Confidence Intervals and Hypothesis Tests

Size: px
Start display at page:

Download "Confidence Intervals and Hypothesis Tests"

Transcription

1 Confidence Intervals and Hypothesis Tests STA 281 Fall Background The central limit theorem provides a very powerful tool for determining the distribution of sample means for large sample sizes. In particular, if X 1,, Xn are independent and identically distributed (iid) with mean E[Xi]=µ and variance V[Xi]=σ 2 AND n is large, then ( ) For the remainder of this handout, all sample sizes should be assumed to be greater than 30 so this result holds. A special case of this result is that if X 1,, Xn Bern(p), then ( ) Results for two samples may be obtained using the formulas for linear combinations of normal distributions. Thus, if are iid with mean and variance while are iid with mean and variance (and of course the X and Y samples are independent), then ( ) This also has a special case for proportions. If and, then ( ) These formulas are the four fundamental results that motivate all of the confidence interval and hypothesis testing theory we will investigate in this course. 2 What are Confidence Intervals and Hypothesis Tests? Inference is the use of data to draw conclusions about population parameters. Probability theory assumes we have X 1,, Xn Bern(0.4) and then specifies the likelihood of generating 0 through n successes. Thus probability theory assumes we know the parameter p and specifies how our data should appear. Inference is concerned about the reverse problem. We already have X 1,, Xn Bern(p), but we don t know p. Our goal is to use the data to determine p. The first thing to note is that we will NEVER be able to determine p exactly with only a finite amount of data. Suppose n=1000 and we observe that X 1,, Xn Bern(p) has 800 successes. What is p? Unfortunately, no value of p in (0,1) can be completely excluded based on this data. It is possible to see the observed data when p=0.01 (not likely, but possible). For any value of p in (0,1), the observed value is possible. Thus, we are forced to make probabilistic statements about p. Intuitively, while p=0.01 cannot be excluded, our observed data (800 successes in 1000 trials) is so unlikely when p=0.01 that for all practical purposes we can exclude p=0.01. These are the kind of inferences we will pursue.

2 We will focus on making two kinds of inferences, confidence intervals and hypothesis tests in several scenarios. Not coincidentally, these scenarios correspond to the situations where we applied the central limit theorem in section 1. Specifically, we will make inferences on means for one or two samples, and on proportions for one or two samples. The two kinds of inferences correspond to two common questions asked in scientific experiments. The first, confidence intervals, answers the question I have no idea what µ (p) is, how do I use the data to estimate it? The second, hypothesis tests, answers the question I have a specific value of µ (p) in mind. Is the data consistent with that particular value of µ (p)? 3 Point Estimates (our best guess) Fundamental to answering both these questions is the notion of a point estimate. A point estimate takes the observed data and produces a single value (or guess) of the parameter. Returning to our example where we had 1000 Bernoulli trials and observed 800 successes, we have already established we are not pleased with p=0.01. If we had to guess a single number, what would we guess? The most common choice is, which in this example is 800/1000=0.8. This guess is justified by the central limit theorem, which states that the expected value of is p. While may not be equal to p in any particular situation, has a distribution that is centered around the true value. Thus, if we are estimating a proportion p, we estimate it with. For a mean µ, use. These extend to the two sample case, so the difference of two proportions is estimated by and the difference of two means is estimated by. Not coincidentally, the center of the distributions of all these guesses is the quantity we are trying to guess. 4 Confidence Intervals Our best guess is a good start for inference, but it isn t ideal. Specifically, our best guess is basically guaranteed to be wrong. If X 1,, Xn N(0,1), then N(0,1), which is a continuous distribution. Although the distribution of is centered at µ=0, the probability that will exactly equal 0 is 0. OK, that doesn t sound great, but it s not terrible. While might not be exactly correct, its key advantage is that it should be close to µ, and the larger the sample size, the closer to µ it should be (this can be observed by noting the variance of, σ 2 /n, tends to 0 as n increases). In fact, the central limit theorem allows us to quantify just how close our point estimate should be to the correct answers. In general, a confidence interval is Thus, for each situation, the only thing to do is find the best guess, and then use the central limit theorem to compute the standard deviation of that best guess.

