Hypothesis testing. Null and alternative hypotheses

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Hypothesis testing. Null and alternative hypotheses"

Transcription

1 Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate that the populatio mea is equal to some specified value ad the use sample iformatio to decide whether the hypothetical value ca be rejected or ot i the light of sample evidece. The decisio will deped o (1) the size of the differece betwee the hypothetical populatio mea ad the sample mea, () the size of the samplig error associated with the sample mea, ad (3) the degree of certaity the decisio-maker requires before rejectig the iitial hypothesis. Null ad alterative hypotheses First we set up what is kow as the ull hypothesis, H 0, about the populatio parameter, e.g. we may claim that the populatio mea µ is equal to some value µ 0, say. This is usually writte as H 0 :µ=µ 0. We the stipulate a alterative hypothesis, H 1, which may state, e.g., that the populatio mea is ot equal to µ 0, H 1 :µ µ 0. The purpose of hypothesis testig is to see if we have sufficiet evidece to reject the ull hypothesis. Typically, the ull hypothesis says that there is othig uusual or importat about the data we are cosiderig; for example, if we were lookig at the average test scores of childre who have received a particular teachig method, the ull hypothesis would be that the mea is equal to the atioal average. If we are testig a ew drug, ad are lookig at the proportio of people takig the drug whose coditio improves, we would take as our ull the proportio who improve with a placebo, or with a previous drug. If we are lookig for a relatioship betwee two variables, the ull hypothesis is usually that there is o relatioship, that is that the regressio coefficiet betwee them is 0. The alterative hypothesis is thus that there is somethig iterestig or differet about the populatio for example that the average test score from the ew teachig method is ot equal to the atioal average, or that the proportio who improve with the ew drug is ot equal to the previous rate, or that there is a relatioship betwee the two variables, so that the regressio coefficiet is ot equal to 0.

2 We treat H 0 as our default positio, ad we usually require quite strog evidece to reject the ull hypothesis typically 90%, 95% or 99%, depedig o the cotext. Test statistic Havig set up our ull ad alterative hypotheses, we look for a suitable test statistic that will give us evidece for or agaist the two hypotheses. For example, if we are lookig for evidece about the populatio mea (H 0 :µ=µ 0 vs. H 1 :µ µ 0 ), we will most likely use a statistic based o the sample mea, X. From our work i sectio 4, a suitable statistic (assumig we ow the stadard deviatio σ of the populatio) is X µ 0 Z = - that is, we measure X -µ 0 i terms of the Stadard Error ( σ / ) of X as a estimator for µ, which is equal to σ/. For large samples, 30, we kow that the distributio of X is ormal, so that Z will be a stadard ormal variable, that is Z N(0,1). The larger is ( X -µ 0 ), the bigger is Z, ad the less credible it is that H 0 is correct. So essetially what we are tryig to do is to measure whether the sample mea, X, is sigificatly differet from µ 0. Decisio rule We ow have to decide how large Z must be for us to reject H 0. This is related to the risk we are prepared to take of a icorrect decisio. I decidig whether to accept or reject a ull hypothesis, there are two types of error we may make: A Type 1 error is to reject the ull hypothesis whe it is correct. A Type error is to accept the ull hypothesis whe it is icorrect. We usually specify our decisio rule i terms of the probability of a type 1 error we are prepared to accept, deoted α. Depedig o α, we ca calculate critical values of the test statistic Z, so that if Z lies beyod the critical values, we reject H 0, while if Z lies withi the critical values, we accept H 0. Thus, i the case of the populatio mea, if our acceptable level of Type 1 error is α=0.05, the the critical values of the test statistic will be

