Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:"

Transcription

1 Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day to day ad ofte modify our questio to somethig like how may calories do I cosume o a typical day? or o average, how much time do I sped talkig per day? I this sectio we will study three ways of measurig cetral tedecy i data, the mea, the media ad the mode Each measure give us a sigle value (the mode may give more tha oe) that might be cosidered typical As we will see however, ay oe of these values ca give us a skewed picture if the give data has certai characteristics A populatio of books, cars, people, polar bears, all games played by Babe Ruth throughout his career etc is the etire collectio of those objects For ay give variable uder cosideratio, each member of the populatio has a particular value of the variable associated to them, for example the umber of home rus scored by Babe Ruth for each game played by him durig his career These values are called data ad we ca apply our measures of cetral tedecy to the etire populatio, to get a sigle value (maybe more tha oe for the mode) measurig cetral tedecy for the etire populatio Whe we calculate the mea, media ad mode usig the data from the etire populatio, we call the results the populatio mea, the populatio media ad the populatio mode A sample is a subset of the populatio, for example, we might collect the data o the umber of home rus scored i a radom sample of 20 games played by Babe Ruth If we calculate the mea, media ad mode usig the data from a sample, the results are called the sample mea, sample media ad sample mode The Mea: The populatio mea of m umbers x 1, x 2,, x m (the data for every member of a populatio of sixe m) is deoted by µ ad is computed as follows: µ = x 1 + x x m m The sample mea of the umbers x 1, x 2,, x (data for a sample of size from the populatio) is deoted by x ad is computed similarly: x = x 1 + x x Example Cosider the followig set of data, showig the umber of times a sample of 5 studets check their per day: 1, 3, 5, 5, 3 Here = 5 ad x 1 = 1, x 2 = 3, x 3 = 5, x 4 = 5 ad x 5 = 3 Calculate the sample mea x 1

2 Example The followig data shows the results for the umber of books that a radom sample of 20 studets were carryig i their book bags: 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4 The the mea of the sample is the average umber of books carried per studet: x = Not that the mea here caot be a observatio i our data = 25 Calculatig the mea more efficietly: We ca calculate the mea above more efficietly here by usig frequecies We ca see from the calculatio above that x = The frequecy distributio for the data is: 0 + (1 2) + (2 8) + (3 4) + (4 5) 20 # Books Frequecy # Books Frequecy = x = Sum 20 The geeral case ca be dealt with as follows: If our frequecy/relative frequecy table for our sample of size, looks like the oe below, (where the observatios are deoted 0 i, the correspodig frequecies, f i ad the relative frequecies f i /): Observatio Frequecy Relative Frequecy 0 i f i f i / 0 1 f 1 f 1 / 0 2 f 2 f 2 / 0 3 f 3 f 3 / 0 R f R f R / the, x = 0 1 f f R f R = 0 1 f f f3 + 0 R fr We ca also use our table with a ew colum to calculate: 2

3 Outcome Frequecy Outcome Frequecy 0 i f i 0 i f i 0 1 f f f f f f 3 0 R f R 0 R f R SUM = x Alteratively we ca use the relative frequcies, istead of dividig by the at the ed Outcome Frequecy Relative Frequecy Outcome Relative Frequecy 0 i f i f i / 0 i f i / 0 1 f 1 f 1 / 0 1 f 1 / 0 2 f 2 f 2 / 0 2 f 2 / 0 3 f 3 f 3 / 0 3 f 3 / 0 R f R f R / 0 R f R / SUM = x You ca of course choose your favorite method for calculatio from the three methods listed above Example The umber of goals scored by the 32 teams i the 2014 world cup are show below: 18, 15, 12, 11, 10, 8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1 Make a frequecy table for the data ad takig the soccer teams who played i the world cup as a populatio, calculate the populatio mea, µ 3

4 Estimatig the mea from a histogram If we are give a histogram (showig frequecies) or a frequecy table where the data is already grouped ito categories ad do ot have access to the origial data, we ca estimate the mea usig the midpoits of the itervals which serve as categories for the data Suppose there are k categories (show as the bases of the rectagles) with midpoits m 1, m 2,, m k respectively ad the frequecies of the correspodig itervals are f 1, f 2,, f k, the the mea of the data set is approximately where = f 1 + f f k m 1 f 1 + m 2 f m k f k Example Approximate the mea for the set of data used to make the followig histogram, showig the time (i secods) spet waitig by a sample of customers at Grigotts Wizardig bak midpoits: approximatio of mea: Time spet waitig (i secods) The Media The Media of a set of quatitative data is the middle umber whe the measuremets are arraged i ascedig order To Calculate the Media: of the data by M 1 If is odd, M is the middle umber 2 If is eve, M is the average of the two middle umbers Arrage the measuremets i ascedig order We deote the media Example The umber of goals scored by the 32 teams i the 2014 world cup are show below: 18, 15, 12, 11, 10, 8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1 Fid the media of the above set of data 4

