Statistics Part IV Confidence Limits and Hypothesis Testing. Joe Nahas University of Notre Dame
|
|
- Homer Armstrong
- 5 years ago
- Views:
Transcription
1 Statistics Part IV Confidence Limits and Hypothesis Testing Joe Nahas University of Notre Dame
2 Statistic Outline (cont.) 3. Graphical Display of Data A. Histogram B. Box Plot C. Normal Probability Plot D. Scatter Plot E. MatLab Plotting 4. Confidence Limits and Hypothesis Testing A. Student s t Distribution i. Who is Student ii. Definitions B. Confidence Limits for the Mean C. Equivalence of two Means D. Equivalence of two Variances 2 2
3 Student s t Distribution Suppose a random sample of size n is drawn from a normal N(μ,σ) population. If x is the estimate of the mean from the sample, and s is the sample standard deviation, then t = x μ s / n has the t distribution with n 1 degrees of freedom. There is a different t distribution for each sample size n as specified in the degrees of freedom. WE ARE NOT GOING TO PROVE THIS! NIST ESH
4 Who was Student? The t distribution was discovered by William S. Gossett, a statistician employed by the Guinness Brewing Company. He was trying to determine how accurate the data from his small samples were. Guinness previously had problems with proprietary information being published so it required Gossett not to publish his discoveries under his own name. Guinness did not want its competitors to know that it was using statistics to improve its beer. He published the t distribution under the pen name Student in The distribution is usually referred to as Student s t Distribution 4
5 Student s t distribution The probability density function for the t distribution is: where B is the Beta function and ν is a positive integer shape parameter. The Beta function is: f (x) = Β(α,β) = 1 + x 2 ν Β(0.5,0.5ν) ν 1 0 (ν +1) 2 t α 1 (1 t) β 1 dt The t distribution is equal to the Cauchy distribution for ν = 1. The t distribution approaches the normal distribution for large ν. 5
6 t Probability Density Function Large tails for ν = 1 Approaches Normal Distribution for large ν 6
7 Confidence Limits for the Mean By definition t = x μ s / n So s μ = x t α,ν n The probably value of μ is distributed around x. NIST ESH
8 t distribution Table Instructions NIST ESH
9 t distribution table NIST ESH
10 t distribution table NIST ESH
11 Confidence Limits of the Mean Confidence Limits are a two sided test. i.e. the real mean can be greater than or less then the estimate. Example: n = 195 m = s = % confidence interval α = 0.05 t 1 α/2,194 = Lower Limit = m t * s / sqrt(n) = * / sqrt(195) = Upper Limit = m + t * s / sqrt(n) = * / sqrt(195) = % of area 95% of area 2.5% of area 11
12 t distribution in Excel Values of t can be obtained using the TINV function in Excel. TINV(probability, degrees of freedom) probability = α for a two sided distribution e.g. instead of 1 α/2 = in table, use α = 0.05 degrees of freedom = ν = n 1 For the previous example: =tinv(0.05, 194) returns For a one sided distribution, use 2*α Hint: Before using tinv, try duplicating an example in the NIST ESH. 12
13 Are two Means Possibly Equal We have two estimates of the mean, m 1 and m 2 with m 1 > m 2. We have two estimates of the standard deviation, s 1 and s 2. We have two sample sizes, n 1 and n 2. This is a one sided test. Null Hypothesis: μ 1 = μ 2. Test Statistic x T = 1 x 2 s 2 1 / n 1 + s 2 2 / n 2 Similar to t = x μ s / n NIST ESH
14 Are two Means Possibly Equal Reject the Null Hypothesis if: where t 1 α,ν is the critical value of the t distribution with ν degrees of freedom where ν = T > t 1 α,ν (s 1 2 / n 1 + s 2 2 / n 2 ) 2 (s 1 2 / n 1 ) 2 /(n 1 1) + (s 2 2 / n 2 ) 2 /(n 2 1) 14
15 Equal Variances If equal variances are assumed: T = x 1 x 2 s p 1/n 1 + 1/n 2 and where s p is the pooled estimate of the standard deviation: s p = (n 1 1)s (n 2 1)s 2 2 n 1 + n 2 2 ν = n 1 + n
16 Example Mileage Data from US and Japanese cars in 1990s n 1 = 79 m 1 = s 1 = n 2 = 249 m 2 = s 2 = Assuming variances are equal T = s p = ν = 326 For 95% confidence, α = 0.05 t 0.95,ν=326 = Since T > t, the hypothesis that the means are equal is rejected! 95% of area 5 % of area 16
17 Large n What happens to the confidence limits as n gets large? μ = lim x (x t α,ν sn ) = x 17