3 4.1 Formulas Single Proportion We have X 1,, Xn Bern(p). The best guess of p is. Looking at the central limit theorem result, the variance of is p(1-p)/n. This is an obvious difficulty, since p is unknown (it is what we are trying to estimate!). However, it turns out that is a sufficiently good guess of p that we can replace p with in the variance, resulting in the confidence interval Single Means We have X 1,, Xn iid with mean µ=e[xi] and variance σ 2 =V[Xi]. The best guess of µ is, which has variance σ 2 /n, resulting in the interval If σ 2 is unknown, then it must be estimated from the data. It turns out that s 2, defined as [ ] is a reasonable guess of σ 2, and thus should be used in place of σ 2 when necessary Difference between two proportions We have and, and are interested in estimating. The best guess of is. The variance of this best guess depends on the unknown quantities px and py, but as with a single proportion these can be replaced with and in the variance, resulting in the interval Difference between two means We have iid with mean µx and variance, and iid with mean µy and variance, and are interested in estimating. The best guess of is. Using the central limit theorem to find the variance of this best guess, we find the confidence interval is As with estimating a single mean, replace with and with as necessary.

4 5 Hypothesis Tests When we have a specific value of the parameter in mind and want to verify whether that parameter value is reasonable for the data, we use a hypothesis test. The specific value of the parameter we have in mind is recorded in the null hypothesis, H0, which might state p=0.2, or µ=5, or =3. The point is that a specific value of the parameter is chosen. A hypothesis test is conducted by observing the difference between our best guess of the parameter and the null value (the value specified in the null hypothesis). This difference must then be standardized. The standardization consists of finding the standard deviation of the best guess under the assumption H0 is true. This results in the test statistic The test statistic merely measures how many standard deviations the best guess is from the null value. If the best guess is too far away, the null hypothesis is rejected, otherwise the null hypothesis is accepted. Too far away in this context is determined by. We reject H0 if or if. Otherwise we do not reject H0. Note that when H0 is true, we have constructed a procedure that rejects H0 with probability α. Thus, we can control the probability of falsely rejecting H Formulas Single Proportion Suppose we have X 1,, Xn Bern(p) and are testing H0: p=p0. Our best guess of p is. When H0 is true,, thus the test statistic is Single Mean Suppose we have X 1,, Xn iid with mean µ and variance σ 2, and we are testing H0: µ=µ0. The best guess of µ is, which under the null hypothesis is distributed N(µ=µ0, σ 2 /n). Thus the test statistic is As with confidence intervals, replace σ 2 with s 2 if the variance is unknown.

5 5.1.3 Difference between two proportions We have and, and are interested in testing H0: =0. There is one trick to this. We have to compute the standard deviation of under the assumption that =0. We cannot just plug in and into the usual variance formula, because it may not be true that =0. We use where Difference between two means We have iid with mean µx and variance, and iid with mean µy and variance, and are interested in testing H0: =d0 against H1: d0. The most common instance of this occurs when d0=0, when the null hypothesis simplifies to H0:. Our best guess of is, and thus the test statistic is where and should be replaced with and as necessary.

EC2001 Econometrics 1 Dr. Jose Olmo Room D309

EC2001 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 information

A Primer on Statistical Inference using Maximum Likelihood

A Primer on Statistical Inference using Maximum Likelihood A Primer on Statistical Inference using Maximum Likelihood November 3, 2017 1 Inference via Maximum Likelihood Statistical inference is the process of using observed data to estimate features of the population.

