AS-Level Maths: Statistics 1 for Edexcel

Size: px
Start display at page:

Download "AS-Level Maths: Statistics 1 for Edexcel"

Transcription

1 1 of 6 AS-Level Maths: Statstcs 1 for Edecel S1. Calculatng means and standard devatons Ths con ndcates the slde contans actvtes created n Flash. These actvtes are not edtable. For more detaled nstructons, see the Gettng Started presentaton.

2 Contents Means Calculatng means Calculatng standard devatons Codng of 6

3 Mean 3 of 6 The mean s the most wdely used average n statstcs. It s found by addng up all the values n the data and dvdng by how many values there are. Notaton: If the data values are mean s Ths s the mean symbol 1,, 3,..., n... n n n 1 3, then the Note: The mean takes nto account every pece of data, so t s affected by outlers n the data. The medan s preferred over the mean f the data contans outlers or s skewed. Ths symbol means the total of all the values

4 Mean 4 of 6 If data are presented n a frequency table: Value Frequency f 1 1 f n f n then the mean s f f... f 1 1 n n f f f

5 5 of 6 Mean Eample: The table shows the results of a survey nto household sze. Fnd the mean sze. Household sze, Frequency, f f TOTAL To fnd the mean, we add a 3 rd column to the table. Mean = = 3.01

6 Contents Standard devaton Calculatng means Calculatng standard devatons Codng 6 of 6

7 Standard devaton 7 of 6 There are three commonly used measures of spread (or dsperson) the range, the nter-quartle range and the standard devaton. The standard devaton s wdely used n statstcs to measure spread. It s based on all the values n the data, so t s senstve to the presence of outlers n the data. The varance s related to the standard devaton: varance = (standard devaton) The followng formulae can be used to fnd the varance and s.d. varance ( ) ( ) s.d. n n

8 Standard devaton 8 of 6 Eample: The md-day temperatures (n C) recorded for one week n June were: 1, 3, 4, 19, 19, 0, 1 Frst we fnd the mean: C Total: ( ) ( ) varance n So varance = 7 = So, s.d. = 1.77 C (3 s.f.)

9 Standard devaton 9 of 6 There s an alternatve formula whch s usually a more convenent way to fnd the varance: ( ) varance n But, ( ) ( ) n n n n Therefore, varance n and s.d. n

10 Standard devaton 10 of 6 Eample (contnued): Lookng agan at the temperature data for June: 1, 3, 4, 19, 19, 0, 1 We know that C 7 Also, = 3109 So, 3109 varance n 7 s.d C Note: Essentally the standard devaton s a measure of how close the values are to the mean value.

11 Calculatng standard devaton from a table 11 of 6 When the data s presented n a frequency table, the formula for fndng the standard devaton needs to be adjusted slghtly: s.d. f f Eample: A class of 0 students were asked how many tmes they eercse n a normal week. Fnd the mean and the standard devaton. Number of tmes eercse taken Frequency

12 Calculatng standard devaton from a table 1 of 6 No. of tmes eercse taken, Frequency, f f f TOTAL: The table can be etended to help fnd the mean and the s.d f 116 s.d f 0

13 13 of 6 Calculatng standard devaton from a table If data s presented n a grouped frequency table, t s only possble to estmate the mean and the standard devaton. Ths s because the eact data values are not known. An estmate s obtaned by usng the md-pont of an nterval to represent each of the values n that nterval. Eample: The table shows the annual mleage for the employees of an nsurance company. Estmate the mean and standard devaton. Annual mleage, Frequency 0 < < 10, ,000 < 15, ,000 < 0, ,000 < 30,000 3

14 Calculatng standard devaton from a table 14 of 6 Mleage Frequency, f Md-pont, f f ,500, , , ,50,000 10,000 15, , ,000,187,500,000 15,000 0, ,500 87,500 1,531,50,000 0,000 30, ,000 75,000 1,875,000,000 TOTAL ,000 6,587,500, ,000 10,667 mles 45 6,587,500,000 s.d. 10, mles

15 Notes about standard devaton 15 of 6 Here are some notes to consder about standard devaton. In most dstrbutons, about 67% of the data wll le wthn 1 standard devaton of the mean, whlst nearly all the data values wll le wthn standard devatons of the mean. Values that le more than standard devatons from the mean are sometmes classed as outlers any such values should be treated carefully. Standard devaton s measured n the same unts as the orgnal data. Varance s measured n the same unts squared. Most calculators have a bult-n functon whch wll fnd the standard devaton for you. Learn how to use ths faclty on your calculator.

