Stat 225 Lecture Notes Week 7, Chapter 8 and 11
|
|
- Julia Cobb
- 5 years ago
- Views:
Transcription
1 Normal Distributio Stat 5 Lecture Notes Week 7, Chapter 8 ad Please also prit out the ormal radom variable table from the Stat 5 homepage. The ormal distributio is by far the most importat distributio that we will study. It is the distributio that is most used i daily life to model real world situatios. It was first used bi 733 by Abraham DeMoive ad used Carl Friedrich Gauss to predict the locatio of astroomical bodies. We call it ormal because it is applicable to most ormal pheomea for a large populatio/ Situatio: Much data arises from repeated measuremets o the same variable. Examples Test scores for all Stat 5 courses. Height of 000 male football players. Sales receipts for 000 customers at a grocery store. PDF: A cotiuous radom variable is said to have a Normal distributio with mea μ ad variace >0, if it had the PDF x μ f ( x) = e for < x < π Note: A ormal distributios ca be completely determied by the mea ad stadard deviatio. Notatio: or ~ Normal( μ, ) ~ N( μ, ) Shape:
2 Expectatio ad Variace E ( ) = μ Var( ) = Characteristics: This distributio is ofte also referred to as the Gaussia distributio. The PDF has the characteristic bell shape ad symmetric about the mea (higher desity i the middle tha the tails). Sice this distributio is symmetric it has the property that the mea is equal to the media. The mea μ determies the locatio of the curve, ad determies the shape (scale or sharpess) of the curve (see the attached graph). Area uder the curve from egative ifiity to x is P( < x) = F(x). Fact: Liear combiatios of idepedet Normal radom variables are themselves Normal. Let ~ Normal( μ, ) ad ~ Normal( μ, ) ad a, b, c costats, the a + b ~ Normal( aμ + b, a ) a + b + c ~ Normal( aμ + bμ + c, a + b ) Stadard Normal Distributio: A ormal radom variable with mea μ = 0 ad variace = is called a Stadard Normal Radom Variable. Notatio: Z ~ Normal( μ = 0, = ) Commet: Ay Normal( μ, ) radom variable ca be coverted to a Stadard Normal radom variable. We refer to this process is called stadardizatio. Corollary: Process of Stadardizatio. If ~ Normal( μ, ), the what should a ad b be so that Z = a + b ~ Normal(0,)? a = b = μ μ Z=
3 Empirical Rule for a Stadard Normal Variable Withi stadard deviatio there is 68% of the data. Withi stadard deviatios there is 95% of the data. Withi 3 stadard deviatios there is 99.7% of the data. Computig Probabilities How do you compute the probabilities for a Normal radom variable? The PDF is ot easy to itegrate ad there does ot exist a closed form of the CDF for geeral μ ad, but the CDF values of a stadard ormal radom variable ca be foud i tables (ad with may calculators ). CDF: The otatio used for the CDF of a stadard Normal distributio is Φ( z) = P( Z z). You ca use Excel (commad: NORMDIST(x, mea, stadard deviatio) ) to compute CDF values for geeral ormal radom variables. Commets: If we wat to compute probabilities for ~ Normal( μ, ) (without usig a computer), the we first have to stadardize ito μ Z = ad the read the values from the stadard Normal table. 3
4 Examples. Fid the followig probabilities give Z ~ Normal( μ = 0, = ) a) P( Z.75) b) P( Z ) i. Usig the empirical rule ii. Usig the table c) P( Z >.) d) P(. Z ) e) What is a if P(Z > a) = 0.44? f) What is a if P( a Z a) = 0.4?. For ~ Normal( μ =, = 4) a) Fid P( < ) b) Fid P( < < ) c) What is b so that P( < b) = 0.08? d) The middle 90% observatios of fall betwee which two values? 4
5 e) Fid P ( >.44) f) What is P(3 + <5.5)? g) What is distributio of Y = 3 +? 3. Let ~ Normal( μ =, = ) ad ~ Normal( μ = 3, = ) be idepedet. Fid: c) P + ) ( b) P( + > ) The table lists values of the stadard Normal CDF, deoted Φ (z) for may differet values of z. Give z, you ca read off the CDF value Φ (z). But the table works i reverse, too. 3. For ~ Normal( μ =, = 3) fid x, such that P ( x) =
