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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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: Samples ad Hypothesis Testig. Test for populatio mea of a Normal distributio. Cotigecy Tables ad the Chi-squared Test Topic 4: Bivariate Data Purchasers have the licece to make multiple copies for use withi a sigle establishmet November, 5 MEI, Oak House, 9 Epsom Cetre, White Horse Busiess Park, Trowbridge, Wiltshire. BA4 XG. Compay No Eglad ad Wales Registered with the Charity Commissio, umber 589 Tel: Fax:

2 Summary S Topic The Poisso Distributio Chapter Pages -4 Example. Page 4 Exercise A Q. 7 Chapter Pages 5-6 Example. Page 6 Exercise A Q. 4, 5, Chapter Pages -5 Exercise B Q. 4, 8 Chapter Pages 8- Exercise C Q. (i), 3, 7 Exercise D Q. 4, 5 The Poisso distributio is a discrete radom variable X where r λ λ P( X r) e r! The parameter, λ, is the mea of the distributio. We write X ~ Poisso(λ ). The distributio may be used to model the umber of occureces of a evet i a give iterval provided the occurreces are: (i) radom, (ii) idepedet, (iii) occurrig at a fixed average rate. Mea, E(X) λ, Variace, Var(X) λ Mea Variace is a quick way of seeig if a Poisso model might be appropriate for some data. It is possible to ca.culate terms of the Poisso distributio by a recurrece relatioship. r λ λ E.g. P( X r) e ; r! r + λ λ λ P( X r + ) e P( X r) ( r + )! ( r + ) Care eeds to be take over the cumulatio of errors. Use of cumulative probability tables Cumulative Poisso probability tables are o pages 4-4 of the Studets' Hadbook ad are available i the examiatios. They give cumulative probabilities, i.e. P(X r). So P( X r) P( X r) P( X r ) For λ.8 (Page 4), the secod ad third etry of the tables give P( X ).468, P( X ).736 i.e. P( X ) P( X ) P( X ) Sum of Poisso distributios Two or more Poisso distributios ca be combied by additio providig they are idepedet of each other. If X ~ Poisso(λ) ad Y ~ Poisso(μ), the X + Y ~ Poisso(λ +μ). N.B. You may oly add two Poisso distributios i this way if they are idepedet of each other. There is o correspodig result for subtractio. Approximatio to the biomial distributio The Poisso distributio may be used as a approximatio to the biomial distributio B(, p) whe (i) is large, (ii) p is small (so the evet is rare). The λ p. Note that the Normal distributio is likely to be a good approximatio if p is large. Statistics Versio B: page Competece statemets P, P, P3, P4, P5 E.g. The mea umber of telephoe calls to a office is every miutes. The probability distributio (Poisso()) is as follows: P( X ) e.353 P( X ) e.77 P( X ) e e.77! 3 P( X 3) e.84 3! 4 P( X 4) e.9 4! P( X > 4) sum of above.57 E.g. For λ, fid P(3) from tables. 3-8 Note that P(3) e.353 3! 6.84 From tables, P(3) P( X 3) P( X ) E.g. Vehicles passig alog a road were couted ad categorised as either private or commercial; o average there were 3 private ad commercial vehicles passig a poit every miute. It is assumed that the distributios are idepedet of each other. Fid the probability that i a give miute there will be (i) o private vehicles, (ii) o vehicles, (iii) exactly oe vehicle. For the private vehicles, λ p 3 ad for commercial vehicles λ c. The distributio may be modelled by the Poisso distributio so that for private vehicles, X ~ Poisso(3) ad for commercial vehicles, Y ~ Poisso(). For all vehicles, Z X + Y ~ Poisso(3+) Poisso(5) (i) P(X ) e (ii) P(Z ) e 5.67 (iii) P(Z ) 5e Note that P( veh) P( priv). P( comm) + P( priv). P ( comm) 3e 3.e + e 3.e ( + 3)e 5 5e E.g. Some equivalet values: For X~B(4,.3), p.; P() (.97) ; For X~Poisso(.), P() e For X~B(8,.), p.6; P() (.98) 8.99.; For X~Poisso(.6), P() e -.6..