More information

Computational Perception. Bayesian Inference

Computational Perception. Bayesian Inference Computational Perception 15-485/785 January 24, 2008 Bayesian Inference The process of probabilistic inference 1. define model of problem 2. derive posterior distributions and estimators 3. estimate parameters

More information

Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing

Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Agenda Introduction to Estimation Point estimation Interval estimation Introduction to Hypothesis Testing Concepts en terminology

More information

Nonparametric hypothesis tests and permutation tests

Nonparametric hypothesis tests and permutation tests Nonparametric hypothesis tests and permutation tests 1.7 & 2.3. Probability Generating Functions 3.8.3. Wilcoxon Signed Rank Test 3.8.2. Mann-Whitney Test Prof. Tesler Math 283 Fall 2018 Prof. Tesler Wilcoxon

More information

Introductory Econometrics. Review of statistics (Part II: Inference)

Introductory Econometrics. Review of statistics (Part II: Inference) Introductory Econometrics Review of statistics (Part II: Inference) Jun Ma School of Economics Renmin University of China October 1, 2018 1/16 Null and alternative hypotheses Usually, we have two competing

More information

Introduction 1. STA442/2101 Fall See last slide for copyright information. 1 / 33

Introduction 1. STA442/2101 Fall See last slide for copyright information. 1 / 33 Introduction 1 STA442/2101 Fall 2016 1 See last slide for copyright information. 1 / 33 Background Reading Optional Chapter 1 of Linear models with R Chapter 1 of Davison s Statistical models: Data, and

More information

STAT 135 Lab 5 Bootstrapping and Hypothesis Testing

STAT 135 Lab 5 Bootstrapping and Hypothesis Testing STAT 135 Lab 5 Bootstrapping and Hypothesis Testing Rebecca Barter March 2, 2015 The Bootstrap Bootstrap Suppose that we are interested in estimating a parameter θ from some population with members x 1,...,

More information

One-sample categorical data: approximate inference

One-sample categorical data: approximate inference One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution

More information

Conditional probabilities and graphical models

Conditional probabilities and graphical models Conditional probabilities and graphical models Thomas Mailund Bioinformatics Research Centre (BiRC), Aarhus University Probability theory allows us to describe uncertainty in the processes we model within

More information

Gov Univariate Inference II: Interval Estimation and Testing

Gov Univariate Inference II: Interval Estimation and Testing Gov 2000-5. Univariate Inference II: Interval Estimation and Testing Matthew Blackwell October 13, 2015 1 / 68 Large Sample Confidence Intervals Confidence Intervals Example Hypothesis Tests Hypothesis

More information

Topic 15: Simple Hypotheses

Topic 15: Simple Hypotheses Topic 15: November 10, 2009 In the simplest set-up for a statistical hypothesis, we consider two values θ 0, θ 1 in the parameter space. We write the test as H 0 : θ = θ 0 versus H 1 : θ = θ 1. H 0 is

More information

Harvard University. Rigorous Research in Engineering Education

Harvard University. Rigorous Research in Engineering Education Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Let s motivate our discussion by considering an example first. What happens when we add the first n positive odd integers? The table below shows what results for the first few values

More information

Summary: the confidence interval for the mean (σ 2 known) with gaussian assumption

Summary: the confidence interval for the mean (σ 2 known) with gaussian assumption Summary: the confidence interval for the mean (σ known) with gaussian assumption on X Let X be a Gaussian r.v. with mean µ and variance σ. If X 1, X,..., X n is a random sample drawn from X then the confidence

More information

Data Analysis and Statistical Methods Statistics 651

Data 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 information

Introduction: MLE, MAP, Bayesian reasoning (28/8/13)

Introduction: MLE, MAP, Bayesian reasoning (28/8/13) STA561: Probabilistic machine learning Introduction: MLE, MAP, Bayesian reasoning (28/8/13) Lecturer: Barbara Engelhardt Scribes: K. Ulrich, J. Subramanian, N. Raval, J. O Hollaren 1 Classifiers In this

More information

Introductory Econometrics

Introductory Econometrics Session 4 - Testing hypotheses Roland Sciences Po July 2011 Motivation After estimation, delivering information involves testing hypotheses Did this drug had any effect on the survival rate? Is this drug

More information

(a) (3 points) Construct a 95% confidence interval for β 2 in Equation 1.