5 Example A sample of 5 studets were asked how much moey they were carryig ad the results are show below: $75, $2, $5, $0, $5 Fid the mea ad media of the above set of data Notice that the media gives us a more represetative picture here, sice the mea is skewed by the outlier $75 The Mode Defiitio The mode of a set of measuremets is the most frequetly occurrig value; it is the value havig the highest frequecy amog the measuremets Example Fid the mode of the data collected o the amout of moey carried by the 5 studets i the example above: $75, $2, $5, $0, $5 You fid that i some cases the mode is ot uique: Example What is the mode of the data o the umber of goals scored by each team i the world cup of 2006? 13, 12, 11, 10, 9, 9, 7, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 0 Note The mode ca be computed for qualitative data The mode is ot ofte used as a measure of ceter for quatitative data The Histogram ad the mea, media ad mode With large sets of data ad arrow class widths, the histogram looks roughly like a smooth curve The mea, media ad mode, have a graphical iterpretatio i this case The mea is thep balace 315 poit of the histogram of the data, whereas the media is the poit o the x-axis such that half of the area uder the histogram lies to the right of the media ad half of the area lies to its left The mode occurs at the data poit where the graph reaches its highest poit This of course524 may ot Which be uique Measure of Ceter to Use? Bell-shaped, Symmetric Bimodal mea=media=mode mea=media two modes Skewed Right 5 Skewed Left

6 mea=media=mode two modes Skewed Right mode mea media Skewed Left Skewed Left mode mea media mea mode media mea mode media Skewed Data Defiitio A data set is said to be skewed if oe tail of the distributio has more extreme observatios tha the other tail Mea, Media, ad Mode The mea is sesitive to extreme observatios, but the media is ot(check out the example below) Example Cosider the data from the above example The most cocerig commo themeasure amout of ceter moeyis carried the mea, by the which loca five studets i the sample balacig poit of the distributio The mea equals the sum Mea, Media, ad Mode $75, $2, $5, observatios, $0, $5 divided by how may there are The mea is a We have already calculated the mea ad the media extreme of the observatios data, which (outliers we foud ad values to be : which mea are = far i the $174, media The most = commo $5 distributio that is skewed) So the mea teds to be a good measure of ceter is the mea, locatig which the ceter locates of the a distributio that is uimodal ad roug Now cosider balacig the same poit set of the of data distributio with thethe largest mea amout with equals o of outliers the moey sum of replaced the by $5,000, that is suppose our dataobservatios, was divided by how may there are The mea is also affected by extreme observatios (outliers ad $5, 000, values $2, which are far i the tail of a The $5, media $0, $5 is a more robust measure of ceter, that is, it is distributio that is skewed) So the mea teds to be a good choice for What is by extreme values The media is the middle observatio wh locatig the ewthe mea ceter adof media? a distributio that is uimodal ad roughly symmetric, with o outliers We ca The see from media theis histograms, a more robust that measure for data of skewed ceter, that to the is, right, it is ot the iflueced mea is larger tha the media ad for by data extreme skewedvalues to left, The themedia mea is is less the tha middle the observatio media whe the data are Differet Measures Ca Give Differet Impressios The famous trio, the mea, the media, ad the mode, represet three differet methods for fidig a so-called ceter value These three values may be the same for a set of data but it is very likely that they will have three differet values Whe they are differet, they ca lead to differet iterpretatios of the data beig summarized Cosider the aual icomes of five families i a eighborhood: What is the typical icome for this group? $12, 000 $12, 000 $30, 000 $51, 000 $100, 000 The mea icome is: $41,000, The media icome is: $30,000, The modal icome is: $12,000 If you were tryig to promote that this is a affluet eighborhood, you might prefer to report the mea icome If you were a Sociologist, tryig to report a typical icome for the area, you might report the media icome If you were tryig to argue agaist a tax icrease, you might argue that icome is too low to afford a tax icrease ad report the mode 6

3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average

3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average 5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives

More information

Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics deals with the description or simple analysis of population or sample data. Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Section 7-3 Estimating a Population. Requirements

Section 7-3 Estimating a Population. Requirements Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

sum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by

sum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by Statistics Module Revisio Sheet The S exam is hour 30 miutes log ad is i two sectios Sectio A 3 marks 5 questios worth o more tha 8 marks each Sectio B 3 marks questios worth about 8 marks each You are

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Statistical Methods. Chapter 1: Overview and Descriptive Statistics

Statistical Methods. Chapter 1: Overview and Descriptive Statistics Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

More information

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

More information

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median. Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

More information

Standard Errors and Confidence Intervals

Standard Errors and Confidence Intervals Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition 7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

More information

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the

More information

Confidence Intervals and Sample Size

Confidence Intervals and Sample Size 8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

Measures of Central Tendency

Measures of Central Tendency Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the

More information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

More information

23.3 Sampling Distributions

23.3 Sampling Distributions COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

x : X bar Mean (i.e. Average) of a sample

x : X bar Mean (i.e. Average) of a sample A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

CHAPTER 11 Financial mathematics

CHAPTER 11 Financial mathematics CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

More information

Hypothesis Tests Applied to Means

Hypothesis Tests Applied to Means The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

This is arithmetic average of the x values and is usually referred to simply as the mean.