3 Summary S Topic The Normal Distributio Chapter Pages 3-44 Example. Page 35 Exercise A Q. 4 Chapter Pages 49, 5 Example.3 Page 49 Exercise B Q. Chapter Pages 5-5 Exercise B Q. 4 Chapter Pages 5-54 Exercise B Q. Exercise C Q., 6, 7 The Normal distributio, N(μ, ), is a cotiuous, symmetric distributio with mea μ ad stadard deviatio. The stadard Normal distributio N(,) has mea ad stadard deviatio. P(X<x ) is represeted by the area uder the curve below x. (It is a special case of a cotiuous probability desity fuctio which is a topic i Statistics 3.) The area uder the stadard Normal distributio curve ca be foud from tables. To fid the area uder ay other Normal distributio curve, the values eed to be stadardised by the formula x μ z Modellig May distributios i the real world, such as adult heights or itelligece quotiets, ca be modelled well by a Normal distributio with appropriate mea ad variace. Give the mea μ ad stadard deviatio, the Normal distributio N(μ, ) may ofte be used. Whe the uderlyig distributio is discrete the the Normal distributio may ofte be used, but i this case a cotiuity correctio must be applied. This requires us to take the mid-poit betwee successive possible values whe workig with cotiuous distributio tables. E.g. P(X) > 3 meas P(X > 3.5) if X ca take oly iteger values. The Normal approximatio to the biomial distributio. This is a valid process provided (i) is large, (ii) p is ot too close to or. Mea p, Variace pq. The approximatio will be N(p, pq). A cotiuity correctio must be applied because we are approximatig a discrete distributio by a cotiuous distributio. The Normal approximatio to the Poisso distributio. This is a valid process provided λ is sufficietly large for the distributio to be reasoably symmetric. A good guidelie is if λ is at least. For a Poisso distributio, mea variace λ The approximatio will be N(λ,λ). As with the Biomial Distributio, a cotiuity correctio must be applied because we are approximatig a discrete distributio by a cotiuous distributio. Statistics Versio B: page 3 Competece statemets N, N, N3, N5 For N(,), P(Z < z ) ca be foud from tables (Studets Hadbook, page 44) E.g. P(Z <.7).758 P(Z >.7) For N(,9) [μ, 3] P(X < 5) P(Z < z ) where z (5 ) / E.g. The distributio of masses of adult males may be modelled by a Normal distributio with mea 75 kg ad stadard deviatio 8 kg. Fid the probability that a ma chose at radom will have mass betwee 7 kg ad 9 kg. We require P(7< X < 9) P( z < Z < z) 7 75 where z ad z P(7 < X < 9) P( Z <.875) P <.65 P( Z <.875) P( <.65).9696 (.734).736 E.g. Fid the probability that whe a die is throw 3 times there are at least sixes. Usig the biomial distributio requires P(3 sixes) + P(9 sixes) P( sixes). However, usig N(p, pq) where 3 ad p / 6, gives N(5, 4.67). P(X > 9.5) P(Z > z ) ( Z ) ( Z ) where z (9.5 5) /.4.5 N.B. a cotiuity correctio is applied because the origial distributio (biomial) is beig approximated by a cotiuous distributio (Normal). E.g. A large firm has 5 telephoe lies. O average, 4 lies are i use at oce ad the distributio may be modelled by Poisso(4). Fid the probability of there ot beig eough lies. The distributio is Poisso(4). Approximate by N(4,4). The we require P( X > 5) P( Z > z) where z.66 4 P( X > 5)