18 Are Two Variances Equal? Null Hypothesis, H0: σ 12 = σ 2 2 Alternative Hypotheses, Ha: σ 12 < σ 2 2 for a lower one tailed test σ 12 > σ 2 2 for an upper one tailed test σ 12 σ 2 2 for a two tailed test Test Statistic: F = s 12 /s 2 2 Where s 12 and s 22 are the sample variances with sample sizes of N 1 and N 2 respectively Significance Level: α NIST ESH
19 Are Two Variances Equal? (cont.) The Hypothesis that the two variances, σ12, and σ22, are equal is rejected if: for an upper one tailed test F > F α, N1 1,N 2 1 F < F 1 α, N1 1,N 2 1 for a lower one tailed test F > F α, N1 1,N 2 1 or for a two tailed test F α, N1 1,N 2 1 F < F 1 α, N1 1,N 2 1 where is the critical value of the F distribution with N 1 1 and N 2 1 degrees of freedom and a significance level of α. NIST ESH
20 F Distribution 20
21 F Distribution NIST ESH
22 Using Excel for F Dist Use finv(α, N 1, N 1) function in Excel. Use exampel in NIST ESH to check usage. 22
23 Ceramic Data Example Is 65.5 significantly different from 61.9? From Excel NIST ESH
Statistics Part I Introduction. Joe Nahas University of Notre Dame
Statistics Part I Introduction Joe Nahas University of Notre Dame A Very Simple Example: A Pair of Die A pair of six sided die Values for each die: 1, 2, 3, 4, 5, 6. Values for the pair: 2, 3, 4, 5, 6,
More informationChapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.
Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,
More informationStatistical Foundations:
Statistical Foundations: t distributions, t-tests tests Psychology 790 Lecture #12 10/03/2006 Today sclass The t-distribution t ib ti in its full glory. Why we use it for nearly everything. Confidence
More informationCBA4 is live in practice mode this week exam mode from Saturday!
Announcements CBA4 is live in practice mode this week exam mode from Saturday! Material covered: Confidence intervals (both cases) 1 sample hypothesis tests (both cases) Hypothesis tests for 2 means as
More informationVisual interpretation with normal approximation
Visual interpretation with normal approximation H 0 is true: H 1 is true: p =0.06 25 33 Reject H 0 α =0.05 (Type I error rate) Fail to reject H 0 β =0.6468 (Type II error rate) 30 Accept H 1 Visual interpretation
More informationP-values and statistical tests 3. t-test
P-values and statistical tests 3. t-test Marek Gierliński Division of Computational Biology Hand-outs available at http://is.gd/statlec Statistical test Null hypothesis H 0 : no effect Significance level
More informationEC2001 Econometrics 1 Dr. Jose Olmo Room D309
EC2001 Econometrics 1 Dr. Jose Olmo Room D309 J.Olmo@City.ac.uk 1 Revision of Statistical Inference 1.1 Sample, observations, population A sample is a number of observations drawn from a population. Population:
More informationSampling distribution of t. 2. Sampling distribution of t. 3. Example: Gas mileage investigation. II. Inferential Statistics (8) t =
2. The distribution of t values that would be obtained if a value of t were calculated for each sample mean for all possible random of a given size from a population _ t ratio: (X - µ hyp ) t s x The result
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationHypothesis Testing One Sample Tests
STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
More information2008 Winton. Statistical Testing of RNGs
1 Statistical Testing of RNGs Criteria for Randomness For a sequence of numbers to be considered a sequence of randomly acquired numbers, it must have two basic statistical properties: Uniformly distributed
More informationMBA 605, Business Analytics Donald D. Conant, Ph.D. Master of Business Administration
t-distribution Summary MBA 605, Business Analytics Donald D. Conant, Ph.D. Types of t-tests There are several types of t-test. In this course we discuss three. The single-sample t-test The two-sample t-test
More informationReview: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.
1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately
More informationHYPOTHESIS TESTING. Hypothesis Testing
MBA 605 Business Analytics Don Conant, PhD. HYPOTHESIS TESTING Hypothesis testing involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population.
More informationCIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8
CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval
More informationProblem 1 (20) Log-normal. f(x) Cauchy
ORF 245. Rigollet Date: 11/21/2008 Problem 1 (20) f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 4 2 0 2 4 Normal (with mean -1) 4 2 0 2 4 Negative-exponential x x f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.5
More informationProblem Set 4 - Solutions
Problem Set 4 - Solutions Econ-310, Spring 004 8. a. If we wish to test the research hypothesis that the mean GHQ score for all unemployed men exceeds 10, we test: H 0 : µ 10 H a : µ > 10 This is a one-tailed
More informationPopulation Variance. Concepts from previous lectures. HUMBEHV 3HB3 one-sample t-tests. Week 8
Concepts from previous lectures HUMBEHV 3HB3 one-sample t-tests Week 8 Prof. Patrick Bennett sampling distributions - sampling error - standard error of the mean - degrees-of-freedom Null and alternative/research
More informationSlides for Data Mining by I. H. Witten and E. Frank
Slides for Data Mining by I. H. Witten and E. Frank Predicting performance Assume the estimated error rate is 5%. How close is this to the true error rate? Depends on the amount of test data Prediction
More informationINTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
More informationHypothesis Testing in Action: t-tests
Hypothesis Testing in Action: t-tests Mark Muldoon School of Mathematics, University of Manchester Mark Muldoon, January 30, 2007 t-testing - p. 1/31 Overview large Computing t for two : reprise Today
More informationThe t-test Pivots Summary. Pivots and t-tests. Patrick Breheny. October 15. Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/18
and t-tests Patrick Breheny October 15 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/18 Introduction The t-test As we discussed previously, W.S. Gossett derived the t-distribution as a way of
More information4.1 Hypothesis Testing
4.1 Hypothesis Testing z-test for a single value double-sided and single-sided z-test for one average z-test for two averages double-sided and single-sided t-test for one average the F-parameter and F-table
More informationTwo-Sample Inferential Statistics
The t Test for Two Independent Samples 1 Two-Sample Inferential Statistics In an experiment there are two or more conditions One condition is often called the control condition in which the treatment is
More informationLecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences
More informationThe Student s t Distribution
The Student s t Distribution What do we do if (a) we don t know σ and (b) n is small? If the population of interest is normally distributed, we can use the Student s t-distribution in place of the standard
More informationChapter 9. Inferences from Two Samples. Objective. Notation. Section 9.2. Definition. Notation. q = 1 p. Inferences About Two Proportions
Chapter 9 Inferences from Two Samples 9. Inferences About Two Proportions 9.3 Inferences About Two s (Independent) 9.4 Inferences About Two s (Matched Pairs) 9.5 Comparing Variation in Two Samples Objective
More informationChapter 7 Comparison of two independent samples
Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N
More informationChapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
More informationSection 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples
Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means
More information+ Specify 1 tail / 2 tail
Week 2: Null hypothesis Aeroplane seat designer wonders how wide to make the plane seats. He assumes population average hip size μ = 43.2cm Sample size n = 50 Question : Is the assumption μ = 43.2cm reasonable?
More informationHypothesis Tests and Estimation for Population Variances. Copyright 2014 Pearson Education, Inc.
