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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all the umbers ad divide by how may there are. b) Media - Put all the umbers i order ad fid the middle oe. If there are two, the fid the mea of them. c) Mode - The umber that appears most ofte (there ca be more tha oe). 4) How to calculate the rage: The differece betwee the biggest umber ad the smallest umber. 5) How to use a frequecy table to fid averages: NUMBER TALLY FREQUENCY NUMBER x FREQUENCY Total - 8 Total - 13 MEAN - 13 divided by 8 gives 1.65 MEDIAN - workig dow the table gives ad, the mea of which is. MODE - it is clear that is the mode. RANGE = 1 6) How to use multiple bar charts ad compoet bar charts. 7) How to use grouped data. 8) How to draw a histogram, ad a frequecy polygo (lies to joi the middle of each bar together). 9) How to calculate averages for grouped data:

2 GROUP MID-POINT FREQUENCY MID-POINT x FREQUENCY > 0 <= > 50 <= > 100 <= ,375 Total - 5 Total - 15 MEAN - estimated at 15 / 5 = 85 MODAL GROUP - > 100 <= 150 MEDIAN - the umber will be i the > 50 <= 100 group. The media is the 13 th umber, ad there are the 7 th up to the 14 th umbers i this group. There are 8 umbers, so the 13 th umber will be 7/8 ito the group. It covers 50, so 7/8 of 50 = This is the estimated media. 10) How to write out questioaires. 11) That a sample ca be used i a survey whe you caot ask everyoe. It should be selected at radom, for if groups of people are ot icluded the it will be biased. The bigger a sample is, the better. If it is ot possible to use radom samplig, the systematic samplig should be used. This is whe the ames are i a list, ad you choose, for istace, every 5 th perso o the list. 1) That the expected frequecy is the amout of times you would expect somethig to happe, e.g. if you tossed a coi 100 times, you would expect 50 heads ad 50 tails. 13) That the relative frequecy is the relative umber of times you would expect somethig to happe, e.g. you would expect a coi to show heads ½ of the time ad tails ½ of the time whe it is tossed. This is the probability of a evet happeig, ad the sample space is the list of evets that could happe (i.e. heads or tails). 14) That the absolute error of a measuremet is the maximum error that ca be made, e.g. a measuremet to the earest millimetre has a absolute error of ½ a millimetre. The relative error is the absolute error divided by the measuremet take, ad the percetage error is the relative error times ) That data ca be either quatitative (whe it is umbers) or qualitative (whe it is somethig else, like a descriptio). 16) Quatitative data ca be either discrete (whe it is oly certai umbers ad ca t be i betwee them) or cotiuous (whe it ca be absolutely ay umber ad to ay degree of accuracy, like a measuremet for istace). 17) That you must use the class boudaries of data to fid the mid-poit of each group. 18) That to estimate the mode with grouped data, you eed to fid the modal group o the histogram, ad draw a lie from the top left had corer of the bar to the top left

3 had corer of the bar o its right, ad do the same for the top right had corer ad the bar o the left. You ca the draw a lie straight dow to the x-axis, ad take the readig as a estimate of the mode. 19) How to draw a cumulative frequecy curve. 0) That to fid the media o a cumulative frequecy curve, you fid the middle value o the y-axis, draw a horizotal lie to the curve, the a vertical lie straight dow to the x-axis to take a readig. 1) That the lower quartile has a y value of the total umber of values divided by 4, ad the upper quartile y value is the total umber of values mius the lower quartile y value. You ca draw these o the graph i the same way as the media. The iterquartile rage is the upper quartile mius the lower quartile, ad it shows how spread out the data is. ) That the semi-iterquartile rage is half the iterquartile rage. Sometimes data is split ito deciles (10 equal parts) or percetiles (100 equal parts) istead. 3) How to draw a scatter graph ad lie of best fit. 4) That the lie of best fit ca show the correlatio of the data. If it is slopig upwards away from the origi, the it is positively correlated. If it is slopig the other way, the it is egatively correlated. If you caot draw a lie of best fit, the the data is ucorrelated. 5) That if you wat to fid a value beyod the lie of best fit, the it is called extrapolatig. Fidig a value betwee the kow values is called iterpolatig. 6) That a stratified radom sample is whe you split people for a survey ito groups, ad choose people at radom from each group. The umber chose though, depeds o the size of the group. 7) That quota samplig is whe you choose certai types of people to ask. It is ot radom though, so it ca be biased. 8) That a evet is aythig that happes whe you are calculatig probabilities. Several evets make up a outcome. 9) That evets are idepedet whe they do t affect each other. Evets are depedat if they affect each other. 30) How to draw a tree diagram.