16 Eamnaton-style queston Eamnaton-style queston: The ages of the people n a cnema queue one Monday afternoon are shown n the stem-and-leaf dagram: 16 of 6 3 means 3 years old a) Eplan why the dagram suggests that the mean and standard devaton can be sensbly used as measures of locaton and spread respectvely. b) Calculate the mean and the standard devaton of the ages. c) The mean and the standard devaton of the ages of the people n the queue on Monday evenng were 9 and 6. respectvely. Compare the ages of the people queung at the cnema n the afternoon wth those n the evenng.

17 Eamnaton-style queston a) The mean and the standard devaton are approprate, as the dstrbuton of ages s roughly symmetrcal and there are no outlers b) 597 so, of 6 3 means 3 years old , so, s.d c) The cnemagoers n the evenng had a smaller mean age, meanng that they were, on average, younger than those n the afternoon. The standard devaton for the ages n the evenng was also smaller, suggestng that the evenng audence were closer together n age.

18 18 of 6 Combnng sets of data Sometmes n eamnaton questons you are asked to pool two sets of data together. Eample: S male and fve female students st an A-level eamnaton. The mean marks were 5% and 57% for the males and females respectvely. The standard devatons were 14 and 18 respectvely. Fnd the combned mean and the standard devaton for the marks of all 11 students.

19 Combnng sets of data 19 of 6 Let Let y,..., 1 6,..., y 1 5 be the marks for the 6 male students. be the marks of the 5 female students. To fnd the overall mean, we frst need to fnd the total marks for all 11 students. As As y 57 y Therefore y So the combned mean s: % 11 54

20 Combnng sets of data 0 of 6 To fnd the overall standard devaton, we need to fnd the total of the marks squared for all 11 students. Notce that the formula s.d. n rearranges to gve As s.d. 14 As s.d. y 18 Therefore, n( s.d. ) 6( 14 5 ) 17,400 y 5( ) 17,865 y 35,65 So the combned s.d. s: 35, % 11 (to 3 s.f.)

21 Contents Codng Calculatng means Calculatng standard devatons Codng 1 of 6

22 Codng of 6 Codng s a technque that can smplfy the numercal effort requred n fndng a mean or standard devaton. Enter some data below, and see how t changes when you add or multply by dfferent numbers.

23 Codng 3 of 6 Addng So, f a number b s added to each pece of data, the mean value s also ncreased by b. The standard devaton s unchanged. Multplyng If each pece of data s multpled by a, the mean value s multpled by a. The standard devaton s also multpled by a. More formally, f s.d. y a b y a b y as.d. then:

24 Codng 4 of 6 Eample: Fnd the mean and the standard devaton of the values n the table. Use the transformaton below to help you. 1 y 5 10 Frequency y Usng the gven transformaton, add a y column to the table.

25 Codng 5 of 6 y Frequency, f y f y f Total To fnd the mean: y To fnd the s.d.: f y 85 s.d. y f 0

26 Codng 6 of 6 You have now found the mean and standard devaton of y. To fnd them for the values, you must reverse the codng. 1 We can rearrange: y 5 10 to get: 10y50 Therefore the mean of s: 10y And the standard devaton of s: = 10.9 Note how the codng helped to smplfy the calculatons by makng the numbers smaller.

Chapter 3 Describing Data Using Numerical Measures

Chapter 3 Describing Data Using Numerical Measures Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The

More information

Definition. Measures of Dispersion. Measures of Dispersion. Definition. The Range. Measures of Dispersion 3/24/2014

Definition. Measures of Dispersion. Measures of Dispersion. Definition. The Range. Measures of Dispersion 3/24/2014 Measures of Dsperson Defenton Range Interquartle Range Varance and Standard Devaton Defnton Measures of dsperson are descrptve statstcs that descrbe how smlar a set of scores are to each other The more

More information

Statistical Inference. 2.3 Summary Statistics Measures of Center and Spread. parameters ( population characteristics )

Statistical Inference. 2.3 Summary Statistics Measures of Center and Spread. parameters ( population characteristics ) Ismor Fscher, 8//008 Stat 54 / -8.3 Summary Statstcs Measures of Center and Spread Dstrbuton of dscrete contnuous POPULATION Random Varable, numercal True center =??? True spread =???? parameters ( populaton

More information

Q1: Calculate the mean, median, sample variance, and standard deviation of 25, 40, 05, 70, 05, 40, 70.