Hypothesis Tests and Estimation for Population Variances 11-1 Learning Outcomes Outcome 1. Formulate and carry out hypothesis tests for a single population variance. Outcome 2. Develop and interpret confidence
More informationConfidence Intervals for the Sample Mean
Confidence Intervals for the Sample Mean As we saw before, parameter estimators are themselves random variables. If we are going to make decisions based on these uncertain estimators, we would benefit
More informationConfidence intervals
Confidence intervals We now want to take what we ve learned about sampling distributions and standard errors and construct confidence intervals. What are confidence intervals? Simply an interval for which
More informationHypothesis Testing in Action
Hypothesis Testing in Action Jonathan Bagley School of Mathematics, University of Manchester Jonathan Bagley, September 23, 2005 The t-tests - p. 1/23 Overview Today we ll examine three data sets and use
More informationChapter 23: Inferences About Means
Chapter 3: Inferences About Means Sample of Means: number of observations in one sample the population mean (theoretical mean) sample mean (observed mean) is the theoretical standard deviation of the population
More informationAdvanced Experimental Design
Advanced Experimental Design Topic Four Hypothesis testing (z and t tests) & Power Agenda Hypothesis testing Sampling distributions/central limit theorem z test (σ known) One sample z & Confidence intervals
More informationHypothesis Testing hypothesis testing approach formulation of the test statistic
Hypothesis Testing For the next few lectures, we re going to look at various test statistics that are formulated to allow us to test hypotheses in a variety of contexts: In all cases, the hypothesis testing
More informationECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12
ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12 Winter 2012 Lecture 13 (Winter 2011) Estimation Lecture 13 1 / 33 Review of Main Concepts Sampling Distribution of Sample Mean
More informationLecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 9 Inferences from Two Samples 9-1 Overview 9-2 Inferences About Two Proportions 9-3
More informationPreliminary Statistics Lecture 5: Hypothesis Testing (Outline)
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 5: Hypothesis Testing (Outline) Gujarati D. Basic Econometrics, Appendix A.8 Barrow M. Statistics
More informationChapter 12: Estimation
Chapter 12: Estimation Estimation In general terms, estimation uses a sample statistic as the basis for estimating the value of the corresponding population parameter. Although estimation and hypothesis
More informationInference About Means and Proportions with Two Populations. Chapter 10
Inference About Means and Proportions with Two Populations Chapter 10 Two Populations? Chapter 8 we found interval estimates for the population mean and population proportion based on a random sample Chapter
More informationIntroduction to Business Statistics QM 220 Chapter 12
Department of Quantitative Methods & Information Systems Introduction to Business Statistics QM 220 Chapter 12 Dr. Mohammad Zainal 12.1 The F distribution We already covered this topic in Ch. 10 QM-220,
More informationChapter 5 Confidence Intervals
Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is
More informationStudent s t-distribution. The t-distribution, t-tests, & Measures of Effect Size
Student s t-distribution The t-distribution, t-tests, & Measures of Effect Size Sampling Distributions Redux Chapter 7 opens with a return to the concept of sampling distributions from chapter 4 Sampling
More informationRelating Graph to Matlab
There are two related course documents on the web Probability and Statistics Review -should be read by people without statistics background and it is helpful as a review for those with prior statistics
More informationAcknowledge error Smaller samples, less spread
Hypothesis Testing with t Tests Al Arlo Clark-Foos kf Using Samples to Estimate Population Parameters Acknowledge error Smaller samples, less spread s = Σ ( X M N 1 ) 2 The t Statistic Indicates the distance
More informationTables Table A Table B Table C Table D Table E 675
BMTables.indd Page 675 11/15/11 4:25:16 PM user-s163 Tables Table A Standard Normal Probabilities Table B Random Digits Table C t Distribution Critical Values Table D Chi-square Distribution Critical Values
More informationChapter 24. Comparing Means. Copyright 2010 Pearson Education, Inc.
Chapter 24 Comparing Means Copyright 2010 Pearson Education, Inc. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationQuestions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.
Chapter 7 Reading 7.1, 7.2 Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.112 Introduction In Chapter 5 and 6, we emphasized
More informationPurposes of Data Analysis. Variables and Samples. Parameters and Statistics. Part 1: Probability Distributions
Part 1: Probability Distributions Purposes of Data Analysis True Distributions or Relationships in the Earths System Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Motivations for the ANOVA We defined the F-distribution, this is mainly used in
More informationDistribution-Free Procedures (Devore Chapter Fifteen)
Distribution-Free Procedures (Devore Chapter Fifteen) MATH-5-01: Probability and Statistics II Spring 018 Contents 1 Nonparametric Hypothesis Tests 1 1.1 The Wilcoxon Rank Sum Test........... 1 1. Normal
More informationHypothesis Testing. ECE 3530 Spring Antonio Paiva
Hypothesis Testing ECE 3530 Spring 2010 Antonio Paiva What is hypothesis testing? A statistical hypothesis is an assertion or conjecture concerning one or more populations. To prove that a hypothesis is
More informationChapter 23. Inference About Means
Chapter 23 Inference About Means 1 /57 Homework p554 2, 4, 9, 10, 13, 15, 17, 33, 34 2 /57 Objective Students test null and alternate hypotheses about a population mean. 3 /57 Here We Go Again Now that
More informationHow do we compare the relative performance among competing models?
