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

Size: px
Start display at page:

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

Transcription

1 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 as attempts to aswer a questio like "Does the mea blood pressure of OC users differ from that of o-users? (Are x ad y differet?)" This speculatio, that maybe there really is o differece (maybe oral cotraceptive use has o effect o blood pressure), is a hypothesis H : X Y or H : X Y The alterative to H is aother hypothesis, H1 : X Y or H1 : X Y The alterative hypothesis says that the mea blood pressure of OC users really is differet from that of o-users. (It does't say which mea is bigger, or how big the differece is, oly that there is a differece.) Authors: Blume, Greevy Bios 311 Lecture Notes Page 1 of 18

2 I this example, the parameter of iterest, x - y, has a wide rage of possible values, ad x - y =, is special because it represets "o differece" (betwee OC users ad o-users), or "o effect" (of OC use). H is called the ull hypothesis. It states that the parameter x - y equals the special value, zero. H 1, the alterative hypothesis, cosists of the complemet of H. It states that the parameter does ot equal zero. I our example we had observatios o 8 OC users ad 21 o-users. The immediate questio is What do these observatios tell us?" or Is it right to say that these data represet strog evidece that there is ideed a differece betwee the mea blood pressure levels of OC users ad ousers? or Do these observatios justify rejectig H? Authors: Blume, Greevy Bios 311 Lecture Notes Page 2 of 18

3 (Note that: we caot directly aswer the questio Which of these hypotheses is true?" because we oly have a sample of the populatio to work with.) We could aswer the previous questios usig our ew cofidece iterval tools: Costruct a 95% cofidece iterval for x - y, ad aswer "Yes, our data show that there is a differece" if the value x - y = is ot i the iterval. That is, reject the hypothesis of o differece (H ) if the value x - y = is ot i the cofidece iterval. But if the value x - y = is iside the 95% CI, our aswer is "These observatios do ot allow us to reject H." We are ot sayig that the observatios are evidece supportig H. (Remember that our best estimate of x - y was 5.42, ot.) But the data do ot represet strog eough evidece agaist H to justify its rejectio. Authors: Blume, Greevy Bios 311 Lecture Notes Page 3 of 18

4 This procedure is essetially a classical hypothesis test: Whe H specifies the value of a parameter, (as it does i our example) (i) Costruct a 1(1-)% CI for the parameter. (ii) Reject H if the hypothesized value is ot i the iterval. Recall our earlier iterpretatio of the CI as: "the values of the parameter that are cosistet with the observatios." What we're doig here is checkig to see if the value specified by H is "cosistet with the observatios". If it is ot, the we reject H. A more mathematically direct approach has prove popular may areas of sciece. Let s look at hypothesis testig more formally. Authors: Blume, Greevy Bios 311 Lecture Notes Page 4 of 18

5 A More Direct Approach To Hypothesis Testig I the blood pressure example, if the two variaces are assumed to be equal the T* X Y S m 2 P m2 X 1 1 m Y ~ t Uder our Null hypothesis (the mea pressures of OC users ad o-users are o differet), H : x - y =, m=21, =8 so T* ~ t SP 8 21 has Studet's t distributio with 27 df (uder H ). For a observed sample we ca calculate the value of this statistic. (T * is called the test statistic ) If it falls i oe of the tails of the t-distributio (say either T * > 2.52 or T * < -2.52) the we reject H. Authors: Blume, Greevy Bios 311 Lecture Notes Page 5 of 18

6 A popular explaatio for the reasoig behid this procedure is as follows: If H is true the T * will usually (95% of the time) fall betwee ad If it falls outside this iterval, the either (a) H is true, ad we just happeed to observe a relatively rare sample where T * falls this far from its expected value,, or (b) H is false; the expected value of T * is really ot. Values of T * that are far from zero (large values of T * ) are improbable uder H. If we observe such a value, we reject H. I this example we chose 2.52 as the critical value for our test, ad used the rule "Reject H if T * exceeds 2.52" This rule will sometimes lead us to reject H whe it is i fact true. It is easy to calculate the probability of makig this error: P * * T 2.52 H 1 P 2.52 T Authors: Blume, Greevy Bios 311 Lecture Notes Page 6 of 18