6 4. I a study of elite distace ruers, the mea weight was reported to be 63. kg with a stadard deviatio of 4.8. Assumig the distributio of the weights is ormal, sketch the desity curve of the weight distributio, with horizotal axis marked first i stadard deviatios, ad below marked i kgs. a) Fid the percet of ruers whose weight was greater tha 69 kg. b) Te elite ruers are weighted oe by oe, what is the probability that at least of them exceed 69 kg? Practice Problems. Usig the table, fid the proportio of observatios from a stadard ormal distributio that satisfies each of the followig statemets: a) For Z ~ Normal ( μ = 0, = ) fid P(Z = 0.5) b) For Z ~ Normal ( μ = 0, = ) fid P (.5 < Z < 0.8) c) For Z ~ Normal ( μ = 0, = ) fid P( Z >.5) d) For ~Normal ( μ =, = 4 ) fid P( > ) 6
7 e) For ~ Normal ( μ = 3, = 4 ) fid P( < ). Let ~Normal ( μ = 0, = ) ad ~Normal ( μ =, = 4 ) a) Fid P( - 3 > 7) b) Fid P( - > ) 3. The graduate Record Examiatios are widely used to help predict the performace of applicats to graduate schools. The rage of possible scores o a GRE is 00 to 900. The psychology departmet at a uiversity fids that the scores of its applicats o the quatitative GRE are approximately ormal with mea = 544 ad stadard deviatio = 03. Use the table to fid the relative frequecy (percet) of applicats whose score is a) less tha 500 b) betwee 500 ad 700. c) What miimum score would a studet eed i order to score better tha 77% of those takig the test? 7
8 4. A college exam for a course i the history of alie ecouters has a mea of 70 ad a stadard deviatio of 0. The professor determies that ay studet who scores i the bottom 0 th percetile will fail the course. a) Determie the scores o the exam that will produce a failig grade. b) Give a studet s score is greater tha 80, what is the probability that his core is less tha A luch stad i the busiess district has a mea daily gross icome of $40 with a stadard deviatio of $50. Assume that the daily gross icomes are ormally distributed. What is the relative frequecy (percet) correspodig to a daily gross icome of $495 or more? 8
9 Normal Approximatio to the Biomial Whe is the ormal approximatio is used? Assume I toss 00 fair cois for 0000 times. Each time I record the umber of heads. Totally there are 0,000 records. If these 0,000 are put ito oe histogram, we will see that the distributio looks like the ormal distributio (bell-shaped ad symmetric). I aother word, if is the umber of heads amog 00, the follows ormal distributio approximately. Situatio: For a radom variable ~Biomial(, p) as, the PMF of the Biomial becomes very similar i shape to the Normal PDF. Due to this similarly we ca use a Normal radom variable to approximate a biomial radom variable if is large eough ad p reasoably close to 0.5. The followig graphs show the Biomial PMF of for p = 0.5 ad selected values of. Superimposed is the cotiuous curve of a Normal PDF. 9
10 As you ca see above, for large values of ad moderate (close to 0.5) values of p i the PMF os is very close to a Normal PDF with μ = p ad = p( p). Approximatio Criteria We will use the criteria p > 5 ad ( p) > 5 as a rule of thumb for whe to use a Normal Approximatio for a Biomial radom ~ Biomial (, p) variable with ~ Normal( μ = p, = p( p)). The larger is, the more precise this approximatio will be. Cotiuity Correctio There is oe importat detail that we have to take care of whe computig Normal approximatio of Biomial radom variables. ~ Biomial (, p) is a discrete radom variable where as a Normal radom variable is a cotiuous radom variable. For a Biomial radom variable there is a differece betwee P ( < x) ad P( x) where for a Normal radom variable there is o differece i these two terms. This meas that we have to either iclude or exclude the a -bar i our computatio of the probability. We ca do that be goig a step of ½ to the left or right depedig o whether we wat to iclude or exclude the bar. This procedure is called a Cotiuity Correctio. 0
11 Cotiuity Correctio Formula For limits a ad b we have, for example, P ( a b + μ a μ b) Φ Φ The Cotiuity Correctio ca also be used to compute probabilities like P( a < b), but the the directio of the step eeds to be chaged. Example: It is kow that a machie has the probability 0. to make defective fuses. a) What is the probability that amog 000 such fuses, at least 60 ad at most 80 are defective oes? b) What is the probability that amog 000 such fuses, more tha 60 ad less tha 80 are defective oes? c) What is probability that over 80 fuses are defective? d) What is the probability that there are exactly 80 defective fuses?