(a) (3 points) Construct a 95% confidence interval for β 2 in Equation 1. Problem 1 (21 points) An economist runs the regression y i = β 0 + x 1i β 1 + x 2i β 2 + x 3i β 3 + ε i (1) The results are summarized in the following table: Equation 1. Variable Coefficient Std. Error

More information

Introduction to Bayesian Learning. Machine Learning Fall 2018

Introduction to Bayesian Learning. Machine Learning Fall 2018 Introduction to Bayesian Learning Machine Learning Fall 2018 1 What we have seen so far What does it mean to learn? Mistake-driven learning Learning by counting (and bounding) number of mistakes PAC learnability

More information

One important way that you can classify differential equations is as linear or nonlinear.

One important way that you can classify differential equations is as linear or nonlinear. In This Chapter Chapter 1 Looking Closely at Linear First Order Differential Equations Knowing what a first order linear differential equation looks like Finding solutions to first order differential equations

More information

ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12

ECO220Y 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 information

Loglikelihood and Confidence Intervals

Loglikelihood and Confidence Intervals Stat 504, Lecture 2 1 Loglikelihood and Confidence Intervals The loglikelihood function is defined to be the natural logarithm of the likelihood function, l(θ ; x) = log L(θ ; x). For a variety of reasons,

More information

appstats27.notebook April 06, 2017

appstats27.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 information

Induction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2

Induction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2 Induction 0-8-08 Induction is used to prove a sequence of statements P(), P(), P(3),... There may be finitely many statements, but often there are infinitely many. For example, consider the statement ++3+

More information

An Introduction to Laws of Large Numbers

An Introduction to Laws of Large Numbers An to Laws of John CVGMI Group Contents 1 Contents 1 2 Contents 1 2 3 Contents 1 2 3 4 Intuition We re working with random variables. What could we observe? {X n } n=1 Intuition We re working with random

More information

(1) Introduction to Bayesian statistics

(1) Introduction to Bayesian statistics Spring, 2018 A motivating example Student 1 will write down a number and then flip a coin If the flip is heads, they will honestly tell student 2 if the number is even or odd If the flip is tails, they

More information

COMP2610/COMP Information Theory

COMP2610/COMP Information Theory COMP2610/COMP6261 - Information Theory Lecture 9: Probabilistic Inequalities Mark Reid and Aditya Menon Research School of Computer Science The Australian National University August 19th, 2014 Mark Reid

More information

Statistical Inference. Hypothesis Testing

Statistical Inference. Hypothesis Testing Statistical Inference Hypothesis Testing Previously, we introduced the point and interval estimation of an unknown parameter(s), say µ and σ 2. However, in practice, the problem confronting the scientist

More information

Hypothesis testing (cont d)

Hypothesis testing (cont d) Hypothesis testing (cont d) Ulrich Heintz Brown University 4/12/2016 Ulrich Heintz - PHYS 1560 Lecture 11 1 Hypothesis testing Is our hypothesis about the fundamental physics correct? We will not be able

More information

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Lecture No. # 36 Sampling Distribution and Parameter Estimation

More information

Chapter 26: Comparing Counts (Chi Square)

Chapter 26: Comparing Counts (Chi Square) Chapter 6: Comparing Counts (Chi Square) We ve seen that you can turn a qualitative variable into a quantitative one (by counting the number of successes and failures), but that s a compromise it forces

More information

Lectures 5 & 6: Hypothesis Testing

Lectures 5 & 6: Hypothesis Testing Lectures 5 & 6: Hypothesis Testing in which you learn to apply the concept of statistical significance to OLS estimates, learn the concept of t values, how to use them in regression work and come across

More information

Lecture 30. DATA 8 Summer Regression Inference

Lecture 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 information

Lecture 4: September Reminder: convergence of sequences

Lecture 4: September Reminder: convergence of sequences 36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 4: September 6 In this lecture we discuss the convergence of random variables. At a high-level, our first few lectures focused