7 By choosig this critical value, we cotrol the error probability. If we use a larger critical value, say t = 2.771, the we reduce the probability that we will reject H whe it is really true: P * * T H 1 P T This is a example of aother geeral scheme for hypothesis testig: Istead of costructig a cofidece iterval, choose a observable "test statistic" (T * ) for which (i) the probability distributio is kow (at least approximately) whe H is true, ad (ii) the larger the value of the test statistic, the stroger the evidece agaist H. The we choose some critical value, ad reject H if the observed value of the test statistic exceeds the critical value. Authors: Blume, Greevy Bios 311 Lecture Notes Page 7 of 18

8 I our OC example the test statistic was T * ~ t SP 8 21 Whe H is true, T * has a Studet's t distributio with 27 df. Sice the distributio of the test statistic whe H is true is kow, we ca choose the critical value to fix the error probability (reject H whe H is really true) at whatever value we like. Popular choices are.5 ad.1 (1/2 ad 1/1). This error probability is ofte called the "sigificace level" of the test ad ofte represet by =P(reject H H true) (Remember, we use the fact that H is true to costruct the test statistic.) Authors: Blume, Greevy Bios 311 Lecture Notes Page 8 of 18

9 Testig a Hypothesis About a Sigle Mea Assume that the data have ormal probability distributio, variace kow. The Null hypothesis is H : =. The Test statistic: T * X Notice the form of T *? * X T E X H Var X This is a appropriate test statistic for this hypothesis, because (i) its distributio whe H is true is kow (stadard ormal), ad (ii) the larger the value of the test statistic, the farther the sample mea, X, is from, ad it seems appropriate to say that the farther X is from, the stroger the evidece that is ot the expected value of X. Authors: Blume, Greevy Bios 311 Lecture Notes Page 9 of 18

10 If we choose the critical value 1.96, the we have P reject H X H P 1.96 H P Z This gives a test of H at the 5% sigificace level. To test H at the 1% sigificace level, we must use a larger critical value, P reject H X H P H P Z To reject H at the 1% level requires stroger evidece (a more extreme value of the test statistic) tha is required to reject at the 5% level. At the 5% level we reject the hypothesis that H : =E[X]= if X falls further tha 1.96 SE's away from the hypothesized value,. At the 1% level we oly reject if X falls further tha SE's away from. Authors: Blume, Greevy Bios 311 Lecture Notes Page 1 of 18

11 Example We kow (Pagao ad Gauvreau) that the distributio of serum cholesterol i adult males i the U.S. has a mea of 211 mg/1ml ad a stadard deviatio of mg/1ml. We wat to lear about me who are hypertesive smokers. Specifically, is the distributio of serum cholesterol amog such me ay differet from that i the geeral populatio? Let's suppose the stadard deviatio is the same as i the geeral populatio ( mg/1ml), ad test whether the mea, μ, is the same or ot. H : = 211 mg/1ml H 1 : 211 mg/1ml Our test statistic will be T * X 211 which, uder H, has a stadard ormal distributio. Authors: Blume, Greevy Bios 311 Lecture Notes Page 11 of 18

12 Thus to test at the 5% level, we will reject H if the absolute value of the test statistic exceeds This meas that we will reject uless X which implies that we will reject uless the 95% CI cotais the hypothesized value, 211: X X 1.96??? Authors: Blume, Greevy Bios 311 Lecture Notes Page 12 of 18

13 I Pagao's example, the sample size is = 12 ad the sample mea is 217, so the test statistic's value is T *.45 Sice this is much less tha the critical value, 1.96, we do ot come close to rejectig H. (The 95% CI is (191, 243), ad the hypothesized value, 211, is well withi the iterval.) If we take a much larger sample of hypertesive smokers, say 25 istead of 12, ad fid the same value for the sample mea, X 25 = 217, the test statistic will be T * 2.6 Sice this is greater tha the critical value, 1.96, the result would ow be "Reject H at the 5% level." Authors: Blume, Greevy Bios 311 Lecture Notes Page 13 of 18

