672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x
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 informationNumber 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 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 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 informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
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 informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. BetaBinomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 12631277 HIKARI Ltd, www.mhikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
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 informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
More informationCURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:
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
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
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 informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationy ij = µ + α i + ɛ ij,
STAT 4 ANOVA Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i
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 informationEcon 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement.
Eco 371 Exam #1 Multiple Choice (5 poits each): For each of the followig, select the sigle most appropriate optio to complete the statemet 1) The probability of a outcome a) is the umber of times that
More informationSampling Error. Chapter 6 Student Lecture Notes 61. Business Statistics: A DecisionMaking Approach, 6e. Chapter Goals
Chapter 6 Studet Lecture Notes 61 Busiess Statistics: A DecisioMakig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 61 Chapter Goals After completig this chapter, you should
More informationFirst Year Quantitative Comp Exam Spring, Part I  203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I1 Part I  203A A radom variable X is distributed with the margial desity: >
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
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 informationOBJECTIVES. Chapter 1 INTRODUCTION TO INSTRUMENTATION FUNCTION AND ADVANTAGES INTRODUCTION. At the end of this chapter, students should be able to:
OBJECTIVES Chapter 1 INTRODUCTION TO INSTRUMENTATION At the ed of this chapter, studets should be able to: 1. Explai the static ad dyamic characteristics of a istrumet. 2. Calculate ad aalyze the measuremet
More informationTables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc.
Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER Orgaizig ad Summarizig Data Relative frequecy frequecy um of all frequecie Cla midpoit: The um of coecutive lower
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 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 informationModeling and Performance Analysis with DiscreteEvent Simulation
Simulatio Modelig ad Performace Aalysis with DiscreteEvet Simulatio Chapter 5 Statistical Models i Simulatio Cotets Basic Probability Theory Cocepts Useful Statistical Models Discrete Distributios Cotiuous
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Lecture 16
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Lecture 16 Variace Questio: Let us retur oce agai to the questio of how may heads i a typical sequece of coi flips. Recall that we
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationA goodnessoffit test based on the empirical characteristic function and a comparison of tests for normality
A goodessoffit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
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 informationClosed book and notes. No calculators. 60 minutes, but essentially unlimited time.
IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of
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 informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
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 informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe, ad two, sided tolerace itervals based o either a distributiofree (oparametric) method or a method based o a ormality assumptio (parametric). A twosided
More informationSimple Linear Regression
Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio
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 informationStatistics 20: Final Exam Solutions Summer Session 2007
1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets
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 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 information