Analysis of Experimental Data

Size: px
Start display at page:

Download "Analysis of Experimental Data"

Transcription

1 Aalysis of Experimetal Data ± 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 accurate ad precise. Accuracy refers to the proximity of a measuremet to the true value of a quatity. Precisio refers to the proximity of several measuremets to each other, that is, the reproducibility of a measuremet or set of measuremets. For Example: Two studets, Raffaella ad Barbara, measured the temperature of boilig water, which by defiitio should be 00 C uder atmosphere of pressure. Each studet made 0 temperature measuremets, show below as red (Raffaella) ad blue (Barbara) dots. The average of Raffaella's temperature measuremets is 00. C ad the average of Barbara's is also 00. C. Give the actual value for b.p. both studets had good accuracy O the other had, you ca see from the figure that the precisio of Raffaella's measuremets was far better tha Barbara's. So, what caused the two studets to get values that were ot equal to the true boilig poit of water. Ad eve if they did t get the true value, why did t they get the same umber every time they took the measuremet of the same sample? The aswer: Experimetal Errors Experimetal Errors There are three types of experimetal errors affectig values:. Systematic error errors affectig the accuracy i measuremets which have a defiite value that ca, i priciple, be measured ad accouted for. Systematic errors ca be corrected for but oly after the cause is determied. Examples may iclude a icorrect calibratio of a balace that always reads.000 g higher tha the actual mass or the presece of a iterferig substace i calorimic studies. I our example, systematic error is why the studets did ot get exactly 00.0 o C for their measuremets.. Radom error errors affectig precisio i every measuremet which fluctuate radomly ad do ot have a defiite value; They caot be positively idetified. To further uderstad radom errors, cosider the weight of a object obtaied by doig five differet weighigs o a four place aalytical balace. trial : g trial : g trial 3: g trial 4: g trial 5: g The first three digits are the same i all cases. The last digit has a ucertaity associated with it. This ucertaity is a fuctio of the type of sample, the coditios uder which it is beig weighed, the balace, ad the perso doig the weighig.

2 Eve whe all factors are optimized, there will still be some variatio i the weight. This variatio or ucertaity is the result of pushig the balace to its limit. We could cut the last figure off; the all the weights would be the same, but the weight would be kow oly to the earest milligram. We obtai more iformatio if we keep that last figure but remai aware of its ucertaity. That ucertaity arises because of radom error; ad is idicative of the precisio of the measuremet. I our example, radom error is why the studets did ot get the same measuremet every time. 3. Gross error errors producig values that are drastically differet from all other data. These errors are the result of a mistake i the procedure, either by the experimeter or by a istrumet. A example would be misreadig the umbers or miscoutig the scale divisios o a buret or istrumet display. A istrumet might produce a gross error if a poor electrical coectio causes the display to read a occasioal icorrect value. If you are aware of a mistake at the time of the procedure, the experimetal result should be discouted ad the experimet repeated correctly. Gross errors result i values that make o sese ad greatly affect the overall precisio of the experimet. It is impossible to perform a chemical aalysis i such a way that the results are totally free of errors. All oe ca hope is to miimize these errors ad to estimate their size with acceptable accuracy. Rarely is it easy to estimate the errors of experimetal data; However, we must make such estimates because data of ukow precisio ad accuracy are worthless. The questio ow becomes, how do we describe the accuracy ad precisio quatitatively? Estimatig Accuracy i Measuremet Accuracy ca be estimated easily whe a true value is kow. For example the desity of iro is 7.87 g/cm 3. I the laboratory, you fid the mass ad volume of a iro block to calculate the desity as 7.3 g/cm 3. There is obvious systematic error i accuracy sice you did ot derive the theoretical desity of pure iro. Note: Sice all values are limited by the techology to describe them a better term for true value is accepted value. Estimatig Accuracy i Measuremet Absolute error (AE) The differece betwee a experimetal value ad accepted value A.E. = exp kow % Absolute Error (%AE) exp kow %AE 00 kow % Accuracy Percetage your value differs from 00. % Accuracy = 00 - %AE Back to our example. There are two ways we could describe the accuracy of the data collected. ) we could calculate the accuracy for every data poit idividually. ) we could calculate the accuracy of each experimeter by treatig the data sets as sigle values, or average values. I our example, the average value for each studets results were 00. o C ad 00. o C..Calculate the absolute error, % absolute error ad % accuracy for each experimeter.