Q1: Calculate the mean, median, sample variance, and standard deviation of 25, 40, 05, 70, 05, 40, 70. Q1: Calculate the mean, medan, sample varance, and standard devaton of 5, 40, 05, 70, 05, 40, 70. Q: The frequenc dstrbuton for a data set s gven below. Measurements 0 1 3 4 Frequenc 3 5 8 3 1 a) What

More information

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of

More information

THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for

More information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there

More information

18.1 Introduction and Recap

18.1 Introduction and Recap CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng

More information

e i is a random error

e i is a random error Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown

More information

Cathy Walker March 5, 2010

Cathy Walker March 5, 2010 Cathy Walker March 5, 010 Part : Problem Set 1. What s the level of measurement for the followng varables? a) SAT scores b) Number of tests or quzzes n statstcal course c) Acres of land devoted to corn

More information

x = , so that calculated

x = , so that calculated Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to

More information

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA 4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected

More information

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9 Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,

More information

Economics 130. Lecture 4 Simple Linear Regression Continued

Economics 130. Lecture 4 Simple Linear Regression Continued Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do

More information

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise. Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the

More information

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott

More information

/ n ) are compared. The logic is: if the two

/ n ) are compared. The logic is: if the two STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence

More information

Notes prepared by Prof Mrs) M.J. Gholba Class M.Sc Part(I) Information Technology

Notes prepared by Prof Mrs) M.J. Gholba Class M.Sc Part(I) Information Technology Inverse transformatons Generaton of random observatons from gven dstrbutons Assume that random numbers,,, are readly avalable, where each tself s a random varable whch s unformly dstrbuted over the range(,).

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

Check off these skills when you feel that you have mastered them. List and describe two types of distributions for a histogram.

Check off these skills when you feel that you have mastered them. List and describe two types of distributions for a histogram. Chapter Objectves Check off these sklls when you feel that you have mastered them. Construct a hstogram for a small data set. Lst and descrbe two types of dstrbutons for a hstogram. Identfy from a hstogram

More information

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6 Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.

More information

6.3.4 Modified Euler s method of integration

6.3.4 Modified Euler s method of integration 6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from

More information

The Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions

The Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions 5-6 The Fundamental Theorem of Algebra Content Standards N.CN.7 Solve quadratc equatons wth real coeffcents that have comple solutons. N.CN.8 Etend polnomal denttes to the comple numbers. Also N.CN.9,

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used

More information

Exercises. 18 Algorithms

Exercises. 18 Algorithms 18 Algorthms Exercses 0.1. In each of the followng stuatons, ndcate whether f = O(g), or f = Ω(g), or both (n whch case f = Θ(g)). f(n) g(n) (a) n 100 n 200 (b) n 1/2 n 2/3 (c) 100n + log n n + (log n)

More information

= z 20 z n. (k 20) + 4 z k = 4

= z 20 z n. (k 20) + 4 z k = 4 Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

The optimal delay of the second test is therefore approximately 210 hours earlier than =2.

The optimal delay of the second test is therefore approximately 210 hours earlier than =2. THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

Statistics Chapter 4

Statistics Chapter 4 Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment

More information

F statistic = s2 1 s 2 ( F for Fisher )

F statistic = s2 1 s 2 ( F for Fisher ) Stat 4 ANOVA Analyss of Varance /6/04 Comparng Two varances: F dstrbuton Typcal Data Sets One way analyss of varance : example Notaton for one way ANOVA Comparng Two varances: F dstrbuton We saw that the

More information

PHYS 450 Spring semester Lecture 02: Dealing with Experimental Uncertainties. Ron Reifenberger Birck Nanotechnology Center Purdue University