14 Whe the observatios lead to rejectio of H at a specified level, such as 5%, they are said to be "statistically sigificat at the 5% level." (Whe they do't, they are "ot statistically sigificat.") We saw that a sample of size 12 with mea 217 is ot statistically sigificat for testig H : =211 at the 5% level ( σ = kow ). A sample of 25 with the same mea, 217, is statistically sigificat at the 5% level (but ot at the 1% level, sice the test statistic's value, 2.6, is less tha the 1% critical value, ) The 95% CI is 217 ± 1.96(/ 25 ), or ( 211.3, ), which excludes the hypothesized value, 211, ad The 99% CI is 217 ± 2.576(/ 25 ), or ( 29.5, ) which icludes that value. Authors: Blume, Greevy Bios 311 Lecture Notes Page 14 of 18

15 Textbooks ofte stress the importace of selectig the sigificace level (choosig the value for the error probability, α) before `lookig at the data. We will see later why they stress this. I practice it is very commo ot to select a fixed α at all. Istead, what is ofte doe is that for the value of the test statistic that is actually observed, the probability of observig a value that large or larger (uder H ) is calculated ad reported. This quatity is called the p-value. It is a measure of how extreme the test statistic is, ad it is used as a measure of how strog the evidece agaist H is. We've bee cosiderig a test procedure that chooses a test statistic, T *, ad a fixed error probability,. The it fids the critical value, say t, for which P(T * > t H )=. This procedure rejects H if the observed value of the test statistic, say T obs *, exceeds the critical value, t. This is equivalet to fidig the p-value, P(T * > T obs * H ), ad the rejectig H if this probability is less tha. Authors: Blume, Greevy Bios 311 Lecture Notes Page 15 of 18

16 For the hypertesive smokers' cholesterol levels (whe = 25) the test statistic was T * 25 X ad the observed value of the test statistic was T * obs The probability (uder H ) of observig a value of the test statistic this extreme is 25 X 25 P(T * > T * obs H )= P 2.6 H PZ Sice this is greater tha.1, but smaller tha.5, it shows that our observatios represet evidece strog eough to justify rejectio at the 5% sigificace level, but ot strog eough for the 1% level. Authors: Blume, Greevy Bios 311 Lecture Notes Page 16 of 18

17 A ote o the P-value The p-value P(T * > T obs * H ), is somethig very familiar. It is just a probability statemet about the sample average. P(T * > T * obs H )= P P X P X X H Which says that the probability of observig a sample mea greater tha or equal to the 217 that was observed is oly.39 if the true mea is really 211 (the ull hypothesis is true). Authors: Blume, Greevy Bios 311 Lecture Notes Page 17 of 18

18 Two Types of Errors We have see that for testig the hypothesis that the mea of a ormal probability distributio has a specified value, H : = (agaist the alterative hypothesis that the mea is ot ), we reject H at sigificace level =.5 if T * X 1.96 This procedure ca lead us to reject the hypothesis H whe it is really true. The probability of this error is.5, ad is ofte called the Type I error, sigificace level, or the size of the test. But what if the hypothesis H is false ad we fail to reject H? This error is called the Type II error ad represet by the symbol. Type I error : =P(reject H H true) Type II error: =P(fail to reject H H false) Power: 1-= P(reject H H false) (Loose defiitio we ll be more specific later) Authors: Blume, Greevy Bios 311 Lecture Notes Page 18 of 18

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

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

More information

Data Analysis and Statistical Methods Statistics 651

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

More information

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

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

More information

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

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

More information

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

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

More information

MA238 Assignment 4 Solutions (part a)

MA238 Assignment 4 Solutions (part a) (i) Sigle sample tests. Questio. MA38 Assigmet 4 Solutios (part a) (a) (b) (c) H 0 : = 50 sq. ft H A : < 50 sq. ft H 0 : = 3 mpg H A : > 3 mpg H 0 : = 5 mm H A : 5mm Questio. (i) What are the ull ad alterative

More information

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. 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 information

Frequentist Inference

Frequentist 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 information

Statistics 511 Additional Materials

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

More information

Power and Type II Error

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

More information

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

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

More information

University of California, Los Angeles Department of Statistics. Hypothesis testing