4 Summary S Topic 3 Samples ad Hypothesis Testig : Estimatig the populatio mea of a Normal distributio Pages 68-7 Example 3. Page 7 Pages 7-73 Exercise 3A Q. (i),(iii) Pages Example 3.3 Page 74 Exercise 3A Q. 6 The distributio of sample meas If a populatio may be modelled by a Normal distributio ad samples of size are take from the populatio, the the distributio of meas of these samples is also Normal. ( μ ) If the paret populatio is N, the the samplig distributio of meas is N μ,. Hypothesis test for the mea usig the Normal distributio Tests o the mea usig a sigle sample. H is μ μ where μ is some specified value. H may be oe tailed: μ <μ or μ >μ or two tailed: μ μ. I other words, give the mea of the sample take we ask the questio, Could the mea of the paret populatio be what we thik it is? Suppose the paret populatio is N μ,, the the ( ) samplig distributio of meas is N μ,.the critical values are therefore μ ± k where the value of k depeds o the level of sigificace ad whether it is a oe or two-tailed test. x μ We therefore calculate the value z ad compare it to the value foud i tables. Alteratively, if the value of the mea of the sample lies iside the acceptace regio the we would accept H, but if it lay i the critical regio the we would reject H i favour of H. Alteratively, calculate the probability that the value is greater tha the value foud ad see if it less tha the sigificace level be used. Kow ad estimated stadard deviatio The hypothesis test described above requires the value of the stadard deviatio of the paret populatio. I reality the stadard deviatio of the paret populatio will usually ot be kow ad will have to be estimated from the sample data. If the sample size is sufficietly large, the s.d. of the sample may be used as the s.d. of the paret populatio. A good guidelie is to require 5. E.g. If the paret populatio is N(, 6) ad a sample of size 5 has mea 8.6, the this value comes from the samplig distributio of meas which is N(,.64). E.g. It is thought that the paret populatio is Normally distributed with mea. A radom sample of 5 data items has a sample mea of 4. ad s.d Is there ay evidece at the.% sigificace level that the mea of the populatio is ot? H : μ H: μ (Note that although the mea of the sample is greater tha the proposed mea, we do ot have μ> because of the wordig of the questio.) x μ 4. z Critical value from tables for two-tailed,.% sigificace level is 3.7 Sice 3.578>3.7 we reject Hi favour of H. There is evidece that the mea of the populatio is ot. E.g. A populatio has variace 6. It is required to test at the.5% level of sigificace whether the mea of the populatio could be or whether it is less tha this. A radom sample size 5 has a mea of 8.6. H: μ H: μ < k.58 (for.5% level, -tailed test) 4 Critical value is Sice 8.6 > we accept H ;there is o evidece at the.5% level of sigificace that the mea is less tha. Alteratively, if the mea is the the samplig distributio of meas is N(,.64) 8.6 The P( X 8.6) Φ Φ(.75) Sice.4 >.5 we accept H Statistics Versio B: page 4 Competece statemets N6

5 Summary S Topic 3 Samples ad Hypothesis Testig : Cotigecy Tables ad the Chi-squared Test Pages 8-85 Cotigecy Tables Suppose the elemets of a populatio have sets of distict characteristics {X, Y}, each set cotaiig a fiite umber of discrete characteristics X {x, x,.,x m } ad Y {y,y,.,y } the each elemet of the populatio will have a pair of characteristics (x i,y j ). The frequecy of these m pairs (x i,y j ) ca be tabulated ito a m cotigecy table. y y y E.g. a group of 5 studets was selected at radom from the whole populatio of studets at a College. Each was asked whether they drove to College or ot ad whether they lived more tha or less tha km from the College. The results are show i this table. Nearer tha km Further tha km Drives 7 8 x f, f, f, x f, f, f, Does ot drive Chapter Pages 87-9 Page 85 Exercise 3B Q. 4, 5 Exercise 3C Q., 8 x m f m, f m, f m, The margial totals are the sum of the rows ad the sum of the colums ad it is usual to add a row ad a colum for these. The requiremet is to determie the extet to which the variables are related. If they are ot related but idepedet, the theoretical probabilities ca be estimated from the sample data. You ow have two tables, oe cotaiig the actual (observed) frequecies ad the other cotaiig the estimated expected frequecies based o the assumptio that the variables are idepedet. The hypothesis test H : The variables are ot associated. H : The variables are associated. The χ Statistic (Chi-squared statistic) This statistic measures how far apart are the set of observed ad expected frequecies. X ( observed frequecy - expected frequecy) ( fo fe) f e expected frequecy Degrees of freedom The distributio depeds o the umber of free variables there are, called the degrees of freedom, ν. This is the umber of cells less the umber of restrictios placed o the data. For a table such as the example give the umber of cells to be filled is 4, but the overall total is 5 which is a restrictio ad the proportios for each variable were also estimated from the data, givig two further restrictios. So the umber of degrees of freedom i the example is. I geeral the umber of degrees of freedom for a m table is (m )( ). Statistics Versio B: page 5 Competece statemets H, H E.g. If drivig to College ad the distace lived are ot associated evets the if oe studet is chose at radom the estimated probabilities are 8 4 P(drives), P(lives further tha km) ad P(drives ad lives further tha km) So out of 5 people we would expect I a similar way the etries i the other three boxes are calculated to give the followig: Nearer tha km Further tha km Drives Does ot drive We test the hypotheses: H : the two evets are ot associated H : The two evets are associated. ( fo fe) X fe ( 4.56) ( ) ( 5.44 ) ( 7.56) If the test is at the 5% level the the tables o page 45 of the MEI Studets Hadbook gives the critical value of 3.84 (v ). Sice 4.4 > 3.84 we reject the ull hypothesis, H, ad coclude that there is evidece that the two evets are associated. If the test were at the % sigificace level the we would coclude that there was ot eough evidece to reject the ull hypothesis.