More information

Stat 206: Estimation and testing for a mean vector,

Stat 206: Estimation and testing for a mean vector, Stat 206: Estimation and testing for a mean vector, Part II James Johndrow 2016-12-03 Comparing components of the mean vector In the last part, we talked about testing the hypothesis H 0 : µ 1 = µ 2 where

More information

Basic Probability Reference Sheet

Basic Probability Reference Sheet February 27, 2001 Basic Probability Reference Sheet 17.846, 2001 This is intended to be used in addition to, not as a substitute for, a textbook. X is a random variable. This means that X is a variable

More information

2008 Winton. Statistical Testing of RNGs

2008 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 information

Topic 3: Hypothesis Testing

Topic 3: Hypothesis Testing CS 8850: Advanced Machine Learning Fall 07 Topic 3: Hypothesis Testing Instructor: Daniel L. Pimentel-Alarcón c Copyright 07 3. Introduction One of the simplest inference problems is that of deciding between

More information

1 Hypothesis testing for a single mean

1 Hypothesis testing for a single mean This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

Scribe to lecture Tuesday March

Scribe to lecture Tuesday March Scribe to lecture Tuesday March 16 2004 Scribe outlines: Message Confidence intervals Central limit theorem Em-algorithm Bayesian versus classical statistic Note: There is no scribe for the beginning of

More information

ECO375 Tutorial 4 Introduction to Statistical Inference

ECO375 Tutorial 4 Introduction to Statistical Inference ECO375 Tutorial 4 Introduction to Statistical Inference Matt Tudball University of Toronto Mississauga October 19, 2017 Matt Tudball (University of Toronto) ECO375H5 October 19, 2017 1 / 26 Statistical

More information

Section 3.1: Direct Proof and Counterexample 1

Section 3.1: Direct Proof and Counterexample 1 Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion

More information

Confidence Intervals

Confidence Intervals Quantitative Foundations Project 3 Instructor: Linwei Wang Confidence Intervals Contents 1 Introduction 3 1.1 Warning....................................... 3 1.2 Goals of Statistics..................................

More information

Bayesian Inference. STA 121: Regression Analysis Artin Armagan

Bayesian Inference. STA 121: Regression Analysis Artin Armagan Bayesian Inference STA 121: Regression Analysis Artin Armagan Bayes Rule...s! Reverend Thomas Bayes Posterior Prior p(θ y) = p(y θ)p(θ)/p(y) Likelihood - Sampling Distribution Normalizing Constant: p(y

More information

STATS 200: Introduction to Statistical Inference. Lecture 29: Course review

STATS 200: Introduction to Statistical Inference. Lecture 29: Course review STATS 200: Introduction to Statistical Inference Lecture 29: Course review Course review We started in Lecture 1 with a fundamental assumption: Data is a realization of a random process. The goal throughout

More information

COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017

COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017 COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University FEATURE EXPANSIONS FEATURE EXPANSIONS

More information

Hypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =

Hypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n = Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,

More information

Summer HSSP Lecture Notes Week 1. Lane Gunderman, Victor Lopez, James Rowan

Summer HSSP Lecture Notes Week 1. Lane Gunderman, Victor Lopez, James Rowan Summer HSSP Lecture Notes Week 1 Lane Gunderman, Victor Lopez, James Rowan July 6, 014 First Class: proofs and friends 1 Contents 1 Glossary of symbols 4 Types of numbers 5.1 Breaking it down...........................