University of California, Los Angeles Department of Statistics. Hypothesis testing Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Elemets of a hypothesis test: Hypothesis testig Istructor: Nicolas Christou 1. Null hypothesis, H 0 (claim about µ, p, σ 2, µ

More information

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

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10 DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set

More information

Stat 200 -Testing Summary Page 1

Stat 200 -Testing Summary Page 1 Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece

More information

Chapter 5: Hypothesis testing

Chapter 5: Hypothesis testing Slide 5. Chapter 5: Hypothesis testig Hypothesis testig is about makig decisios Is a hypothesis true or false? Are wome paid less, o average, tha me? Barrow, Statistics for Ecoomics, Accoutig ad Busiess

More information

Final Examination Solutions 17/6/2010

Final 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 information

HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018

HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 We are resposible for 2 types of hypothesis tests that produce ifereces about the ukow populatio mea, µ, each of which has 3 possible

More information

Sample Size Determination (Two or More Samples)

Sample Size Determination (Two or More Samples) Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie

More information

Comparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading

Comparing 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 information

Successful HE applicants. Information sheet A Number of applicants. Gender Applicants Accepts Applicants Accepts. Age. Domicile

Successful HE applicants. Information sheet A Number of applicants. Gender Applicants Accepts Applicants Accepts. Age. Domicile Successful HE applicats Sigificace tests use data from samples to test hypotheses. You will use data o successful applicatios for courses i higher educatio to aswer questios about proportios, for example,

More information

Notes on Hypothesis Testing, Type I and Type II Errors

Notes on Hypothesis Testing, Type I and Type II Errors Joatha Hore PA 818 Fall 6 Notes o Hypothesis Testig, Type I ad Type II Errors Part 1. Hypothesis Testig Suppose that a medical firm develops a ew medicie that it claims will lead to a higher mea cure rate.

More information

1 Constructing and Interpreting a Confidence Interval

1 Constructing and Interpreting a Confidence Interval Itroductory Applied Ecoometrics EEP/IAS 118 Sprig 2014 WARM UP: Match the terms i the table with the correct formula: Adrew Crae-Droesch Sectio #6 5 March 2014 ˆ Let X be a radom variable with mea µ ad

More information

Chapter 22: What is a Test of Significance?

Chapter 22: What is a Test of Significance? Chapter 22: What is a Test of Sigificace? Thought Questio Assume that the statemet If it s Saturday, the it s the weeked is true. followig statemets will also be true? Which of the If it s the weeked,

More information

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

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

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST 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 information

1036: Probability & Statistics

1036: 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 information

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test.

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test. Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal

More information

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

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

More information

Common Large/Small Sample Tests 1/55

Common 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 information

We have also learned that, thanks to the Central Limit Theorem and the Law of Large Numbers,

We have also learned that, thanks to the Central Limit Theorem and the Law of Large Numbers, Cofidece Itervals III What we kow so far: We have see how to set cofidece itervals for the ea, or expected value, of a oral probability distributio, both whe the variace is kow (usig the stadard oral,

More information

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 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 information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tau.edu/~suhasii/teachig.htl Suhasii Subba Rao Exaple The itroge cotet of three differet clover plats is give below. 3DOK1 3DOK5 3DOK7

More information

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 3 (This versio August 17, 014) 015 Pearso Educatio, Ic. Stock/Watso

More information

1 Constructing and Interpreting a Confidence Interval

1 Constructing and Interpreting a Confidence Interval Itroductory Applied Ecoometrics EEP/IAS 118 Sprig 2014 WARM UP: Match the terms i the table with the correct formula: Adrew Crae-Droesch Sectio #6 5 March 2014 ˆ Let X be a radom variable with mea µ ad

More information

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

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis

Section 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 information

Chapter 23: Inferences About Means

Chapter 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 information

Math 140 Introductory Statistics

Math 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 information

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

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

More information

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

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

More information

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

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

More information

Chapter 8: Estimating with Confidence

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

More information

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

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

More information

Introductory statistics

Introductory statistics CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key

More information

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

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

More information

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests 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 information

z is the upper tail critical value from the normal distribution

z is the upper tail critical value from the normal distribution Statistical Iferece drawig coclusios about a populatio parameter, based o a sample estimate. Populatio: GRE results for a ew eam format o the quatitative sectio Sample: =30 test scores Populatio Samplig

