Confidence Intervals QMET103

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Confidence Intervals QMET103"

Transcription

1 Cofidece Itervals QMET103 Library, Teachig ad Learig

2 CONFIDENCE INTERVALS provide a iterval estimate of the ukow populatio parameter. What is a cofidece iterval? Statisticias have a habit of hedgig their bets. They always isert qualifiers ito reports, war about all sorts of assumptios, ad ever admit to aythig more extreme tha probable. There's a famous sayig: "Statistics meas ever havig to say you're certai." Statemets must be qualified, of course, because we are always dealig with imperfect iformatio. I particular, it is ofte ecessary to make statemets about a populatio usig iformatio from a sample. No matter how carefully this sample is selected to be a fair ad ubiased represetatio of the populatio, relyig o iformatio from a sample will always lead to some level of ucertaity. So, a cofidece iterval is a iterval withi which we ca estimate, with some cofidece, that the true populatio parameter will lie. Itroductio Suppose we were iterested i aswerig a simple research questio such as: "What is the mea umber of digits that ca be remembered?" Havig specified the populatio of people to be: Licol Uiversity studets, we take a sample of 10. The umber of digits remembered for these 10 studets is: 4, 4, 5, 5, 5, 6, 6, 7, 8, 9. From these results we fid the estimated value of, that is x, to be 5.9 ad s But this will certaily ot be a perfect estimate. It is boud to be at least either a little too high or a little too low. For the estimate of to be of value, we eed to have some idea of how precise it is. That is, how close to is the estimate likely to be? A excellet way to specify the precisio is to costruct a cofidece iterval. Sice we kow that approximately 68% of a distributio lies withi 1 s.d. of the mea, we could say that we are 68% certai that the populatio mea lies withi a iterval of x 1s.d. That is, we could be about 68% cofidet that the true mea umber of digits that ca be remembered lies betwee or betwee 4.24 ad Ad, sice we kow that approximately 95% of a distributio lies withi 2 s.d. of the mea, we could say that we are about 95% certai that the populatio mea lies withi a iterval of x 2s.d. or betwee i.e. betwee 0.92 ad Similarly if approximately 99% of a distributio lies withi 3 s.d. of the mea, we could say that we are about 99% certai that the populatio mea lies withi a iterval of x 3s.d. or betwee i.e. betwee 4.24 ad Iterpretatio: A 95% cofidece iterval estimate meas that if all possible samples are take, 95% of them would iclude the true populatio mea somewhere i their iterval. Or we ca be 95% cofidet the iterval cotais the true populatio mea. (Other cofidece itervals used more frequetly are 90% CI or 99% CI). 2

3 How is it calculated? The formula for a cofidece iterval is sample statistic Z s.e.( populatio parameter) score or sample statistic t s.e.( sample statistic) score Each situatio eeds careful cosideratio, ad the followig decisios made: Is the sample statistic a mea or proportio? Is there oe sample or two? What is the stadard error (s.e.) of the sample statistic? Should a t or a Z score be used? A flow diagram may help to see the process. (see later) Notice: The formula cosists of three parts, separated by ad. I all s, the expressio after the -sig is the stadard error. That is, you are give a sample mea ad the populatio stadard deviatio or variace. Use x Z score 1. From the sample, calculate x or ote it if is give. 2. Look up a Z score from the stadard table (*see below). Note the level of cofidece required. 2 or kow 3. Calculate the stadard error of the sample statistic. For a mea, the s.e. is Example: For a set of data, x 85,,ad, fid a 95% x 85 Z score : 95% 095. ; ; s. z That is, Z Hece x Z score , Iterpretatio: We ca be 95% cofidet the true populatio mea lies betwee ad Note o use of calculator: The i the formula meas you must do two calculatios. Use the replay key o your calculator for this. Calculate the lower value i the, usig the mius ( - ) key The, with the right had > (which takes you to the begiig of the calculatio) scroll across util the cursor is over the ( ). Chage to + ad press =. You ow have the upper value of the iterval. 3

4 2 If the oly iformatio give is mea ad sample stadard deviatio orvariace, ukow a t score is used istead of a Z score x t 1 Use - 1. From the sample, calculate x. (This may be give to you.) s 2. Calculate the degrees of freedom. For a oe sample mea, this is Look up a t score from the t- table (*see below). Note level of cofidece required ad use the correct degrees of freedom (df). 5. Calculate the stadard error of the sample statistic. s se. For a oe sample mea, mea Example: For a set of data, x, s,ad, se ( mea) degrees of freedom fid a 95% ad 95% upper tail = So from table: t score : df t.100 t.050 t.025 t.010 t.005 t.001 t Hece, the = That is, t.025 = =( , ) That is, we ca be 95% cofidet that the true populatio mea lies betwee approximately ad Note that this iterval is oly slightly greater tha the oe calculated previously usig populatio s.d ad a Z score. 4