3 Ofte, you will eed the average for a data set to gai a better estimate of experimetal error. There are two commo ways of expressig a average: the mea ad the media. The mea (χ) is the arithmetic average of the results, or: xi x x... x Mea x i. If the aalysis of acetamiophe tablets resulted i the presece of 48mg, 479mg, 44mg, ad 435mg active igrediet i four differet trials, what was the mea value? The media is the value that lies i the middle amog the results. Half of the measuremets are above the media ad half below. If there are ad eve umber of values, the media is the average of the two middle results. 3. What the is the media for the values foud i aalyzig the acetamiophe tablets? There are ofte advatages for usig the media i place of the mea whe a average is desired. If a small umber of measuremets are made, oe value ca greatly affect the mea. 4. Compare the mea to media i our sample set. Which oe would be a more realistic average? Whe Should a Value be Omitted as a Outlier Whe Fidig a Average? There are may differet ways to determie whether or ot a value is a outlier due to gross error. Some methods iclude: Q test 0% assumptio Stadard deviatio (4σ approximatio) Five umber summary (Box Plot) We will utilize the five umber summary. The Five Number Summary For ay group of umbers you believe may cotai a outlier: I. Arrage umbers i ascedig or descedig order ad fid the media. II. Calculate Q by fidig the middle value for the umbers left of the media. III.Calculate Q 3 by fidig the middle value for the umbers right of the media. IV.Calculate the Ier Quartile Rage (IQR) by subtractig Q from Q 3. V. Multiply IQR by.5 VI.Subtract modified IQR from Q. Aythig less tha this value is a outlier. VII.Add the modified IQR to Q 3. Aythig greater tha the result is a outlier. 5. Determie if outliers exist i the followig set of data. The fid both the mea ad media of the appropriate data., 4, 56, 6, 6, 69, 73, 07. Estimatig Precisio i Measuremet I chemistry you are ofte lookig for a ukow value ad have othig to compare your experimetal values to. I quatitative work, precisio is ofte used as a idicatio of accuracy; we assume that the average of a series of precise measuremets (which should average out the radom errors because of their equal probability of beig high or low), is accurate, or close to the true value. Agai, you may be tryig to determie the true/accepted value. 3

4 Estimatig Precisio i Measuremet Relative error (RE) The differece betwee a experimetal value ad the average (mea or media) for a set of experimetal values R.E. = exp avg % Relative Error (%RE) exp avg %RE 00 avg % Precisio The closeess of a value to a set of values i terms of percetage. % Precisio = 00 - %RE Agai, to our example. Whe lookig at the precisio of the experimet we could describe the precisio i two ways. ) the precisio of each data poit collected i respect to the data for each studet usig the previous treatmets. ) estimate the precisio of each studets experimet by ivestigatig the ucertaity of their measuremets (discussed later) 6. Calculate the relative error, % relative error ad % precisio for the third measuremet take by each studet. Whe dealig with precisio, the greater the umber of trials, the better the estimatio of the overall rage. We have estimated values for accuracy ad precisio but what cofidece is there i these values obtaied experimetally? You ca see the results above are spread over a rage of values. The width of this spread is a measure of the ucertaity caused by radom errors. The elemet of ucertaity i experimetal data ca be quatified ad should be reported alog with the actual experimetal value itself writte as the experimetal value ± some degree of ucertaity. I this case Raffaella would report a boilig poit of 00. ± 0.3 C, ad Barbara would report 00. ±.4 C. The value of the ucertaity gives oe a idea of the precisio iheret i a measuremet of a experimetal quatity; here, Raffaella is more certai of her values tha Barbara. There are may ways to quatify ucertaity, ragig from very simple techiques to highly sophisticated methods. The method used will deped upo how may measuremets of a sigle quatity are made ad o how crucial the reportig of the value of ucertaity is with regard to the iterpretatio of the experimetal data. We will cosider:. The Graduatio Method. Rage 3. Sample Stadard Deviatio 4. Cofidece Limits The Graduatio Method Whe a measuremet is made directly by the studet i lab, the ucertaity must be approximated. Usig the graduatio the ucertaity i a sigle measuremet is estimated by a value oe-half of the smallest level of graduatio i the measurig istrumet. For example, if a sigle measuremet of the legth of a object is to be made usig a meter stick marked with millimeter graduatios, the legth should be reported ±.5 mm (or ±.005 m). [some professors prefer a 0% rule] We will use ½ the lowest graduatio i our lab. You must use your ow judgmet i choosig the precisio usig the Graduatio Method. If i doubt, always be coservative; i.e. report the largest of possible ucertaities (50% vs. 0%). 7. Use a ruler to measure the width of your text. Report your measuremet with the correct ucertaity accordig to the Graduatio Method. 8. Make a temperature readig ad report with the correct ucertaity. 9. Repeat for the volume of water i a accurately read 50 ml volumetric flask. This is how all measuremets must be recorded i your lab reports. Some istrumets or glassware may have the ucertaity prited o the tool ad should be used i place of the graduatio method. 4