More information

A sequential hypothesis test based on a generalized Azuma inequality 1

A sequential hypothesis test based on a generalized Azuma inequality 1 A sequential hypothesis test based on a generalized Azuma inequality 1 Daniël Reijsbergen a,2, Werner Scheinhardt b, Pieter-Tjerk de Boer b a Laboratory for Foundations of Computer Science, University

More information

The paradox of knowability, the knower, and the believer

The paradox of knowability, the knower, and the believer The paradox of knowability, the knower, and the believer Last time, when discussing the surprise exam paradox, we discussed the possibility that some claims could be true, but not knowable by certain individuals

More information

Relating Graph to Matlab

Relating 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 information

STA121: Applied Regression Analysis

STA121: Applied Regression Analysis STA121: Applied Regression Analysis Linear Regression Analysis - Chapters 3 and 4 in Dielman Artin Department of Statistical Science September 15, 2009 Outline 1 Simple Linear Regression Analysis 2 Using

More information

The t-distribution. Patrick Breheny. October 13. z tests The χ 2 -distribution The t-distribution Summary

The t-distribution. Patrick Breheny. October 13. z tests The χ 2 -distribution The t-distribution Summary Patrick Breheny October 13 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction Introduction What s wrong with z-tests? So far we ve (thoroughly!) discussed how to carry out hypothesis

More information

Mathematical Statistics

Mathematical Statistics Mathematical Statistics MAS 713 Chapter 8 Previous lecture: 1 Bayesian Inference 2 Decision theory 3 Bayesian Vs. Frequentist 4 Loss functions 5 Conjugate priors Any questions? Mathematical Statistics

More information

Math 475, Problem Set #8: Answers

Math 475, Problem Set #8: Answers Math 475, Problem Set #8: Answers A. Brualdi, problem, parts (a), (b), and (d). (a): As n goes from to 6, the sum (call it h n ) takes on the values,, 8, 2, 55, and 44; we recognize these as Fibonacci

More information

Physics 403. Segev BenZvi. Credible Intervals, Confidence Intervals, and Limits. Department of Physics and Astronomy University of Rochester

Physics 403. Segev BenZvi. Credible Intervals, Confidence Intervals, and Limits. Department of Physics and Astronomy University of Rochester Physics 403 Credible Intervals, Confidence Intervals, and Limits Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Summarizing Parameters with a Range Bayesian

More information

Discrete Distributions

Discrete Distributions Discrete Distributions STA 281 Fall 2011 1 Introduction Previously we defined a random variable to be an experiment with numerical outcomes. Often different random variables are related in that they have

More information

STA Module 10 Comparing Two Proportions

STA Module 10 Comparing Two Proportions STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare

More information

Introduction to Econometrics

Introduction to Econometrics Introduction to Econometrics Lecture 2 : Causal Inference and Random Control Trails(RCT) Zhaopeng Qu Business School,Nanjing University Sep 18th, 2017 Zhaopeng Qu (Nanjing University) Introduction to Econometrics

More information

Simple Linear Regression for the Climate Data

Simple Linear Regression for the Climate Data Prediction Prediction Interval Temperature 0.2 0.0 0.2 0.4 0.6 0.8 320 340 360 380 CO 2 Simple Linear Regression for the Climate Data What do we do with the data? y i = Temperature of i th Year x i =CO

More information

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Statistical Inference. Why Use Statistical Inference. Point Estimates. Point Estimates. Greg C Elvers

Statistical Inference. Why Use Statistical Inference. Point Estimates. Point Estimates. Greg C Elvers Statistical Inference Greg C Elvers 1 Why Use Statistical Inference Whenever we collect data, we want our results to be true for the entire population and not just the sample that we used But our sample

More information

Lecture 11 - Tests of Proportions

Lecture 11 - Tests of Proportions Lecture 11 - Tests of Proportions Statistics 102 Colin Rundel February 27, 2013 Research Project Research Project Proposal - Due Friday March 29th at 5 pm Introduction, Data Plan Data Project - Due Friday,

More information

Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman.

Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman. Math 224 Fall 2017 Homework 1 Drew Armstrong Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman. Section 1.1, Exercises 4,5,6,7,9,12. Solutions to Book Problems.