5 Practise Questios oe sample 1. A machie maufactures bolts to a set legth with variace of 6.25 mm. A radom sample of 20 bolts is checked ad foud to have a mea legth of 75.2 mm. Fid the 99% cofidece iterval for the mea legth of the bolts people were asked to measure their pulse rates after completig a 3 km ru. The mea was 105 beats ad the stadard deviatio was 8 beats. Costruct a 95% cofidece iterval for the mea of the populatio of people. 3. A type of golf ball is tested, by droppig it oto a hard surface from a height of 1 metre. The height it bouces is kow to be ormally distributed with a stadard deviatio of 3.6 cm. If a sample of 100 balls are tested ad the mea height of the bouces is 82 cm, fid a. 90% b. 95% c. 99% cofidece itervals for the mea of the bouce of the golf ball. 4. A sample of stalactites (a type of rock formatio) foud i a glow worm cave produced the followig legths i cm: Assumig that this sample came from a ormal populatio, calculate a 95% cofidece iterval for the mea legth of stalactites i the cave. 5. A doctor coducts a small survey with a radom sample of his patiets, measurig their cholesterol levels. Here is his data (the measuremets are i m.mol/l): Fid a 80% cofidece iterval for the mea cholesterol level of his patiets. 6. A major departmet store chai is iterested i estimatio the average amout its credit card users spet o their first visit to the chai s ew store. Fiftee credit cards were radomly sampled ad aalysed to show a mea of $50.50 ad variace 400. Costruct a 95% cofidet iterval for the average amout its credit card users spet o their first visit to the chai s ew store assumig that the amout spet follows a ormal distributio. 7. A race car driver tested his car for the time he takes to accelerate from 0 to 60 km/hr. I 20 such tests he obtaied a average of 4.85 secods with a stadard deviatio of 1.47 secods. What is a 95% cofidece iterval for the acceleratio time? 8. The actual voltages of power packs labelled as 12 volts are as follows: 11.77, 11.90, 11.64, 11.84, 12.13, 11.99, ad Calculate a 99% cofidece iterval for the true voltage i these packs. Whe readig a questio, ote: Has the variace, stadard deviatio or stadard error bee give? Adjust your formula to match what has bee give. Is the iformatio from the populatio or the sample? Remember to use a Z score if it is from the populatio ad t score for a sample. 5

6 p 1 p Use p Z score 1. From the sample, calculate p or ote if give. 2. Look up a Z score from the stadard table (*see below). Note the level of cofidece required. 3. Calculate the stadard error of the sample statistic. For a proportio, the s.e. is p1 p Example: I a Rugby World Cup, a radom sample of supporters were asked, Which coutry do you thik will wi the 2003 Rugby World Cup? The results are summarised: Coutry Number of supporters who thik their coutry will wi Australia 116 Eglad 13 Frace 25 New Zealad 140 Wester Samoa 50 South Africa 47 Wales 24 Udecided 65 Total 480 Calculate a 90% cofidece Iterval for the proportio who had ot yet decided. Solutio: ot decided p ; 90% cofidece Z = ( We ca be 90% cofidet betwee 11% ad 16% of the populatio were udecided. (If you chage the fractios to decimals, there will be a slight roudig error, but this will usually ot be greatly sigificat.) Practice Questios 1. Samples of size are take from populatios with a probability of success p. Use the values of ad p, the sample size ad proportio, give below, to fid cofidece itervals for the populatio proportio with the levels of cofidece idicated. p Cofidece Level a % b % c %. ) ,

7 2. A political cadidate fids that i a radom sample of 0 costituets, 34% support her party. Fid the 95% cofidece iterval for the support she i fact has. 3. Houses o a street are umbered from 1 to 627. Roimata takes a radom sample of 40 houses. She fids that i 25 of them, there are more tha 3 residets. Fid a 90% cofidece iterval for the proportio of all houses i the street havig more tha three residets. 4 A toy maufacturer wats to test for the proportio of faulty toys i a large batch produced by a particular factory. He tests a radom sample of 200 toys ad fids that 25 are faulty. Calculate a 94% cofidece iterval for the proportio of faulty toys i the complete batch. 5. I a survey carried out i Aucklad, 38 people out of a radom sample of 70 people said that they bought the New Zealad Herald regularly. Fid a 99% cofidece iterval for the proportio of people who buy the Herald i Aucklad. Aswers (mea) 1 Populatio variace give, so use Z score, ad calculate stadard dev % Z Hece , Sample stadard deviatio give so use t score , Populatio std.dev. give, so use Z score. (a) 90% Z , (b) 95% , , x 15.17, s , x 5.75, s , Z (c) 99% Z 4. Calculate sample mea ad s.d.: 17 95% df 13 t 2.16 ad 5. Calculate sample mea ad s.d.: % df 19 t ad x 50. 5,s , 15 df 14, p t , That is, we ca be 95% cofidet, credit card users spet o average betwee $40 ad $62. 7