This is arithmetic average of the x values and is usually referred to simply as the mean. prepared by Dr. Adre Lehre, Dept. of Geology, Humboldt State Uiversity http://www.humboldt.edu/~geodept/geology51/51_hadouts/statistical_aalysis.pdf STATISTICAL ANALYSIS OF HYDROLOGIC DATA This hadout

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

Searching Algorithm Efficiencies

Searching Algorithm Efficiencies Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay

More information

AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

More information

TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004

TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004 TIEE Teachig Issues ad Experimets i Ecology - Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013

More information

Section 7.2 Confidence Interval for a Proportion

Section 7.2 Confidence Interval for a Proportion Sectio 7.2 Cofidece Iterval for a Proportio Before ay ifereces ca be made about a proportio, certai coditios must be satisfied: 1. The sample must be a SRS from the populatio of iterest. 2. The populatio

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

Case Study. Contingency Tables. Graphing Tabled Counts. Stacked Bar Graph

Case Study. Contingency Tables. Graphing Tabled Counts. Stacked Bar Graph Case Study Cotigecy Tables Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 4 6, 2011 Case Study Example 9.3 begiig o page 213 of the text describes a experimet i which

More information

Confidence Intervals for One Mean with Tolerance Probability

Confidence Intervals for One Mean with Tolerance Probability Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

More information

Statistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals

Statistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals Statistics 111 - Lecture 14 Itroductio to Iferece Hypothesis Tests Admiistrative Notes Sprig Break! No lectures o Tuesday, March 8 th ad Thursday March 10 th Exteded Sprig Break! There is o Stat 111 recitatio

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

Exploratory Data Analysis

Exploratory Data Analysis 1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

3. Continuous Random Variables

3. Continuous Random Variables Statistics ad probability: 3-1 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Institute for the Advancement of University Learning & Department of Statistics

Institute for the Advancement of University Learning & Department of Statistics Istitute for the Advacemet of Uiversity Learig & Departmet of Statistics Descriptive Statistics for Research (Hilary Term, 00) Lecture 5: Cofidece Itervals (I.) Itroductio Cofidece itervals (or regios)

More information

Chapter 9: Correlation and Regression: Solutions

Chapter 9: Correlation and Regression: Solutions Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because

More information

ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability). INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Joint Probability Distributions and Random Samples

Joint Probability Distributions and Random Samples STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial

More information

Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept

Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics for

More information

ˆ p 2. ˆ p 1. ˆ p 3. p 4. ˆ p 8

ˆ p 2. ˆ p 1. ˆ p 3. p 4. ˆ p 8 Sectio 8 1C The Techiques of Hypothesis Testig A claim is made that 10% of the populatio is left haded. A alterate claim is made that less tha 10% of the populatio is left haded. We will use the techiques

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

More information

Confidence Intervals for the Population Mean

Confidence Intervals for the Population Mean Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios.

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

BASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.

BASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1. BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

More information

A Resource for Free-standing Mathematics Qualifications Working with %

A Resource for Free-standing Mathematics Qualifications Working with % Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%

More information

Checklist. Assignment

Checklist. Assignment Checklist Part I Fid the simple iterest o a pricipal. Fid a compouded iterest o a pricipal. Part II Use the compoud iterest formula. Compare iterest growth rates. Cotiuous compoudig. (Math 1030) M 1030

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

1 Lesson 5 Measures of Central Tendencies

1 Lesson 5 Measures of Central Tendencies 1 Lesso 5 Measures of Cetral Tedecies 1.1 Itroductio We have outlied the correct process for dealig with data. The steps are 1. Orgaize the data. 2. Arrage the data. 3. Preset the data. We have spet some

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

Lecture Notes CMSC 251

Lecture Notes CMSC 251 We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

MATH 361 Homework 9. Royden Royden Royden

MATH 361 Homework 9. Royden Royden Royden MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

More information

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book) MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

More information

Chapter XIV: Fundamentals of Probability and Statistics *

Chapter XIV: Fundamentals of Probability and Statistics * Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive

More information

0,1 is an accumulation

0,1 is an accumulation Sectio 5.4 1 Accumulatio Poits Sectio 5.4 Bolzao-Weierstrass ad Heie-Borel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real

More information

Question 2: How is a loan amortized?

Question 2: How is a loan amortized? Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Figure 40.1. Figure 40.2

Figure 40.1. Figure 40.2 40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information