12 Practice Problems Verify if the assumptios of the Normal approximatio are met for each of the problems below ad aswer the questio asked usig appropriate methods.. Accordig to govermet data, % of America childre uder the age of 6 live i households with icomes less tha the official poverty level. A study of learig i early childhood chooses a SRS of 300 childre. What is the probability that more tha 80 of the childre i the sample are from poverty households?.07. I a test of ESP (extrasesory perceptio), the experimeter looks at cards that are hidde from the subject. Each card is equally likely to cotai a star, a circle, a wavy lie, or a square. A experimeter looks at each of 00 cards i tur, ad the subject tries to read the experimeter's mid ad ame the shape o each. What is the probability that the subject gets 30 or more correct if the subject does ot have ESP ad is just guessig? (Assume the 00 observatios are idepedet.).49. The Purdue quarterback completes 44% of his passes. Fid the probability of the quarterback completig 5 or more of his ext 0 passes..005
13 Cetral Limit Theorem The Cetral Limit Theorem (CLT) is oe of the most importat theorems i probability theory. It was first stated by Abraham De Moivre ad geeralized by Pierre Laplace. Because of this, this theorem is also sometimes called the De Moivre Laplace Theorem. Situatio: Cosider a experimet where a certai variable is measured repeatedly may times. Let i be the result of the i th measuremet. The i s may have some discrete or cotiuous (ot ecessarily kow) distributio. What ca we say about the sum or average of the s? Demostratio: i Suppositio: If a measuremet is repeated may times, the the sum or average _ of the measuremets is _ approximately ormally _ distributed. 3
14 Defiitio: Let,..., be idepedetly ad idetically distributed (iid) radom variables. We defie the sample mea (average) to be Note: If ( ) = μ i = i= i E ad Var( ) = for all, the E( ) = E i = E i i= i= Var( ) = Var i = Var i ( ) Var = i= i= Cetral Limit Theorem: i ( ) = ( ) E( ) = E( ) = μ i ( ) ( ) i = Let,..., be idepedet radom variables with E ( i ) = μ ad Var( i ) = for i =,...,. The for large ( > 30 ), the sum ad the average of the i s has approximately a Normal distributio. Average: = Normal i ~ μ, i= Sum: i ~ Normal( μ, ) i= The larger is, the more closely will the PDF of the sum or average resemble a Normal PDF. If the i ~ N ( μ, ) to start with you do ot eed to meet the >30 criteria for the sum ad average to be ormally distributed. Examples. Oe has 00 light bulbs whose lifetimes are idepedet expoetials with mea 5 hours. If the bulbs are used oe at a time, with a failed bulb beig replaced immediately by a ew oe, what is the probability that there is a still workig bulb after 55 hours. 4
15 . How ofte do you have to roll a fair (six-sided) die, so that the probability that the average of the scores is betwee 3. ad 3.8 is at least 0.9? 3. Assume there are a average of 0.5 typos o each page of a ovel draft with 000 pages writte by a ovelist. Jim is a editor ad is goig to proofread the draft. What is the probability that for the first 50 pages, the average typos he foud o each page is betwee 0.44 ad 0.48? Practice Problems PP. Studet scores o exams give by a certai istructor have a mea 74 ad stadard deviatio 4. The istructor is about to give the exam to a class of size 40. Approximate the probability that the average test scores i the class of size of 40 exceeds 80. PP. A ew elevator i a large hotel is desiged to carry about 30 people, with a total weight of up to 5000 lbs. More tha 5000 lbs overloads the elevator. The average weight of guests at this hotel is 50 lbs with a SD of 55 lbs. Suppose 30 of the hotel's guests get ito the elevator. Assumig the weights of these guests are idepedet radom variables, what is the chace of overloadig the elevator? 5
16 PP3. The distributio of actual weights of 8-oz. chocolate bars produced by a certai machie is ormal with mea 8. ouces ad stadard deviatio 0. ouces. a) If a sample of five of these chocolate bars is selected, the probability that their average weight is less tha 8 oz. b) If a sample of five of these chocolate bars is selected, there is oly a 5% chace that the average weight of the sample of five of the chocolate bars will be below what average value Statistical Iferece: Cofidece Itervals Z-cofidece itervals for the populatio mea Why do we eve bother aalyzig data? We wat to draw coclusios from the data. Populatio: the etire group of idividuals that we wat iformatio about. Sample: a part of the populatio that we actually examie i order to gather iformatio about the whole populatio Parameter: a umber that describes the populatio a fixed umber, but i practice we do ot kow its value Note: The values of the parameters were give or you may figure them out i the previous chapters, so that you kow the exact distributio of the populatio ad you ca calculate the correspodig probabilities, etc. However, i this sectio, we will assume that they are ukow. Example: populatio mea μ. 6