8 7. x 4. 85, s d.f. 19, p t , That is, we ca be 95%cofidet that the true acceleratio time is betwee 4.16 ad 5.54 secods. 8. x ,s (after eterig data i calculator). That is we ca be 99% cofidet that the true voltage i the power packs is betwee ad volts. Aswers (proportio) a , b. C , p 0.34, 95% Z c , , 7 df 6, p t , , , p % Z , , ,

9 To fid a z-score (Usig Z table with upper half shadig oly) Probability Table etry is probability at or below z Z etc To locate the appropriate Z-score For example, to fid the correct Z-score for a 98% cofidece iterval: Chage the %age to a decimal 98% = 0.98 Halve this decimal 0.98/2 = 0.49 Fid this value i the table Closest value =.4901 Use the correspodig Z-score Z-score = 2.33 To fid a z-score (Usig Z table with left had tail shaded) Probability Table etry is probability at or below z Z etc To locate the appropriate Z-score For example, to fid the correct Z-score for a 98% cofidece iterval: Chage the %age to a decimal 98% = 0.98 Halve this decimal 0.98/2 = 0.49 Add = 0.99 Fid this value i the table Use the correspodig Z-score Closest value = Z-score = 2.33

10 10

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

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

Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.

Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63. STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to

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

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:

Mathacle. 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 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

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

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

Statistical Intervals for a Single Sample

Statistical Intervals for a Single Sample 3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea

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

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

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

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

1 Inferential Methods for Correlation and Regression Analysis

1 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 information

AP Statistics Review Ch. 8

AP 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 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

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

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

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

Instructor: 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 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

MBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS

MBACATÓLICA. 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 information

Confidence Interval for one population mean or one population proportion, continued. 1. Sample size estimation based on the large sample C.I.

Confidence Interval for one population mean or one population proportion, continued. 1. Sample size estimation based on the large sample C.I. Cofidece Iterval for oe populatio mea or oe populatio proportio, cotiued 1. ample size estimatio based o the large sample C.I. for p ˆ(1 ˆ) ˆ(1 ˆ) From the iterval ˆ p p Z p ˆ, p Z p p L legh of your 100(1

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

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

Confidence Intervals

Confidence Intervals Cofidece Itervals Berli Che Deartmet of Comuter Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chater 5 & Teachig Material Itroductio

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

CONFIDENCE INTERVALS STUDY GUIDE

CONFIDENCE INTERVALS STUDY GUIDE CONFIDENCE INTERVALS STUDY UIDE Last uit, we discussed how sample statistics vary. Uder the right coditios, sample statistics like meas ad proportios follow a Normal distributio, which allows us to calculate

More information

Homework 5 Solutions

Homework 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 information

MATH/STAT 352: Lecture 15

MATH/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 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

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

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

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

Confidence Intervals for the Population Proportion p

Confidence 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 information

Computing Confidence Intervals for Sample Data

Computing Confidence Intervals for Sample Data Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios

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

Economics Spring 2015

Economics Spring 2015 1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures

More information

S160 #12. Review of Large Sample Result for Sample Proportion

S160 #12. Review of Large Sample Result for Sample Proportion S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Review of Large Sample Result for Sample Proportio Recall that for large sample (ie, sample size is large, say p > 5 ad (1

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

Confidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab)

Confidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab) Cofidece Itervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Aa Phophoem, Ph.D. aa.p@ku.ac.th Itelliget Wireless Network Group (IWING Lab) http://iwig.cpe.ku.ac.th Computer Egieerig Departmet Kasetsart Uiversity,

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

S160 #12. Sampling Distribution of the Proportion, Part 2. JC Wang. February 25, 2016

S160 #12. Sampling Distribution of the Proportion, Part 2. JC Wang. February 25, 2016 S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Outlie 1 Estimatig Proportio Usig Itervals Cofidece Iterval for the Populatio Proportio iclicker Questios 2 JC Wag (WMU) S160

More information

(7 One- and Two-Sample Estimation Problem )

(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 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

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

Topic 10: Introduction to Estimation

Topic 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 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

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

Recall 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 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

BIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov

BIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios

More information

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion Poit Estimatio Poit estimatio is the rather simplistic (ad obvious) process of usig the kow value of a sample statistic as a approximatio to the ukow value of a populatio parameter. So we could for example

More information

STAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence

STAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Itroductory Statistics Chapter 6: Itroductio to Iferece Lecture 18: Estimatio with Cofidece 11/14/06 Lecture 18 1 Itroductio Statistical Iferece

More information

MIT : Quantitative Reasoning and Statistical Methods for Planning I

MIT : Quantitative Reasoning and Statistical Methods for Planning I MIT 11.220 Sprig 06 Recitatio 4 March 16, 2006 MIT - 11.220: Quatitative Reasoig ad Statistical Methods for Plaig I Recitatio #4: Sprig 2006 Cofidece Itervals ad Hypothesis Testig I. Cofidece Iterval 1.