How do we compare the relative performance among competing models? 1 Comparing Data Mining Methods Frequent problem: we want to know which of the two learning techniques is better How to reliably say Model
More informationClassroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats
Classroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats Materials Needed: Bags of popcorn, watch with second hand or microwave with digital timer. Instructions: Follow the instructions on the
More informationz and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests
z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests Chapters 3.5.1 3.5.2, 3.3.2 Prof. Tesler Math 283 Fall 2018 Prof. Tesler z and t tests for mean Math
More informationInferential statistics
Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,
More information1 Binomial Probability [15 points]
Economics 250 Assignment 2 (Due November 13, 2017, in class) i) You should do the assignment on your own, Not group work! ii) Submit the completed work in class on the due date. iii) Remember to include
More informationChapter 9 Inferences from Two Samples
Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review
More informationLecture 9 Two-Sample Test. Fall 2013 Prof. Yao Xie, H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech
Lecture 9 Two-Sample Test Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech Computer exam 1 18 Histogram 14 Frequency 9 5 0 75 83.33333333
More informationTopic 22 Analysis of Variance
Topic 22 Analysis of Variance Comparing Multiple Populations 1 / 14 Outline Overview One Way Analysis of Variance Sample Means Sums of Squares The F Statistic Confidence Intervals 2 / 14 Overview Two-sample
More informationCorrelation Analysis
Simple Regression Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
More informationDesign of Engineering Experiments Part 2 Basic Statistical Concepts Simple comparative experiments
Design of Engineering Experiments Part 2 Basic Statistical Concepts Simple comparative experiments The hypothesis testing framework The two-sample t-test Checking assumptions, validity Comparing more that
More informationNon-parametric Inference and Resampling
Non-parametric Inference and Resampling Exercises by David Wozabal (Last update. Juni 010) 1 Basic Facts about Rank and Order Statistics 1.1 10 students were asked about the amount of time they spend surfing
More informationAn inferential procedure to use sample data to understand a population Procedures
Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 7 Inferences Based on Two Samples: Confidence Intervals & Tests of Hypotheses Content 1. Identifying the Target Parameter 2. Comparing Two Population Means:
More informationME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV
Theory of Engineering Experimentation Chapter IV. Decision Making for a Single Sample Chapter IV 1 4 1 Statistical Inference The field of statistical inference consists of those methods used to make decisions
More informationBiostatistics. Chapter 11 Simple Linear Correlation and Regression. Jing Li
Biostatistics Chapter 11 Simple Linear Correlation and Regression Jing Li jing.li@sjtu.edu.cn http://cbb.sjtu.edu.cn/~jingli/courses/2018fall/bi372/ Dept of Bioinformatics & Biostatistics, SJTU Review
More informationDr. Maddah ENMG 617 EM Statistics 10/12/12. Nonparametric Statistics (Chapter 16, Hines)
Dr. Maddah ENMG 617 EM Statistics 10/12/12 Nonparametric Statistics (Chapter 16, Hines) Introduction Most of the hypothesis testing presented so far assumes normally distributed data. These approaches
More informationStatistical Quality Design & Control Fall 2003 Odette School of Business University of Windsor
Name (print, please) ID Statistical Quality Design & Control 7-5 Fall Odette School of Business University of Windsor Midterm Exam Solution Wednesday, October 15, 5: 6:5 pm Instructor: Mohammed Fazle Baki
More informationChap The McGraw-Hill Companies, Inc. All rights reserved.
11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview
More informationTwo Sample Hypothesis Tests
Note Packet #21 Two Sample Hypothesis Tests CEE 3710 November 13, 2017 Review Possible states of nature: H o and H a (Null vs. Alternative Hypothesis) Possible decisions: accept or reject Ho (rejecting
More informationConfidence Intervals with σ unknown
STAT 141 Confidence Intervals and Hypothesis Testing 10/26/04 Today (Chapter 7): CI with σ unknown, t-distribution CI for proportions Two sample CI with σ known or unknown Hypothesis Testing, z-test Confidence
More informationLAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2
LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept
More informationStats Review Chapter 14. Mary Stangler Center for Academic Success Revised 8/16
Stats Review Chapter 14 Revised 8/16 Note: This review is meant to highlight basic concepts from the course. It does not cover all concepts presented by your instructor. Refer back to your notes, unit
More informationKeller: Stats for Mgmt & Econ, 7th Ed July 17, 2006
Chapter 17 Simple Linear Regression and Correlation 17.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More informationSimple Linear Regression
Simple Linear Regression ST 370 Regression models are used to study the relationship of a response variable and one or more predictors. The response is also called the dependent variable, and the predictors
More informationEconometrics. 4) Statistical inference
30C00200 Econometrics 4) Statistical inference Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Confidence intervals of parameter estimates Student s t-distribution
More informationAre data normally normally distributed?