5 Rage The graduatio method adequately estimates the ucertaity of a sigle value but ca ot be used whe a series of values exists for a sigle observable. I this case the ucertaity ca be crudely approximated by the rage. The rage is give as the differece betwee the maximum ad miimum values of the measured quatity. I the case of the set of five weights give: trial : ±.000 g trial : ±.000 g trial 3: ±.000 g trial 4: g trial 5: g the rage is g g = g. So i our example, the value should be recorded as the average (mea or media) ± rage/, rouded to the correct sig. figs. Or, ± g (R) If you remember our previous example, we did ot have a way to describe the precisio of the two studets data. Errors cause ucertaity, ucertaity affects precisio; therefore, ucertaity ca quatify precisio. The greater the ucertaity, or rage, the less precise the values. Here we see that, crudely, the mass of the sample should be somewhere betwee ad g. We could use rage to estimate the precisio of each studet but rage teds to be to much of a over estimatio. We should seek a better estimatio. Sample Stadard Deviatio The most commo way to describe the ucertaity, or precisio, for a set of data is by the sample stadard deviatio (s). The stadard deviatio is used to describe the likelihood that a value will fall ear the mea for a ormal set of data s i x x i For ormal data sets, 68.3 % of experimetal values have statistical probability of fallig withi oe stadard deviatio (σ) of the mea, 95.5% withi σ ad 99.7% withi 3σ. let s determie the ucertaity of our weights usig stadard deviatio: trial : g trial : g trial 3: g trial 4: g trial 5: g xi x i s s 4 = So the value should be recorded as the average (mea or media) ± the sample deviatio: ± g, or ± g (s) This tells us that there is a 68.3 % chace that the mass is betwee ad g. Our precisio, or ucertaity, for the weights measured has bee foud as: Rage ± g, or ± g (R) SD ± g, or ± g (s) Notice that the stadard deviatio method gives a smaller margi for error, however, it oly describes a 68.3% cofidece. I other words, there is a 68.3 % probability that the true mass of the weights is betwee ad g. 68.3% is a reasoable descriptio of ucertaity but a 95 % cofidece iterval is the miimum stadard for reportig ucertaity i chemistry ad physics. Cofidece Limits The sample stadard deviatio ca be adjusted to ay give cofidece iterval by utilizig the followig equatio: ts Cofidece iterval = N Where s is the sample stadard deviatio, t is a statistical costat (foud o the followig slides) ad N is the umber of samples i the data set. Let us calculate the ucertaity of our weight measuremets with a 95 % cofidece C.I. = ± (.57)(0.000)/ 5 = ± So our ucertaity is ± (95%,N=5) 5

6 Cofidece Limit T values 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% t value Cofidece Level 0.I lab, you have titrated 8 equal volumes of a ukow acid with a stadard base. You have dispesed the followig volumes of titrat: 43.8mL, 43.30mL, 44.0mL, 43.90mL, 5.69mL, 43.87mL, 43.88mL ad 43.70mL. Do the followig aalysis: A. Determie if ay gross errors occurred. B. Fid the mea ad media. C. Choose the best value for the average; justify. D. Estimate the ucertaity of the measuremets usig the Graduatio Method (assume correct measuremets). E. Calculate the precisio of your measuremets i terms of rage ad sample stadard deviatio. F. Adjust the ucertaity to a 99% cofidece level..if the actual amout of titrat added i problem 0 should have bee 43.85mL, what was the absolute error ad % accuracy?.o a further titratio, you obtaied a volume of 44.0mL. What is the relative error ad % precisio i relatioship to the ew average volume added? 6

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

ANALYSIS OF EXPERIMENTAL ERRORS

ANALYSIS OF EXPERIMENTAL ERRORS ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder

More information

Analysis of Experimental Measurements

Analysis of Experimental Measurements Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,

More information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample. Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized

More information

MATH/STAT 352: Lecture 15

MATH/STAT 352: Lecture 15 MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

Computing Confidence Intervals for Sample Data

Computing Confidence Intervals for Sample Data Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios

More information

Error & Uncertainty. Error. More on errors. Uncertainty. Page # The error is the difference between a TRUE value, x, and a MEASURED value, x i :

Error & Uncertainty. Error. More on errors. Uncertainty. Page # The error is the difference between a TRUE value, x, and a MEASURED value, x i : Error Error & Ucertaity The error is the differece betwee a TRUE value,, ad a MEASURED value, i : E = i There is o error-free measuremet. The sigificace of a measuremet caot be judged uless the associate

More information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

1 Inferential Methods for Correlation and Regression Analysis

1 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 information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA 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 large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Chapter 8: Estimating with Confidence

Chapter 8: Estimating with Confidence Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig

More information

Comparing your lab results with the others by one-way ANOVA

Comparing your lab results with the others by one-way ANOVA Comparig your lab results with the others by oe-way ANOVA You may have developed a ew test method ad i your method validatio process you would like to check the method s ruggedess by coductig a simple

More information

A quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population

A quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate

More information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio

More information

Data Description. Measure of Central Tendency. Data Description. Chapter x i

Data Description. Measure of Central Tendency. Data Description. Chapter x i Data Descriptio Describe Distributio with Numbers Example: Birth weights (i lb) of 5 babies bor from two groups of wome uder differet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Chapter 3

More information

Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.

Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63. STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to

More information

1.010 Uncertainty in Engineering Fall 2008

1.010 Uncertainty in Engineering Fall 2008 MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval

More information

Lecture 1 Probability and Statistics

Lecture 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 information

Topic 10: Introduction to Estimation

Topic 10: Introduction to Estimation Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio

More information

Confidence Intervals for the Population Proportion p

Confidence 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 information

Chapter 6 Sampling Distributions

Chapter 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 information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates. 5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Agreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times

Agreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

Read through these prior to coming to the test and follow them when you take your test.

Read through these prior to coming to the test and follow them when you take your test. Math 143 Sprig 2012 Test 2 Iformatio 1 Test 2 will be give i class o Thursday April 5. Material Covered The test is cummulative, but will emphasize the recet material (Chapters 6 8, 10 11, ad Sectios 12.1

More information

Scientific notation makes the correct use of significant figures extremely easy. Consider the following:

Scientific notation makes the correct use of significant figures extremely easy. Consider the following: Revised 08/1 Physics 100/10 INTRODUCTION MEASUREMENT AND UNCERTAINTY The physics laboratory is the testig groud of physics. Physicists desig experimets to test theories. Theories are usually expressed

More information

a. For each block, draw a free body diagram. Identify the source of each force in each free body diagram.

a. For each block, draw a free body diagram. Identify the source of each force in each free body diagram. Pre-Lab 4 Tesio & Newto s Third Law Refereces This lab cocers the properties of forces eerted by strigs or cables, called tesio forces, ad the use of Newto s third law to aalyze forces. Physics 2: Tipler

More information

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process. Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike

More information

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9 BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous

More information

Lecture 1 Probability and Statistics

Lecture 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 information

Economics Spring 2015

Economics Spring 2015 1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures

More information

OBJECTIVES. 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 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 information

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

More information

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Berli Che Deartmet of Comuter Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chater 5 & Teachig Material Itroductio

More information

CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics

CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based

More information

Measures of Spread: Standard Deviation

Measures of Spread: Standard Deviation Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of

More information

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

Instructor: 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 information

MEASURES OF DISPERSION (VARIABILITY)

MEASURES OF DISPERSION (VARIABILITY) POLI 300 Hadout #7 N. R. Miller MEASURES OF DISPERSION (VARIABILITY) While measures of cetral tedecy idicate what value of a variable is (i oe sese or other, e.g., mode, media, mea), average or cetral

More information

Understanding Samples

Understanding Samples 1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We

More information

MBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS

MBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos

More information

Confidence Intervals QMET103

Confidence Intervals QMET103 Cofidece Itervals QMET103 Library, Teachig ad Learig CONFIDENCE INTERVALS provide a iterval estimate of the ukow populatio parameter. What is a cofidece iterval? Statisticias have a habit of hedgig their

More information

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, 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 o-users, x - y. Such studies are sometimes viewed

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 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 information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5 CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio

More information

Continuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised

Continuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for

More information

Estimation of a population proportion March 23,

Estimation of a population proportion March 23, 1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes

More information

Understanding Dissimilarity Among Samples

Understanding Dissimilarity Among Samples Aoucemets: Midterm is Wed. Review sheet is o class webpage (i the list of lectures) ad will be covered i discussio o Moday. Two sheets of otes are allowed, same rules as for the oe sheet last time. Office

More information

The Method of Least Squares. To understand least squares fitting of data.