6 Summary S Topic 4 Bivariate Data Pages 4-9 Pages - Bivariate Data are pairs of values (x, y) associated with a sigle item. e.g. legths ad widths of leaves. The idividual variables x ad y may be discrete or cotiuous. A scatter diagram is obtaied by plottig the poits (x, y ), (x, y ) etc. Correlatio is a measure of the liear associatio betwee the variables. A lie of best fit is a lie draw to fit the set of data poits as closely as possible. This lie will pass through the mea poit ( xy, ) where xis the mea of the xvalues ad yis the mea of the y values. There is said to be perfect correlatio if all the poits lie o a lie. Correlatio ad Regressio If the x ad y values are both regarded as values of radom variables, the the aalysis is correlatio. Choose a sample from a populatio ad measure two attributes. If the x value is o-radom (e.g. time at fixed itervals) the the aalysis is regressio. Choose the value of oe variable ad measure the correspodig value of aother. Notatio for pairs of observatios ( x, y). Sxy xi x yi y, S xi x xi x Syy yi y yi y The alterative form for Sxy is xiy i Sxy xiyi xiyi xy Example The legth (x cm) ad width (y cm) of leaves of a tree were measured ad recorded as follows: x y The scatter graph is draw as show...5 y The mea poit is ( x, y) which is (.9,.5) The lie of best fit is draw through the poit (.9,.5) For the data above: S xy E.g. 5 studets are selected at radom ad their heights ad weights are measured. This will require correlatio aalysis. A ball is bouced 5 times from each of a umber of differet heights ad the height is recorded. This will require regressio aalysis. x 7.4; y 9.; xy Sxy.87 6 x Pages -4 Example 4. Page Exercise 4A Q. Pearso s Product Momet Correlatio Coefficiet provides a stadardised measure of covariace. Sxy r S. S yy xi x yi y xi x yi y The pmcc lies betwee - ad +. i i xy xy xi x yi y For the data above: x 5.76 S.3 y 3.9 S yy.4 r S xy.87 S S yy r ca be foud directly with a appropriate calculator. Statistics Versio B: page 6 Competece statemets b, b, b3

7 Summary S Topic 4 Bivariate Data Pages 8-4 Exercise 4C Q. 3 Pages 3-34 Example 4.3 Page 34 Exercise 4C Q. 5 Statistics Versio B: page 7 Competece statemets b4, b5, b6, b7, b8 Pages 4-44 Exercise 4D Q. 4 Testig a paret populatio correlatio by meas of a sample where r has bee foud The value of r foud for a sample ca be used to test hypotheses about the value of ρ, the correlatio i the paret populatio. Coditios: (i) the values of x ad y must be take from a bivariate Normal distributio, (ii) the data must be a radom sample. A idicatio that a bivariate Normal distributio is a valid model is show by a scatter plot which is roughly elliptical with the poits deser ear the middle. H : ρ There is o correlatio betwee the two variables. H : ρ There is correlatio betwee Or : the two variables ( - tailed test.) H : ρ > There is positive correlatio betwee Or : the two variables (- tailed test.) H : ρ < There is egative correlatio betwee the two variables (- tailed test.) Spearma s coefficiet of rak correlatio If the data do ot look liear whe plotted o a scatter graph (but appear to be reasoably mootomic), or if the rak order istead of the values is give, the the Pearso correlatio coefficiet is ot appropriate. Istead, Spearma s rak correlatio coefficiet should be used. It is usually calculated usig the formula 6 di rs ( ) where d is the differece i raks for each data pair. This coefficiet is used: (i) whe oly raked data are available, (ii) the data caot be assumed to be liear. I the latter case, the data should be raked. Where r s has bee foud the hypothesis test is set up i the same way. The coditio here is that the sample is radom. Make sure that you use the right tables! Tied Raks If two raks are tied i, say, the 3rd place the each should be give the rak 3 /. The least squares regressio lie For each value of x the value of y give ad the value o the lie may be differet by a amout called the residual. If the data pair is (x i,y i ) where the lie of best fit is y a + bx the y i - (a + bx i ) e i givig the residual e i. The least squares regressio lie is the lie that miimises the sum of the squares of the residuals. S xy The equatio of the lie is y y ( x x) S For the data of Example : r.885 We wish to test the hypothesis that there is positive correlatio betwee legths ad widths of the leaves of the tree. H o : ρ There is o correlatio betwee the two variables. H : ρ > There is positive correlatio betwee the two variables (-tailed test). From the Studets Hadbook, the critical value for 8 at 5% level (oe tailed test) is.65. Sice.885>.65 there is evidece that H ca be rejected ad that there is positive correlatio betwee the two variables. Example judges raked 5 competitors as follows: Competitor A B C D E Judge Judge d -3 d x4 d 4 rs.3 5x4 For the data of example : r s.3 H o : ρ There is o correlatio betwee the two variables. H : ρ > There is positive correlatio betwee the two variables (-tailed test). For 5 at the 5% level ( tailed test), the critical value is.9. Sice.3 <.9 we are uable to reject H o ad coclude that there is o evidece to suggest correlatio. E.g. For the data x y x 5, y 7.9, x 55, xy x 3, y S x x Sxy xy x y y 3.58 ( x 3) y.3x+.9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