PHYS 450 Spring semester Lecture 02: Dealing with Experimental Uncertainties. Ron Reifenberger Birck Nanotechnology Center Purdue University PHYS 45 Sprng semester 7 Lecture : Dealng wth Expermental Uncertantes Ron Refenberger Brck anotechnology Center Purdue Unversty Lecture Introductory Comments Expermental errors (really expermental uncertantes)

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

CHAPTER 8. Exercise Solutions

CHAPTER 8. Exercise Solutions CHAPTER 8 Exercse Solutons 77 Chapter 8, Exercse Solutons, Prncples of Econometrcs, 3e 78 EXERCISE 8. When = N N N ( x x) ( x x) ( x x) = = = N = = = N N N ( x ) ( ) ( ) ( x x ) x x x x x = = = = Chapter

More information

Lecture 3 Stat102, Spring 2007

Lecture 3 Stat102, Spring 2007 Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture

More information

STAT 511 FINAL EXAM NAME Spring 2001

STAT 511 FINAL EXAM NAME Spring 2001 STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte

More information

Chapter 6. Supplemental Text Material

Chapter 6. Supplemental Text Material Chapter 6. Supplemental Text Materal S6-. actor Effect Estmates are Least Squares Estmates We have gven heurstc or ntutve explanatons of how the estmates of the factor effects are obtaned n the textboo.

More information

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable

More information

Correlation and Regression

Correlation and Regression Correlaton and Regresson otes prepared by Pamela Peterson Drake Index Basc terms and concepts... Smple regresson...5 Multple Regresson...3 Regresson termnology...0 Regresson formulas... Basc terms and

More information

WINTER 2017 EXAMINATION

WINTER 2017 EXAMINATION (ISO/IEC - 700-00 Certfed) WINTER 07 EXAMINATION Model wer ject Code: Important Instructons to Eamners: ) The answers should be eamned by key words and not as word-to-word as gven n the model answer scheme.

More information

Important Instructions to the Examiners:

Important Instructions to the Examiners: Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as word-to-word as gven n the model

More information

Chapter 13: Multiple Regression

Chapter 13: Multiple Regression Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to

More information

As is less than , there is insufficient evidence to reject H 0 at the 5% level. The data may be modelled by Po(2).

As is less than , there is insufficient evidence to reject H 0 at the 5% level. The data may be modelled by Po(2). Ch-squared tests 6D 1 a H 0 : The data can be modelled by a Po() dstrbuton. H 1 : The data cannot be modelled by Po() dstrbuton. The observed and expected results are shown n the table. The last two columns

More information

ROBUST AND EFFICIENT ESTIMATION OF THE MODE OF CONTINUOUS DATA: THE MODE AS A VIABLE MEASURE OF CENTRAL TENDENCY

ROBUST AND EFFICIENT ESTIMATION OF THE MODE OF CONTINUOUS DATA: THE MODE AS A VIABLE MEASURE OF CENTRAL TENDENCY ROBUST AND EFFICIENT ESTIMATION OF THE MODE OF CONTINUOUS DATA: THE MODE AS A VIABLE MEASURE OF CENTRAL TENDENCY Davd R. Bckel Medcal College of Georga Offce of Bostatstcs and Bonformatcs Ffteenth St.,

More information

Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD

Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD - 08 Date: 0.03.08 MATHEMATICS - PAPER- - SOLUTIONS Q. Attempt any FIVE of

More information

For example, if the drawing pin was tossed 200 times and it landed point up on 140 of these trials,

For example, if the drawing pin was tossed 200 times and it landed point up on 140 of these trials, Probablty In ths actvty you wll use some real data to estmate the probablty of an event happenng. You wll also use a varety of methods to work out theoretcal probabltes. heoretcal and expermental probabltes

More information

Statistics II Final Exam 26/6/18

Statistics II Final Exam 26/6/18 Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the

More information

Lossy Compression. Compromise accuracy of reconstruction for increased compression.

Lossy Compression. Compromise accuracy of reconstruction for increased compression. Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost

More information

UNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours

UNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x

More information

Case A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.