Standard Normal Image source Are data normally normally distributed? Sample mean: 66.78 Sample standard deviation: 3.37 (66.78-1 x 3.37, 66.78 + 1 x 3.37) (66.78-2 x 3.37, 66.78 + 2 x 3.37) (66.78-3 x
More informationTest 3 Practice Test A. NOTE: Ignore Q10 (not covered)
Test 3 Practice Test A NOTE: Ignore Q10 (not covered) MA 180/418 Midterm Test 3, Version A Fall 2010 Student Name (PRINT):............................................. Student Signature:...................................................
More informationHYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă
HYPOTHESIS TESTING II TESTS ON MEANS Sorana D. Bolboacă OBJECTIVES Significance value vs p value Parametric vs non parametric tests Tests on means: 1 Dec 14 2 SIGNIFICANCE LEVEL VS. p VALUE Materials and
More informationLast two weeks: Sample, population and sampling distributions finished with estimation & confidence intervals
Past weeks: Measures of central tendency (mean, mode, median) Measures of dispersion (standard deviation, variance, range, etc). Working with the normal curve Last two weeks: Sample, population and sampling
More informationSolutions to Practice Test 2 Math 4753 Summer 2005
Solutions to Practice Test Math 4753 Summer 005 This test is worth 00 points. Questions 5 are worth 4 points each. Circle the letter of the correct answer. Each question in Question 6 9 is worth the same
More informationCHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities:
CHAPTER 9, 10 Hypothesis Testing Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: The person is guilty. The person is innocent. To
More informationPLSC PRACTICE TEST ONE
PLSC 724 - PRACTICE TEST ONE 1. Discuss briefly the relationship between the shape of the normal curve and the variance. 2. What is the relationship between a statistic and a parameter? 3. How is the α
More informationStatistical inference (estimation, hypothesis tests, confidence intervals) Oct 2018
Statistical inference (estimation, hypothesis tests, confidence intervals) Oct 2018 Sampling A trait is measured on each member of a population. f(y) = propn of individuals in the popn with measurement
More informationHypothesis tests for two means
Chapter 3 Hypothesis tests for two means 3.1 Introduction Last week you were introduced to the concept of hypothesis testing in statistics, and we considered hypothesis tests for the mean if we have a
More informationPOLI 443 Applied Political Research
POLI 443 Applied Political Research Session 4 Tests of Hypotheses The Normal Curve Lecturer: Prof. A. Essuman-Johnson, Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College
More information23. MORE HYPOTHESIS TESTING
23. MORE HYPOTHESIS TESTING The Logic Behind Hypothesis Testing For simplicity, consider testing H 0 : µ = µ 0 against the two-sided alternative H A : µ µ 0. Even if H 0 is true (so that the expectation
More informationLecture 30. DATA 8 Summer Regression Inference
DATA 8 Summer 2018 Lecture 30 Regression Inference Slides created by John DeNero (denero@berkeley.edu) and Ani Adhikari (adhikari@berkeley.edu) Contributions by Fahad Kamran (fhdkmrn@berkeley.edu) and
More informationHypothesis Testing. Mean (SDM)
Confidence Intervals and Hypothesis Testing Readings: Howell, Ch. 4, 7 The Sampling Distribution of the Mean (SDM) Derivation - See Thorne & Giesen (T&G), pp. 169-171 or online Chapter Overview for Ch.
More informationInference for Regression Inference about the Regression Model and Using the Regression Line
Inference for Regression Inference about the Regression Model and Using the Regression Line PBS Chapter 10.1 and 10.2 2009 W.H. Freeman and Company Objectives (PBS Chapter 10.1 and 10.2) Inference about
More informationMidterm Examination. Mth 136 = Sta 114. Wednesday, 2000 March 8, 2:20 3:35 pm
Midterm Examination Mth 136 = Sta 114 Wednesday, 2000 March 8, 2:20 3:35 pm This is a closed-book examination so please do not refer to your notes, the text, or to any other books. You may use a two-sided
More informationFinal Exam - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis March 19, 2010 Instructor: John Parman Final Exam - Solutions You have until 5:30pm to complete this exam. Please remember to put your
More informationChapter 10: Analysis of variance (ANOVA)
Chapter 10: Analysis of variance (ANOVA) ANOVA (Analysis of variance) is a collection of techniques for dealing with more general experiments than the previous one-sample or two-sample tests. We first
More information