3 Z=±1.96, sice we kow from sectio 4 that, if H 0 is true ad µ=µ 0, the P(-1.96<Z<1.96)=0.95. Hece we kow that, if µ=µ 0, there would be a less tha 5% probability of obtaiig a value of greater tha 1.96 or less tha -1.96, so that the probability of a type 1 error i rejectig H 0 is less tha 5%. If we obtai a value of Z betwee the critical values, we coclude that we do ot have sufficiet evidece to reject H 0, so we accept it. The acceptable probability of Type 1 error is also called the sigificace level of the test. If, say, α=5%, ad we reject H 0, we will say that we reject H 0 at the 5% level of sigificace, or that X is sigificatly differet from µ 0 at the 5% level of sigificace, etc. Thus, we set up our decisio rule to give H 0 the beefit of the doubt. We require 95% cofidece to reject it. Note agai that if we reject the ull hypothesis, we are ot sayig there is a 95% probability that µ µ 0. µ is a costat which either is equal to µ 0 or it is t. What we are sayig is that, if µ were equal to µ 0, there would be a 95% chace of obtaiig a test statistic betwee the critical values. Oly 5% of the time would we obtai a value for Z that would lead us to reject H 0. Hece P(Reject H 0 H 0 true) Note that if we were prepared to accept a Type 1 error probability of 10%, we would set our critical values at Z=±1.645, while if we were oly prepared to accept a 1% Type 1 error, we would set critical values of Z=±.58. Power of a test The power of a hypothesis test is the probability β of a Type error. Give two tests of a hypothesis H 0, we say that oe test is more powerful tha the other if, give a specified level of Type 1 error, it has a lower probability of Type error. Example Suppose we kow that average household icome i the populatio is 300 p.w., with stadard deviatio 50 per week. We are tryig to see whether households i a particular tow have a higher or lower average icome. We take a radom sample of 100 households i the tow, ad fid a average icome of 85 p.w. We wish to test the hypothesis that

4 average household icome i the tow is equal to the atioal average, with a 5% level of sigificace. Here H 0 is µ= 300, ad H 1 is µ 300. X µ 0 Our test statistic is Z=, with µ 0 =300, σ=50, ad =100. From the ( σ / ) sample, X =85. Hece, Z=(85-300)/(50/ 100) = -15/5 = -3. Give a 5% sigificace level, the critical values of the Z statistic are ±1.96. Our decisio rule is to accept H 0 if -1.96<Z<1.96, ad reject H 0 otherwise. Hece, we reject H 0, ad coclude that µ 300. I fact, we may coclude that the average household icome i this tow is sigificatly less tha the atioal average, at the 5% (or ideed at the 1%) level of sigificace. Two-tailed ad oe-tailed tests The example above ivolved a two-tailed test of sigificace that is, we were tryig to see if X was sigificatly higher or sigificatly lower tha µ 0. That is, H 1 was specified as µ µ 0. I a oe-tailed test, the alterative hypothesis is H 1 :µ>µ 0, or Hµ<µ 0. This would be appropriate if we had some a priori reaso to believe that we were likely to fid a differece i a particular directio. For example, if we were tryig to see if graduates have the same icome as the rest of the populatio, we might use a 1-tailed test, as we would aturally assume that graduates ted to ejoy a higher icome, so H 1 would be that µ>µ 0, where µ is graduate average icome, ad µ 0 is the average for the whole populatio. Whe we use a 1-tailed test, the critical value of Z is differet. For example, at the 5% level of sigificace, we would use a critical value for Z of 1.645, istead of ±1.96, sice P(Z>1.645 H 0 )=5%. (Hece ±1.645 as the 10% critical value for a -tailed test, sice P(Z< H 0 ) is also 5%, so we have 5% i each tail.) If our alterative hypothesis were µ<µ 0, the our critical value would be Z=-1.645, rejectig H 0 if Z falls below this.

5 1-tailed vs. Two-tailed test f(z).5%.5%.5% Z Proportios The procedure ad ratioale for testig hypotheses about populatio proportios are similar to those used for meas. They are based o the ormal distributio ad apply to large samples, 30. The ull hypothesis is specified i terms of the populatio proportio P, ad the sample proportio, p, ad the stadard error, SE(p)=( P(1-P))/ are used i the test statistic. For example, suppose we wish to test the ull hypothesis that the proportio of households i a certai tow with at least oe wage-earer is We have a radom sample of 100 households, ad the proportio of the sample with at least oe wageearer is p=0.81. We have H 0 : P=P 0 =0.85 H 1 :P Z = P p P 1 P ) 0 ( 0 0 = * = -.04/.0357 = Note that we use the stadard error calculated from the populatio proportio based o the ull hypothesis this is because we are tryig to say If the ull hypothesis were true, how likely would it be to get this