17 Statistic: a umber that describes a sample its value is kow whe we have take a sample, but it ca chage from sample to sample (samplig variability) ofte used to estimate a ukow parameter Example: sample mea =. Statistical Iferece: use a fact about a sample to estimate the truth about the whole populatio Why ca t we just accept our sample mea as the populatio mea μ? Every time whe we use the statistics (sample mea), we get a differet aswer due to samplig variability. Two most commo types of formal statistical iferece: Cofidece Itervals: whe we wat to estimate a populatio parameter Sigificace Tests: whe we wat to assess the evidece provided by the data i favor of some claim about the populatio (yes/o questio about the populatio) Cofidece Itervals allow us to estimate a rage of values for the populatio mea kow how cofidet we should be that the populatio mea is withi that rage The true mea for the populatio exists ad is a fixed umber, but we just do t kow what it is. Usig our sample statistic, we ca create a et to give us a estimate of where to expect the populatio parameter to be. If we just take a sigle sample, our sigle cofidece iterval et may or may ot iclude the populatio parameter. However if we take may samples of the same size ad create a cofidece iterval from each sample statistic, over the log ru 95% of our cofidece itervals will cotai the true populatio parameter, if we are usig a 95% cofidece level. 7
18 If you icrease the size of your sample (), you decrease the size of your et (or your margi of error). If you icrease your cofidece level, the you icrease the size of your et (or your margi of error). A smaller et is good because it gives you more iformatio. It is a smaller rage for where to expect your true populatio parameter. Cofidece itervals look like: estimate ± margi of error Cofidece Iterval for a Populatio Mea μ: ± z * x (sample mea) is the estimate of μ; is the populatio stadard deviatio; x is the stadard deviatio of ; x m= z * x is called margi of error; 8
19 z* is the value o the stadard ormal curve with area C betwee z* ad z*. z* C 90% 95% 99% is our guess for μ, ad the margi of error guess is. C is the cofidece level. z * x shows how accurate we believe our Remember that the mea ad stadard deviatio for a sample mea are: μx = μx x x = Also remember that if is ormally distributed the will be too, ad if is large, the sample mea will be approximately ormally distributed eve if is ot ormally distributed (Cetral Limit Theorem), that is ~ Normal( μ, ) approximately. What if your margi of error is too large? Here are ways to reduce it: Icrease the sample size (bigger ) [Populatio size does ot matter, as log as it is much larger tha the sample size.] Use a lower level of cofidece (smaller C) Reduce [Hard to do] x Choosig sample Size for the desired margi of error m: z * = (Solvig a simple equatio from m= z* ) m 9
20 Be careful: You ca oly use the formula ± z* uder certai circumstaces: Data must be a SRS (simple radom sample) from the populatio. Data must be collected correctly (o bias). The margi of error covers oly radom samplig errors. You must kow the populatio stadard deviatio. Examples:. A questioaire of drikig habits was give to a radom sample of fraterity members, ad each studet was asked to report the # of beers he had druk i the past moth. The sample of 30 studets resulted i a average of beers with stadard deviatio of 9 beers. a) Give a 90% cofidece iterval for the mea umber of beers druk by fraterity members i the past moth. * ± z = ± = ± = (9.3, 4.7) 30 b) Is it true that 90% of the fraterity members each moth drik the umber of beers that lie i the iterval you foud i part (a)? Explai your aswer. No. The C.I. is for, ot for the actual idividuals. c) What is the margi of error for the 90% cofidece iterval? * m= z = d) How may studets should you sample if you wat a margi of error of for a 90% cofidece iterval? z * = = m 0
21 Stat 5 Lecture Notes Chapter 8.5 ad. A sample of STAT 5 studets yields the followig Exam scores: Assume that the populatio stadard deviatio is 0. The sample mea ca be calculated by calculator to be a) Fid the 90% cofidece iterval for the mea score μ for STAT 5 studets. * 0 ± z = ± = ± = (78.08, 87.58) b) Fid the 95% cofidece iterval. * 0 ± z = ± = ± = (77.7, 88.49) c) Fid the 99% cofidece iterval. ± = ± = ± = (75.39, 90.7) d) How do the margis of error i (b), (c), ad (d) chage as the cofidece level icreases? Why? Margi of error will (icrease or decrease?) as cofidece level icreases. wat to be more cofidet that the populatio mea will fall i that rage eeds to be (bigger or small?). We rage. So the How does sigificace level α i a two-sided test relate to cofidece itervals? If you have a -sided test (a claim o if μ is equal a umber μ 0 or ot: H 0 : μ=μ 0 vs H : μ μ 0 ), - α = the cofidece level. For example, a 95% cofidece level would give you α = If you have a -sided test, if the α ad cofidece level add to 00%, you ca reject H 0 if μ 0 (the umber you were checkig) is ot i the cofidece iterval. Graph below shows how to use a -α cofidece iterval to draw coclusio for a -side test.
Expectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationClass 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationAgreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log
More informationOverview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions
Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationUnderstanding Samples
1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We
More informationExam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234
STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More informationTopic 10: Introduction to Estimation
Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More informationComparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading
Topic 15 - Two Sample Iferece I STAT 511 Professor Bruce Craig Comparig Two Populatios Research ofte ivolves the compariso of two or more samples from differet populatios Graphical summaries provide visual
More informationSection 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis
Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests
More informationChapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p
Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationRead through these prior to coming to the test and follow them when you take your test.
Math 143 Sprig 2012 Test 2 Iformatio 1 Test 2 will be give i class o Thursday April 5. Material Covered The test is cummulative, but will emphasize the recet material (Chapters 6 8, 10 11, ad Sectios 12.1
More informationAP Statistics Review Ch. 8
AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationBig Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.
5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
More informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More informationLecture 1. Statistics: A science of information. Population: The population is the collection of all subjects we re interested in studying.
Lecture Mai Topics: Defiitios: Statistics, Populatio, Sample, Radom Sample, Statistical Iferece Type of Data Scales of Measuremet Describig Data with Numbers Describig Data Graphically. Defiitios. Example
More informationChapter 1 (Definitions)
FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9
Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationCommon Large/Small Sample Tests 1/55
Commo Large/Small Sample Tests 1/55 Test of Hypothesis for the Mea (σ Kow) Covert sample result ( x) to a z value Hypothesis Tests for µ Cosider the test H :μ = μ H 1 :μ > μ σ Kow (Assume the populatio
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:
PSet ----- Stats, Cocepts I Statistics 7.3. Cofidece Iterval for a Mea i Oe Sample [MATH] The Cetral Limit Theorem. Let...,,, be idepedet, idetically distributed (i.i.d.) radom variables havig mea µ ad
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notatio Math 113 - Itroductio to Applied Statistics Name : Use Word or WordPerfect to recreate the followig documets. Each article is worth 10 poits ad ca be prited ad give to the istructor
More informationContinuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised
Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for
More informationChapter 23: Inferences About Means
Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationCHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics
CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based
More informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
More informationMath 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency
Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the
More informationClass 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 7 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 013 by D.B. Rowe 1 Ageda: Skip Recap Chapter 10.5 ad 10.6 Lecture Chapter 11.1-11. Review Chapters 9 ad 10
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad o-users, x - y. Such studies are sometimes viewed
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationEco411 Lab: Central Limit Theorem, Normal Distribution, and Journey to Girl State
Eco411 Lab: Cetral Limit Theorem, Normal Distributio, ad Jourey to Girl State 1. Some studets may woder why the magic umber 1.96 or 2 (called critical values) is so importat i statistics. Where do they
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
More informationLecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS
Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece
More information1036: Probability & Statistics
036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0- Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationUnderstanding Dissimilarity Among Samples
Aoucemets: Midterm is Wed. Review sheet is o class webpage (i the list of lectures) ad will be covered i discussio o Moday. Two sheets of otes are allowed, same rules as for the oe sheet last time. Office
More informationStatistics 300: Elementary Statistics
Statistics 300: Elemetary Statistics Sectios 7-, 7-3, 7-4, 7-5 Parameter Estimatio Poit Estimate Best sigle value to use Questio What is the probability this estimate is the correct value? Parameter Estimatio
More information- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion
1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter
More informationChapter 11: Asking and Answering Questions About the Difference of Two Proportions
Chapter 11: Askig ad Aswerig Questios About the Differece of Two Proportios These otes reflect material from our text, Statistics, Learig from Data, First Editio, by Roxy Peck, published by CENGAGE Learig,
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2009 EXAMINATIONS ECO220Y1Y PART 1 OF 2 SOLUTIONS
PART of UNIVERSITY OF TORONTO Faculty of Arts ad Sciece APRIL/MAY 009 EAMINATIONS ECO0YY PART OF () The sample media is greater tha the sample mea whe there is. (B) () A radom variable is ormally distributed
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationStatistical 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 informationSTATS 200: Introduction to Statistical Inference. Lecture 1: Course introduction and polling
STATS 200: Itroductio to Statistical Iferece Lecture 1: Course itroductio ad pollig U.S. presidetial electio projectios by state (Source: fivethirtyeight.com, 25 September 2016) Pollig Let s try to uderstad
More information