More information

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

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

More information

Lecture 6 Simple alternatives and the Neyman-Pearson lemma

Lecture 6 Simple alternatives and the Neyman-Pearson lemma STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull

More information

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n. ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic

More information

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

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

More information

Statistics 20: Final Exam Solutions Summer Session 2007

Statistics 20: Final Exam Solutions Summer Session 2007 1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets

More information

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

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

More information

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes. Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem

More information

Announcements. Unit 5: Inference for Categorical Data Lecture 1: Inference for a single proportion

Announcements. Unit 5: Inference for Categorical Data Lecture 1: Inference for a single proportion Housekeepig Aoucemets Uit 5: Iferece for Categorical Data Lecture 1: Iferece for a sigle proportio Statistics 101 Mie Çetikaya-Rudel PA 4 due Friday at 5pm (exteded) PS 6 due Thursday, Oct 30 October 23,

More information

Lecture Notes 15 Hypothesis Testing (Chapter 10)

Lecture Notes 15 Hypothesis Testing (Chapter 10) 1 Itroductio Lecture Notes 15 Hypothesis Testig Chapter 10) Let X 1,..., X p θ x). Suppose we we wat to kow if θ = θ 0 or ot, where θ 0 is a specific value of θ. For example, if we are flippig a coi, we

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

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

More information

Topic 18: Composite Hypotheses

Topic 18: Composite Hypotheses Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:

More information

This is an introductory course in Analysis of Variance and Design of Experiments.

This 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 information

Two sample test (def 8.1) vs one sample test : Hypotesis testing: Two samples (Chapter 8) Example 8.2. Matched pairs (Example 8.6)

Two sample test (def 8.1) vs one sample test : Hypotesis testing: Two samples (Chapter 8) Example 8.2. Matched pairs (Example 8.6) Hypotesis testig: Two samples (Chapter 8) Medical statistics 00 http://folk.tu.o/eiriksko/medstat0/medstath0.html Two sample test (def 8.) vs oe sample test : Two sample test: Compare the uderlyig parameters

More information

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2 Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:

More information

6 Sample Size Calculations

6 Sample Size Calculations 6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig

More information

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency

Math 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 information

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

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

More information

A Confidence Interval for μ

A Confidence Interval for μ INFERENCES ABOUT μ Oe of the major objectives of statistics is to make ifereces about the distributio of the elemets i a populatio based o iformatio cotaied i a sample. Numerical summaries that characterize

More information

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The

More information

Worksheet 23 ( ) Introduction to Simple Linear Regression (continued)

Worksheet 23 ( ) Introduction to Simple Linear Regression (continued) Worksheet 3 ( 11.5-11.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This

More information

STATISTICAL INFERENCE

STATISTICAL INFERENCE STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

More information

Lecture 2: Monte Carlo Simulation

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

More information

The Hong Kong University of Science & Technology ISOM551 Introductory Statistics for Business Assignment 3 Suggested Solution

The Hong Kong University of Science & Technology ISOM551 Introductory Statistics for Business Assignment 3 Suggested Solution The Hog Kog Uiversity of ciece & Techology IOM55 Itroductory tatistics for Busiess Assigmet 3 uggested olutio Note All values of statistics i Q ad Q4 are obtaied by Excel. Qa. Let be the robability that

More information

Stat 421-SP2012 Interval Estimation Section

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

More information

Chapter 6 Sampling Distributions

Chapter 6 Sampling Distributions Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to

More information

Chapter 1 (Definitions)

Chapter 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 information

Last Lecture. Wald Test

Last Lecture. Wald Test Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig

More information

M1 for method for S xy. M1 for method for at least one of S xx or S yy. A1 for at least one of S xy, S xx, S yy correct. M1 for structure of r

M1 for method for S xy. M1 for method for at least one of S xx or S yy. A1 for at least one of S xy, S xx, S yy correct. M1 for structure of r Questio 1 (i) EITHER: 1 S xy = xy x y = 198.56 1 19.8 140.4 =.44 x x = 1411.66 1 19.8 = 15.657 1 S xx = y y = 1417.88 1 140.4 = 9.869 14 Sxy -.44 r = = SxxSyy 15.6579.869 = 0.76 1 S yy = 14 14 M1 for method