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

Sample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for

Sample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they

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

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

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

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes. Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely

More information

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Confidence Interval Guesswork with Confidence

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Confidence Interval Guesswork with Confidence PSet ----- Stats, Cocepts I Statistics Cofidece Iterval Guesswork with Cofidece VII. CONFIDENCE INTERVAL 7.1. Sigificace Level ad Cofidece Iterval (CI) The Sigificace Level The sigificace level, ofte deoted

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 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 information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians 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 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

Understanding Samples

Understanding 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 information

Stat 139 Homework 7 Solutions, Fall 2015

Stat 139 Homework 7 Solutions, Fall 2015 Stat 139 Homework 7 Solutios, Fall 2015 Problem 1. I class we leared that the classical simple liear regressio model assumes the followig distributio of resposes: Y i = β 0 + β 1 X i + ɛ i, i = 1,...,,

More information

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M. MATH1005 Statistics Lecture 24 M. Stewart School of Mathematics ad Statistics Uiversity of Sydey Outlie Cofidece itervals summary Coservative ad approximate cofidece itervals for a biomial p The aïve iterval

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

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

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2. SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample

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

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

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

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

Chapter 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 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 information

24.1 Confidence Intervals and Margins of Error

24.1 Confidence Intervals and Margins of Error 24.1 Cofidece Itervals ad Margis of Error Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Resource Locker Explore Idetifyig Likely

More information

Class 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 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 information

Statistical Inference About Means and Proportions With Two Populations

Statistical Inference About Means and Proportions With Two Populations Departmet of Quatitative Methods & Iformatio Systems Itroductio to Busiess Statistics QM 220 Chapter 10 Statistical Iferece About Meas ad Proportios With Two Populatios Fall 2010 Dr. Mohammad Zaial 1 Chapter

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

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

Estimating the Population Mean - when a sample average is calculated we can create an interval centered on this average

Estimating the Population Mean - when a sample average is calculated we can create an interval centered on this average 6. Cofidece Iterval for the Populatio Mea p58 Estimatig the Populatio Mea - whe a sample average is calculated we ca create a iterval cetered o this average x-bar - at a predetermied level of cofidece

More information

(6) Fundamental Sampling Distribution and Data Discription

(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 information

WORKING WITH NUMBERS

WORKING WITH NUMBERS 1 WORKING WITH NUMBERS WHAT YOU NEED TO KNOW The defiitio of the differet umber sets: is the set of atural umbers {0, 1,, 3, }. is the set of itegers {, 3,, 1, 0, 1,, 3, }; + is the set of positive itegers;

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

1 Models for Matched Pairs

1 Models for Matched Pairs 1 Models for Matched Pairs Matched pairs occur whe we aalyse samples such that for each measuremet i oe of the samples there is a measuremet i the other sample that directly relates to the measuremet i

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

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22 CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first

More information

Chapter 20. Comparing Two Proportions. BPS - 5th Ed. Chapter 20 1

Chapter 20. Comparing Two Proportions. BPS - 5th Ed. Chapter 20 1 Chapter 0 Comparig Two Proportios BPS - 5th Ed. Chapter 0 Case Study Machie Reliability A study is performed to test of the reliability of products produced by two machies. Machie A produced 8 defective

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

UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION

UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION STATISTICAL METHODS FOR BUSINESS UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION 8..- Itroductio to iterval estimatio 8..- Cofidece itervals. Costructio ad characteristics 8.3.- Cofidece itervals for the mea

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

ANALYSIS OF EXPERIMENTAL ERRORS

ANALYSIS OF EXPERIMENTAL ERRORS ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder

More information

Analysis of Experimental Data

Analysis of Experimental Data Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.

Big 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 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

The standard deviation of the mean

The 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 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

Census. Mean. µ = x 1 + x x n n

Census. Mean. µ = x 1 + x x n n MATH 183 Basic Statistics Dr. Neal, WKU Let! be a populatio uder cosideratio ad let X be a specific measuremet that we are aalyzig. For example,! = All U.S. households ad X = Number of childre (uder age

More information