6 much differece betwee the sample proportio ad populatio proportio?. So we cosider the probability distributio of the test statistic that would apply if the ull hypothesis were true. As 1.10<1.96, the r% level of sigificace -tailed critical value of the Z statistic, we caot reject H 0, i other words the sample proportio is ot sigificatly differet from 0.85 (at the 5% level). We therefore accept H 0. Differece betwee two sample meas So far we have made ifereces o a sigle sample. Now we shall make ifereces from two samples. Typically we shall have two radom samples from two populatios ad we shall be makig ifereces about the differeces betwee the meas of the two populatios usig the differece betwee the two sample meas. For example, we may be iterested i testig whether boys are achievig sigificatly differet results i school tha girls. To be able to aswer such a questio, we first eed to study the samplig distributio of the differece betwee two sample meas. If a radom sample of size 1 is take from oe populatio with mea µ 1 ad variace σ 1, ad aother radom sample of size is take from aother populatio with mea µ ad variace σ, the differece betwee the two sample meas is defied as d=( X 1 X ) where X 1 ad X are idepedet radom variables because they will ot vary from oe set of two samples to aother, ad because chages i X 1 are ot iflueced by chages i X ad vice-versa. E(d) = E( X 1- X ) = E( X 1)-E( X ) = µ 1 -µ = D. i.e. the sample differece (d) is a ubiased estimator of the populatio differece D. Var(d) = Var( X 1 X ) = Var( X 1) + Var( X ) = (σ 1 / 1 ) + (σ+ / ) Sice X 1 ad X are idepedet.

7 σ The stadard error of d is give by SE(d)= 1 σ + ad shows that the larger are the two variaces ad the smaller the sample sizes, the larger will be the samplig error of d. If X 1 ad X are ormally distributed, the X 1 ad X are also ormally distributed. Also, if both samples are large ( 1, 30), the eve if X ad X are ot ormally distributed, the Cetral Limit Theorem esures that X 1 ad X will be approximately ormally distributed. If either of these is true, the d will also be ormally distributed, as the differece betwee two ormal variables. Thus, σ d=( X 1 X ) N[(µ 1 -µ ), 1 σ + ] The cofidece iterval for the differece betwee the populatio meas ca ow be easily calculated. The 95% cofidece iterval is (µ 1 -µ ) = ( X 1 X ) ±1.96 σ 1 σ + The calculated cofidece iterval will cotai the true populatio differece i 95% of samples. Hece, the hypothesis test for the populatio differece ca also be performed i the usual maer. Let H 0 : µ 1 -µ =0, ad H 1 :µ 1 -µ 0. The test statistic is Z = ( X1 X ) 0, σ σ + 1 ad the decisio rule, for a 5% sigificace level, will be to reject H 0 if Z 1.96, otherwise accept H 0. Example A school wats to fid out if there is a differece i test performace betwee boys ad girls. A sample of test scores of 60 boys ad 50 girls is

8 examied. It is foud that the boys have sample mea X 1=54 with stadard deviatio 14, ad the girls have sample mea X =60, with stadard deviatio 9. NB: we shall igore for ow the problem of estimatig the populatio stadard deviatios, ad assume these figures are correct. We set up H 0 : X 1- X =0 H 1 : X 1- X 0. Our test statistic is ( X 1 X ) 0 σ 1 σ + = = -6/ (4.68) = As usual, for a 5% level of sigificace o a two-tailed test, our critical value for Z is ±1.96, so we do ot have sufficiet evidece to reject the ull hypothesis. Girls are doig better, but ot sigificatly better. Differece betwee two sample proportios This ca be tested i a similar maer. Exercise Two differet teachig methods are tried with differet groups of studets o the same course. I the first group, 47 out of 63 studets pass. I the secod group, 66 out of 78 pass. The departmet wats to work out whether oe teachig method is sigificatly better tha the other. Formulate suitable ull ad alterative hypotheses, ad calculate a suitable test statistic, to test this.