Case A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k. THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the Szemeréd-Trotter theorem. The method was ntroduced n the paper Combnatoral complexty

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal

Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal 9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd

More information

Statistics MINITAB - Lab 2

Statistics MINITAB - Lab 2 Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that

More information

STATISTICS AND PROBABILITY

STATISTICS AND PROBABILITY CHAPTER 3 STATISTICS AND PROBABILITY (A) Man Concepts and Results Statstcs Measures of Central Tendency (a) Mean of Grouped Data () () () To fnd the mean of grouped data, t s assumed that the frequency

More information

Multiple Contrasts (Simulation)

Multiple Contrasts (Simulation) Chapter 590 Multple Contrasts (Smulaton) Introducton Ths procedure uses smulaton to analyze the power and sgnfcance level of two multple-comparson procedures that perform two-sded hypothess tests of contrasts

More information

THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions

THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George

More information

Linear Feature Engineering 11

Linear Feature Engineering 11 Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19

More information

Answers Problem Set 2 Chem 314A Williamsen Spring 2000

Answers Problem Set 2 Chem 314A Williamsen Spring 2000 Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method

More information

1 GSW Iterative Techniques for y = Ax

1 GSW Iterative Techniques for y = Ax 1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn

More information

Introduction to Regression

Introduction to Regression Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes

More information

Chapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D

Chapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D Chapter Twelve Integraton 12.1 Introducton We now turn our attenton to the dea of an ntegral n dmensons hgher than one. Consder a real-valued functon f : R, where the doman s a nce closed subset of Eucldean

More information

Chapter 11: Simple Linear Regression and Correlation

Chapter 11: Simple Linear Regression and Correlation Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests

More information

SIMPLE LINEAR REGRESSION

SIMPLE LINEAR REGRESSION Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two

More information

Exercise 1 The General Linear Model : Answers

Exercise 1 The General Linear Model : Answers Eercse The General Lnear Model Answers. Gven the followng nformaton on 67 pars of values on and -.6 - - - 9 a fnd the OLS coeffcent estmate from a regresson of on. Usng b 9 So. 9 b Suppose that now also

More information

BIO Lab 2: TWO-LEVEL NORMAL MODELS with school children popularity data

BIO Lab 2: TWO-LEVEL NORMAL MODELS with school children popularity data Lab : TWO-LEVEL NORMAL MODELS wth school chldren popularty data Purpose: Introduce basc two-level models for normally dstrbuted responses usng STATA. In partcular, we dscuss Random ntercept models wthout

More information

Polynomial Regression Models

Polynomial Regression Models LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance

More information

Primer on High-Order Moment Estimators

Primer on High-Order Moment Estimators Prmer on Hgh-Order Moment Estmators Ton M. Whted July 2007 The Errors-n-Varables Model We wll start wth the classcal EIV for one msmeasured regressor. The general case s n Erckson and Whted Econometrc

More information

A Robust Method for Calculating the Correlation Coefficient

A Robust Method for Calculating the Correlation Coefficient A Robust Method for Calculatng the Correlaton Coeffcent E.B. Nven and C. V. Deutsch Relatonshps between prmary and secondary data are frequently quantfed usng the correlaton coeffcent; however, the tradtonal

More information

Statistics for Economics & Business

Statistics for Economics & Business Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable

More information

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction Prerequste Sklls Ths lesson requres the use of the followng sklls: understandng that multplyng the numerator and denomnator of a fracton by the same quantty produces an equvalent fracton multplyng complex

More information

Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng

More information

experimenteel en correlationeel onderzoek

experimenteel en correlationeel onderzoek expermenteel en correlatoneel onderzoek lecture 6: one-way analyss of varance Leary. Introducton to Behavoral Research Methods. pages 246 271 (chapters 10 and 11): conceptual statstcs Moore, McCabe, and

More information

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models

Statistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables

More information

Bayesian Planning of Hit-Miss Inspection Tests

Bayesian Planning of Hit-Miss Inspection Tests Bayesan Plannng of Ht-Mss Inspecton Tests Yew-Meng Koh a and Wllam Q Meeker a a Center for Nondestructve Evaluaton, Department of Statstcs, Iowa State Unversty, Ames, Iowa 5000 Abstract Although some useful

More information

January Examinations 2015

January Examinations 2015 24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)

More information

A random variable is a function which associates a real number to each element of the sample space

A random variable is a function which associates a real number to each element of the sample space Introducton to Random Varables Defnton of random varable Defnton of of random varable Dscrete and contnuous random varable Probablty blt functon Dstrbuton functon Densty functon Sometmes, t s not enough

More information

AP Physics 1 & 2 Summer Assignment

AP Physics 1 & 2 Summer Assignment AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers

More information

FREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,

FREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced, FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then

More information

CS-433: Simulation and Modeling Modeling and Probability Review

CS-433: Simulation and Modeling Modeling and Probability Review CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

CHAPTER IV RESEARCH FINDING AND DISCUSSIONS

CHAPTER IV RESEARCH FINDING AND DISCUSSIONS CHAPTER IV RESEARCH FINDING AND DISCUSSIONS A. Descrpton of Research Fndng. The Implementaton of Learnng Havng ganed the whole needed data, the researcher then dd analyss whch refers to the statstcal data

More information

: 5: ) A

: 5: ) A Revew 1 004.11.11 Chapter 1: 1. Elements, Varable, and Observatons:. Type o Data: Qualtatve Data and Quanttatve Data (a) Qualtatve data may be nonnumerc or numerc. (b) Quanttatve data are always numerc.

More information

Why Monte Carlo Integration? Introduction to Monte Carlo Method. Continuous Probability. Continuous Probability

Why Monte Carlo Integration? Introduction to Monte Carlo Method. Continuous Probability. Continuous Probability Introducton to Monte Carlo Method Kad Bouatouch IRISA Emal: kad@rsa.fr Wh Monte Carlo Integraton? To generate realstc lookng mages, we need to solve ntegrals of or hgher dmenson Pel flterng and lens smulaton

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results. Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson

More information

Chapter 3. Descriptive Statistics Numerical Methods

Chapter 3. Descriptive Statistics Numerical Methods Chapter 3 Descrptve Statstcs Numercal Methods Our goal? Numbers to help us answer smple questons. What s a typcal value? How varable are the data? How extreme s a partcular value? Gven data on two varables,

More information

Chapter 8 Indicator Variables

Chapter 8 Indicator Variables Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n

More information

Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010

Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010 Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton

More information

Negative Binomial Regression

Negative Binomial Regression STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...

More information

This column is a continuation of our previous column

This column is a continuation of our previous column Comparson of Goodness of Ft Statstcs for Lnear Regresson, Part II The authors contnue ther dscusson of the correlaton coeffcent n developng a calbraton for quanttatve analyss. Jerome Workman Jr. and Howard

More information

Statistical analysis using matlab. HY 439 Presented by: George Fortetsanakis

Statistical analysis using matlab. HY 439 Presented by: George Fortetsanakis Statstcal analyss usng matlab HY 439 Presented by: George Fortetsanaks Roadmap Probablty dstrbutons Statstcal estmaton Fttng data to probablty dstrbutons Contnuous dstrbutons Contnuous random varable X

More information

Linear Correlation. Many research issues are pursued with nonexperimental studies that seek to establish relationships among 2 or more variables

Linear Correlation. Many research issues are pursued with nonexperimental studies that seek to establish relationships among 2 or more variables Lnear Correlaton Many research ssues are pursued wth nonexpermental studes that seek to establsh relatonshps among or more varables E.g., correlates of ntellgence; relaton between SAT and GPA; relaton

More information

STAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression

STAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,

More information

ANSWERS CHAPTER 9. TIO 9.2: If the values are the same, the difference is 0, therefore the null hypothesis cannot be rejected.

ANSWERS CHAPTER 9. TIO 9.2: If the values are the same, the difference is 0, therefore the null hypothesis cannot be rejected. ANSWERS CHAPTER 9 THINK IT OVER thnk t over TIO 9.: χ 2 k = ( f e ) = 0 e Breakng the equaton down: the test statstc for the ch-squared dstrbuton s equal to the sum over all categores of the expected frequency

More information

1 Generating functions, continued

1 Generating functions, continued Generatng functons, contnued. Generatng functons and parttons We can make use of generatng functons to answer some questons a bt more restrctve than we ve done so far: Queston : Fnd a generatng functon

More information

CinChE Problem-Solving Strategy Chapter 4 Development of a Mathematical Model. formulation. procedure

CinChE Problem-Solving Strategy Chapter 4 Development of a Mathematical Model. formulation. procedure nhe roblem-solvng Strategy hapter 4 Transformaton rocess onceptual Model formulaton procedure Mathematcal Model The mathematcal model s an abstracton that represents the engneerng phenomena occurrng n

More information