More information

y ij = µ + α i + ɛ ij,

y ij = µ + α i + ɛ ij, STAT 4 ANOVA -Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i

More information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

More information

LESSON 20: HYPOTHESIS TESTING

LESSON 20: HYPOTHESIS TESTING LESSN 20: YPTESIS TESTING utlie ypothesis testig Tests for the mea Tests for the proportio 1 YPTESIS TESTING TE CNTEXT Example 1: supervisor of a productio lie wats to determie if the productio time of

More information

Topic 9: Sampling Distributions of Estimators

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

More information

Topic 9: Sampling Distributions of Estimators

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

More information

Lesson 2. Projects and Hand-ins. Hypothesis testing Chaptre 3. { } x=172.0 = 3.67

Lesson 2. Projects and Hand-ins. Hypothesis testing Chaptre 3. { } x=172.0 = 3.67 Lesso 7--7 Chaptre 3 Projects ad Had-is Project I: latest ovember Project I: latest december Laboratio Measuremet systems aalysis I: latest december Project - are volutary. Laboratio is obligatory. Give

More information

Econ 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement.

Econ 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement. Eco 371 Exam #1 Multiple Choice (5 poits each): For each of the followig, select the sigle most appropriate optio to complete the statemet 1) The probability of a outcome a) is the umber of times that

More information

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

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

More information

Mathematical Notation Math Introduction to Applied Statistics

Mathematical 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 information

GG313 GEOLOGICAL DATA ANALYSIS

GG313 GEOLOGICAL DATA ANALYSIS GG313 GEOLOGICAL DATA ANALYSIS 1 Testig Hypothesis GG313 GEOLOGICAL DATA ANALYSIS LECTURE NOTES PAUL WESSEL SECTION TESTING OF HYPOTHESES Much of statistics is cocered with testig hypothesis agaist data

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

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ

If, 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

Chapter 13: Tests of Hypothesis Section 13.1 Introduction

Chapter 13: Tests of Hypothesis Section 13.1 Introduction Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed

More information

Estimation of a population proportion March 23,

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

More information

Topic 9: Sampling Distributions of Estimators

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

More information

UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2009 EXAMINATIONS ECO220Y1Y PART 1 OF 2 SOLUTIONS

UNIVERSITY 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 information

This chapter focuses on two experimental designs that are crucial to comparative studies: (1) independent samples and (2) matched pair samples.

This chapter focuses on two experimental designs that are crucial to comparative studies: (1) independent samples and (2) matched pair samples. Chapter 9 & : Comparig Two Treatmets: This chapter focuses o two eperimetal desigs that are crucial to comparative studies: () idepedet samples ad () matched pair samples Idepedet Radom amples from Two

More information

STAC51: Categorical data Analysis

STAC51: Categorical data Analysis STAC51: Categorical data Aalysis Mahida Samarakoo Jauary 28, 2016 Mahida Samarakoo STAC51: Categorical data Aalysis 1 / 35 Table of cotets Iferece for Proportios 1 Iferece for Proportios Mahida Samarakoo

More information

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

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

More information

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

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

More information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-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 information

p we will use that fact in constructing CI n for population proportion p. The approximation gets better with increasing n.

p we will use that fact in constructing CI n for population proportion p. The approximation gets better with increasing n. Estimatig oulatio roortio: We will cosider a dichotomous categorical variable(s) ( classes: A, ot A) i a large oulatio(s). Poulatio(s) should be at least 0 times larger tha the samle(s). We will discuss

More information

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

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

More information

1 Review of Probability & Statistics

1 Review of Probability & Statistics 1 Review of Probability & Statistics a. I a group of 000 people, it has bee reported that there are: 61 smokers 670 over 5 960 people who imbibe (drik alcohol) 86 smokers who imbibe 90 imbibers over 5

More information

Chapter two: Hypothesis testing

Chapter two: Hypothesis testing : Hypothesis testig - Some basic cocepts: - Data: The raw material of statistics is data. For our purposes we may defie data as umbers. The two kids of umbers that we use i statistics are umbers that result

More information