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

One-sample test of proportions

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

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

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

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

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

Practice Problems for Test 3

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

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

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

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

Confidence intervals and hypothesis tests

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

More information

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

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

More information

OMG! Excessive Texting Tied to Risky Teen Behaviors

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

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Lesson 15 ANOVA (analysis of variance)

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

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

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

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

More information

STATISTICAL METHODS FOR BUSINESS

STATISTICAL METHODS FOR BUSINESS STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING 7.1.- Distributios associated with the samplig process. 7.2.- Iferetial processes ad relevat distributios.

More information

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

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

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

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

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

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

Chapter 7 Methods of Finding Estimators

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

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

A Mathematical Perspective on Gambling

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

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Soving Recurrence Relations

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

More information

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

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

More information

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

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book) MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

More information

THE TWO-VARIABLE LINEAR REGRESSION MODEL

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

More information

Central Limit Theorem and Its Applications to Baseball

Central Limit Theorem and Its Applications to Baseball Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead

More information

Maximum Likelihood Estimators.

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

More information

A probabilistic proof of a binomial identity

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

More information

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

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%

0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Sectio 10 Aswer Key: 0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% 1) A simple radom sample of New Yorkers fids that 87 are

More information

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

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

More information

Quadrat Sampling in Population Ecology

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

More information

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

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

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

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

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

More information

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

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

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Topic 5: Confidence Intervals (Chapter 9)

Topic 5: Confidence Intervals (Chapter 9) Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

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

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Predictive Modeling Data. in the ACT Electronic Student Record

Predictive Modeling Data. in the ACT Electronic Student Record Predictive Modelig Data i the ACT Electroic Studet Record overview Predictive Modelig Data Added to the ACT Electroic Studet Record With the release of studet records i September 2012, predictive modelig

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

A Recursive Formula for Moments of a Binomial Distribution

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

More information

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

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

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

3. Greatest Common Divisor - Least Common Multiple

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

More information

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

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

More information

Department of Computer Science, University of Otago

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

More information

The Forgotten Middle. research readiness results. Executive Summary

The Forgotten Middle. research readiness results. Executive Summary The Forgotte Middle Esurig that All Studets Are o Target for College ad Career Readiess before High School Executive Summary Today, college readiess also meas career readiess. While ot every high school

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

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

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

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations 4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

More information

Hypothesis testing using complex survey data

Hypothesis testing using complex survey data Hypotesis testig usig complex survey data A Sort Course preseted by Peter Ly, Uiversity of Essex i associatio wit te coferece of te Europea Survey Researc Associatio Prague, 5 Jue 007 1 1. Objective: Simple

More information

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

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

More information

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Chapter 5: Basic Linear Regression

Chapter 5: Basic Linear Regression Chapter 5: Basic Liear Regressio 1. Why Regressio Aalysis Has Domiated Ecoometrics By ow we have focused o formig estimates ad tests for fairly simple cases ivolvig oly oe variable at a time. But the core

More information

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

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

More information

Lecture 2: Karger s Min Cut Algorithm

Lecture 2: Karger s Min Cut Algorithm priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.

More information

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore

More information

Modified Line Search Method for Global Optimization

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

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k. 18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Incremental calculation of weighted mean and variance

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

More information

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

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

More information

The Stable Marriage Problem

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

More information

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the

More information

1 The Gaussian channel

1 The Gaussian channel ECE 77 Lecture 0 The Gaussia chael Objective: I this lecture we will lear about commuicatio over a chael of practical iterest, i which the trasmitted sigal is subjected to additive white Gaussia oise.

More information

How to use what you OWN to reduce what you OWE

How to use what you OWN to reduce what you OWE How to use what you OWN to reduce what you OWE Maulife Oe A Overview Most Caadias maage their fiaces by doig two thigs: 1. Depositig their icome ad other short-term assets ito chequig ad savigs accouts.

More information