4 31) That the probability of somethig happeig is the umber of your outcomes over the total umber of outcomes. 3) That evets are mutually exclusive if they have o poits i commo, ad evets that have poits i commo are o-mutually exclusive. 33) That you ca use special symbols to represet umbers i statistics: x - represets all the data collected. - stads for the umber of items collected. - (sigma) meas to add all the umbers up. x - (x bar) represets the mea. e.g. x x = 34) That the deviatio from the mea is the umber mius the mea (x - x). The variace of a set of umbers shows how spread out they are: variace = ( x x) 35) That the stadard deviatio ca be used to measure the spread of data. It is useful because it takes all the umbers ito accout: stadard deviatio = ( x x) 36) That the mea deviatio ca also be used, but it does t take ito accout the sig, ad does t do ay squarig. It uses the modulus ( ) to show that it does t matter what the sig is: mea deviatio = x x 37) That to compare two sets of data which are t from the same experimet, you eed to scale oe or both of the sets of data. You first eed to stadardise the data, the you scale it: x x stadardise - s = σ scale - x + ( sσ )

5 38) That the frequecy distributio shows all the possible results, ad the frequecy of each result. 39) That the shape of the frequecy distributio ca be described: SYMMETRICAL - has the mode at the cetre. BIMODAL - has two peaks. SKEW - ot symmetrical. POSITIVELY SKEWED - skewed towards the y-axis. NEGATIVELY SKEWED - skewed away from the y-axis. NORMAL DISTRIBUTION - a special curve with few high ad low values, but with the highest frequecies aroud the middle. 40) How to draw a box plot, ad a stem ad leaf diagram. 41) That to fid the equatio of a lie of best fit, you eed the gradiet of the lie, ad the y-itercept of the lie. The gradiet ca be foud by drawig a lie horizotally across from the y-axis, the goig vertically up or dow agai, to reach the x-axis. The legth of the vertical lie divided by the legth of the horizotal lie gives the gradiet. The y-itercept is the y-coordiate of the poit the lie crosses the y-axis (it may eed to be exteded). This will give the followig equatio: y = (gradiet)x + (y-itercept) OR y = mx + c 4) That the gradiet is called the regressio coefficiet. The higher the regressio coefficiet, the steeper the lie. 43) How to use Spearma s coefficiet of Rak Correlatio (d is the differece betwee raks, ad is the umber of thigs placed i order): 6d ( 1) r = s 1 44) That diagrams ca be misleadig by chagig scales, ad by icreasig the width of bars / pictures, as well as the height. 45) That 5! meas 5 factorial, or 5 x 4 x 3 x x 1. The same applies for ay umber (e.g. 3! = 3 x x 1). 46) That to calculate the umber of ways of choosig several items out of a group of items (e.g. 5 out of 4), you use the followig formula ( is the umber to choose from, ad r is the umber to choose):

6 C r =! r! ( r)! 47) To fid the probabilities of several evets (all the same) happeig oe after aother i a experimet, you use the followig formula: r Probability (r successes) = C p ( 1 p) 48) That a time series is whe you collect data over a series of time. If a graph goes up ad dow quite regularly, it is called variatio. There are three types of variatio: SEASONAL - a regular chage that ca happe over ay period of time (weeks, moths, years etc.). CYCLICAL - a chage that keeps o happeig but is ot regular. RANDOM - a chage that is t seasoal or cyclical. 49) That movig averages are used to show the geeral tred by eveig out seasoal variatio, ad how to use movig averages. 50) That you ca exted the tred lie o a time series graph to predict what could happe i the future. After extedig the tred lie, you eed to add the seasoal variatio. To do this you look at the previous times whe seasoal variatio takes place, ad calculate how far above or below the tred lie you should be. You the fid the mea, ad add it to (or subtract it from) the x value of the tred lie at that poit. 51) That the birth rate is the umber of births for every 1000 i the populatio i oe year. It ca be calculated usig the followig formula: r r total umber of births i the year x 1000 total populatio at the middle of the year 5) How to stadardise the birth rate ad death rate for differet populatios. 53) That to calculate the weighted mea you use the followig formula: weighted mea = w1 score1 + w score + w3 score w + w + w ) That a idex umber is a percetage used to compare prices. A base price is used to compare other prices with, ad is give the idex umber of 100. The formula for calculatig other idex umbers is as follows: idex umber = price x 100

7 base price 55) That a retail price idex is a idex umber for lots of differet items. It ca be used to compare the cost of livig from oe year to aother. The formula to calculate a retail price idex is as follows (weightig = w, ad idex umber = d): retail price idex = ( w d) w

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

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

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

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

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

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

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

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

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

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 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

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

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

Alternatives To Pearson s and Spearman s Correlation Coefficients

Alternatives To Pearson s and Spearman s Correlation Coefficients Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives

More information

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

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

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

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

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

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

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

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

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

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

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

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

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC. SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol

More 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

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

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

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

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

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

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

Covariance and correlation

Covariance and correlation Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative

More information

Simple Linear Regression

Simple Linear Regression Simple Liear Regressio We have bee itroduced to the otio that a categorical variable could deped o differet levels of aother variable whe we discussed cotigecy tables. We ll exted this idea to the case

More information

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective:. Improve calculator skills eeded i a multiple choice statistical eamiatio where the eam allows the studet to use a scietific calculator..