Practice Problems for Test 3

Practice Problems for Test 3 Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all

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

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

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

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

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

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

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

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

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

A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:

A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as: A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 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

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

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

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

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

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

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

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

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

OMG! Excessive Texting Tied to Risky Teen Behaviors

OMG! Excessive Texting Tied to Risky Teen Behaviors BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9

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

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater

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

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

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

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011 15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

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

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

, a Wishart distribution with n -1 degrees of freedom and scale matrix.

, a Wishart distribution with n -1 degrees of freedom and scale matrix. UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

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

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

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

Confidence intervals and hypothesis tests

Confidence intervals and hypothesis tests Chapter 2 Cofidece itervals ad hypothesis tests This chapter focuses o how to draw coclusios about populatios from sample data. We ll start by lookig at biary data (e.g., pollig), ad lear how to estimate

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

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

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

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

Parametric (theoretical) probability distributions. (Wilks, Ch. 4) Discrete distributions: (e.g., yes/no; above normal, normal, below normal)

Parametric (theoretical) probability distributions. (Wilks, Ch. 4) Discrete distributions: (e.g., yes/no; above normal, normal, below normal) 6 Parametric (theoretical) probability distributios. (Wilks, Ch. 4) Note: parametric: assume a theoretical distributio (e.g., Gauss) No-parametric: o assumptio made about the distributio Advatages of assumig

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error STA 2023 Practice Questios Exam 2 Chapter 7- sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (-1) oe p ( 1 p) CI: prop.

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

A Mathematical Perspective on Gambling

A Mathematical Perspective on Gambling A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal

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

TI-83, TI-83 Plus or TI-84 for Non-Business Statistics

TI-83, TI-83 Plus or TI-84 for Non-Business Statistics TI-83, TI-83 Plu or TI-84 for No-Buie Statitic Chapter 3 Eterig Data Pre [STAT] the firt optio i already highlighted (:Edit) o you ca either pre [ENTER] or. Make ure the curor i i the lit, ot o the lit

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

3 Basic Definitions of Probability Theory

3 Basic Definitions of Probability Theory 3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio

More information

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1) BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet

More information

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies

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

3. If x and y are real numbers, what is the simplified radical form

3. If x and y are real numbers, what is the simplified radical form lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

More information

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

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

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

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

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

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

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

THE TWO-VARIABLE LINEAR REGRESSION MODEL

THE TWO-VARIABLE LINEAR REGRESSION MODEL THE TWO-VARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther part

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

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

Confidence Intervals for Linear Regression Slope

Confidence Intervals for Linear Regression Slope Chapter 856 Cofidece Iterval for Liear Regreio Slope Itroductio Thi routie calculate the ample ize eceary to achieve a pecified ditace from the lope to the cofidece limit at a tated cofidece level for

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

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

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

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