The Method of Least Squares. To understand least squares fitting of data. The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve

More information

Exam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234

Exam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234 STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6

More information

Hypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance

Hypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?

More information

Power and Type II Error

Power and Type II Error Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error

More information

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10

DS 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 information

Stat 421-SP2012 Interval Estimation Section

Stat 421-SP2012 Interval Estimation Section Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible

More information

- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion

- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion 1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter

More information

Linear Regression Models

Linear Regression Models Liear Regressio Models Dr. Joh Mellor-Crummey 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 information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 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 information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio

More information

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01 ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly

More information

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc. Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo

More information

Final Review. Fall 2013 Prof. Yao Xie, H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech

Final Review. Fall 2013 Prof. Yao Xie, H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech Fial Review Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech 1 Radom samplig model radom samples populatio radom samples: x 1,..., x

More information

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece

More information

Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p

Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret

More information

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

More information

AP Statistics Review Ch. 8

AP Statistics Review Ch. 8 AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.

More information

Lecture 2: Monte Carlo Simulation

Lecture 2: Monte Carlo Simulation STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?

More information

Parameter, Statistic and Random Samples

Parameter, 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 information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

More information

There is no straightforward approach for choosing the warmup period l.

There is no straightforward approach for choosing the warmup period l. B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.

More information

Lecture 7: Non-parametric Comparison of Location. GENOME 560, Spring 2016 Doug Fowler, GS

Lecture 7: Non-parametric Comparison of Location. GENOME 560, Spring 2016 Doug Fowler, GS Lecture 7: No-parametric Compariso of Locatio GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review How ca we set a cofidece iterval o a proportio? 2 Review How ca we set a cofidece iterval

More information

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x 2 + +

More information

Chapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.

Chapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc. Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more

More information

1 Lesson 6: Measure of Variation

1 Lesson 6: Measure of Variation 1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages

More information

STAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence

STAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Itroductory Statistics Chapter 6: Itroductio to Iferece Lecture 18: Estimatio with Cofidece 11/14/06 Lecture 18 1 Itroductio Statistical Iferece

More information

24.1. Confidence Intervals and Margins of Error. Engage Confidence Intervals and Margins of Error. Learning Objective

24.1. Confidence Intervals and Margins of Error. Engage Confidence Intervals and Margins of Error. Learning Objective 24.1 Cofidece Itervals ad Margis of Error Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Resource Locker LESSON 24.1 Cofidece

More information

CURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:

CURRICULUM 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 information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

Statistical Intervals for a Single Sample

Statistical Intervals for a Single Sample 3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea

More information

Interval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),

Interval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ), Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We

More information

October 25, 2018 BIM 105 Probability and Statistics for Biomedical Engineers 1

October 25, 2018 BIM 105 Probability and Statistics for Biomedical Engineers 1 October 25, 2018 BIM 105 Probability ad Statistics for Biomedical Egieers 1 Populatio parameters ad Sample Statistics October 25, 2018 BIM 105 Probability ad Statistics for Biomedical Egieers 2 Ifereces

More information

Probability, Expectation Value and Uncertainty

Probability, 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 information

Simple Random Sampling!

Simple Random Sampling! Simple Radom Samplig! Professor Ro Fricker! Naval Postgraduate School! Moterey, Califoria! Readig:! 3/26/13 Scheaffer et al. chapter 4! 1 Goals for this Lecture! Defie simple radom samplig (SRS) ad discuss

More information

Chapter 2 Descriptive Statistics

Chapter 2 Descriptive Statistics Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data

More information

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequences A 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 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet

More information

Lecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS

Lecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece

More information

Properties and Hypothesis Testing

Properties 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. Cross-sectioal data. 2. Time series data.

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio

More information

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes. Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely

More information