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

SENIOR CERTIFICATE EXAMINATIONS

SENIOR CERTIFICATE EXAMINATIONS SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P1 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 9 pages ad 1 iformatio sheet. Please tur over Mathematics/P1 DBE/016 INSTRUCTIONS AND INFORMATION

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

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

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More 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

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More 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

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

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

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

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

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

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

The shaded region above represents the region in which z lies.

The shaded region above represents the region in which z lies. GCE A Level H Maths Solutio Paper SECTION A (PURE MATHEMATICS) (i) Im 3 Note: Uless required i the questio, it would be sufficiet to just idicate the cetre ad radius of the circle i such a locus drawig.

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

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

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

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

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

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

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

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

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

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

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

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

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

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

Correlation. example 2

Correlation. example 2 Correlatio Iitially developed by Sir Fracis Galto (888) ad Karl Pearso (8) Sir Fracis Galto 8- correlatio is a much abused word/term correlatio is a term which implies that there is a associatio betwee

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

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information

Compare Multiple Response Variables

Compare Multiple Response Variables Compare Multiple Respose Variables STATGRAPHICS Mobile Rev. 4/7/006 This procedure compares the data cotaied i three or more Respose colums. It performs a oe-way aalysis of variace to determie whether

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

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

Problem Set 1 Oligopoly, market shares and concentration indexes

Problem Set 1 Oligopoly, market shares and concentration indexes Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over Mathematics/P DoE/Feb.

More information

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test) No-Parametric ivariate Statistics: Wilcoxo-Ma-Whitey 2 Sample Test 1 Ma-Whitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo-) Ma-Whitey (WMW) test is the o-parametric equivalet of a pooled

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

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a -page formula sheet. Please tur over Mathematics/P DoE/Exemplar

More information

MEP Pupil Text 9. The mean, median and mode are three different ways of describing the average.

MEP Pupil Text 9. The mean, median and mode are three different ways of describing the average. 9 Data Aalysis 9. Mea, Media, Mode ad Rage I Uit 8, you were lookig at ways of collectig ad represetig data. I this uit, you will go oe step further ad fid out how to calculate statistical quatities which

More information

hp calculators HP 12C Platinum Statistics - correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient

hp calculators HP 12C Platinum Statistics - correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient HP 1C Platium Statistics - correlatio coefficiet The correlatio coefficiet HP1C Platium correlatio coefficiet Practice fidig correlatio coefficiets ad forecastig HP 1C Platium Statistics - correlatio coefficiet

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

Chapter 4. Example: Q: Bob rolls a 6-sided die. What is the sample space of this procedure? A: S = {1, 2, 3, 4, 5, 6}

Chapter 4. Example: Q: Bob rolls a 6-sided die. What is the sample space of this procedure? A: S = {1, 2, 3, 4, 5, 6} Chapter 4 Key Ideas Evets, Simple Evets, Sample Space, Odds, Compoud Evets, Idepedece, Coditioal Probability, 3 Approaches to Probability Additio Rule, Complemet Rule, Multiplicatio Rule, Law of Total

More information

Chapter 5 Discrete Probability Distributions

Chapter 5 Discrete Probability Distributions Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide Chapter 5 Discrete Probability Distributios Radom Variables Discrete Probability Distributios Epected Value ad Variace Poisso Distributio

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

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

Forecasting techniques

Forecasting techniques 2 Forecastig techiques this chapter covers... I this chapter we will examie some useful forecastig techiques that ca be applied whe budgetig. We start by lookig at the way that samplig ca be used to collect

More information

Using Excel to Construct Confidence Intervals

Using Excel to Construct Confidence Intervals OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

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

Notes on Hypothesis Testing

Notes on Hypothesis Testing Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

More information

Math 105: Review for Final Exam, Part II - SOLUTIONS

Math 105: Review for Final Exam, Part II - SOLUTIONS Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio fx) =x 3 l x o the iterval [/e, e ]. a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum or

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

7818 Interval estimation and hypothesis testing - Set

7818 Interval estimation and hypothesis testing - Set 7 7818 Iterval estimatio ad hypothesis testig - Set revised Nov 9, 010 You might wat to read some of the chapter i MGB o Parametric Iterval Estimatio. There are subtle di ereces across questios. uderstad

More information

Chapter 10 Student Lecture Notes 10-1

Chapter 10 Student Lecture Notes 10-1 Chapter 0 tudet Lecture Notes 0- Basic Busiess tatistics (9 th Editio) Chapter 0 Two-ample Tests with Numerical Data 004 Pretice-Hall, Ic. Chap 0- Chapter Topics Comparig Two Idepedet amples Z test for

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

One-sample test of proportions

One-sample test of proportions Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:

More 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

Quadrat Sampling in Population Ecology

Quadrat Sampling in Population Ecology Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information