More information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests

Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests 1999 Prentice-Hall, Inc. Chap. 8-1 Chapter Topics Hypothesis Testing Methodology Z Test

More information

Hypothesis Testing with Z and T

Hypothesis Testing with Z and T Chapter Eight Hypothesis Testing with Z and T Introduction to Hypothesis Testing P Values Critical Values Within-Participants Designs Between-Participants Designs Hypothesis Testing An alternate hypothesis

More information

Inference in Regression Analysis

Inference in Regression Analysis Inference in Regression Analysis Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 4, Slide 1 Today: Normal Error Regression Model Y i = β 0 + β 1 X i + ǫ i Y i value

More information

Institute of Actuaries of India

Institute of Actuaries of India Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the

More information

Preliminary Statistics. Lecture 5: Hypothesis Testing

Preliminary Statistics. Lecture 5: Hypothesis Testing Preliminary Statistics Lecture 5: Hypothesis Testing Rory Macqueen (rm43@soas.ac.uk), September 2015 Outline Elements/Terminology of Hypothesis Testing Types of Errors Procedure of Testing Significance

More information

14.30 Introduction to Statistical Methods in Economics Spring 2009

14.30 Introduction to Statistical Methods in Economics Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Statistical inference

Statistical inference Statistical inference Contents 1. Main definitions 2. Estimation 3. Testing L. Trapani MSc Induction - Statistical inference 1 1 Introduction: definition and preliminary theory In this chapter, we shall

More information

Midterm, Fall 2003

Midterm, Fall 2003 5-78 Midterm, Fall 2003 YOUR ANDREW USERID IN CAPITAL LETTERS: YOUR NAME: There are 9 questions. The ninth may be more time-consuming and is worth only three points, so do not attempt 9 unless you are

More information

Chapter 24. Comparing Means

Chapter 24. Comparing Means Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing

More information

MS&E 226: Small Data

MS&E 226: Small Data MS&E 226: Small Data Lecture 15: Examples of hypothesis tests (v5) Ramesh Johari ramesh.johari@stanford.edu 1 / 32 The recipe 2 / 32 The hypothesis testing recipe In this lecture we repeatedly apply the

More information

STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is

STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is EQUATIONS OF LINES 1. Writing Equations of Lines There are many ways to define a line, but for today, let s think of a LINE as a collection of points such that the slope between any two of those points

More information

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.

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. 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 information

A FLOW DIAGRAM FOR CALCULATING LIMITS OF FUNCTIONS (OF SEVERAL VARIABLES).

A FLOW DIAGRAM FOR CALCULATING LIMITS OF FUNCTIONS (OF SEVERAL VARIABLES). A FLOW DIAGRAM FOR CALCULATING LIMITS OF FUNCTIONS (OF SEVERAL VARIABLES). Version 5.5, 2/12/2008 In many ways it is silly to try to describe a sophisticated intellectual activity by a simple and childish

More information

HST.582J / 6.555J / J Biomedical Signal and Image Processing Spring 2007

HST.582J / 6.555J / J Biomedical Signal and Image Processing Spring 2007 MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Chapter Three. Hypothesis Testing

Chapter Three. Hypothesis Testing 3.1 Introduction The final phase of analyzing data is to make a decision concerning a set of choices or options. Should I invest in stocks or bonds? Should a new product be marketed? Are my products being

More information

Chapter 27 Summary Inferences for Regression

Chapter 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 information

ME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV

ME3620. 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 information

The problem of base rates

The problem of base rates Psychology 205: Research Methods in Psychology William Revelle Department of Psychology Northwestern University Evanston, Illinois USA October, 2015 1 / 14 Outline Inferential statistics 2 / 14 Hypothesis

More information

Statistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018

Statistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 Statistics Boot Camp Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 March 21, 2018 Outline of boot camp Summarizing and simplifying data Point and interval estimation Foundations of statistical

More information

Statistics for IT Managers

Statistics for IT Managers Statistics for IT Managers 95-796, Fall 2012 Module 2: Hypothesis Testing and Statistical Inference (5 lectures) Reading: Statistics for Business and Economics, Ch. 5-7 Confidence intervals Given the sample

More information

Section 1.x: The Variety of Asymptotic Experiences

Section 1.x: The Variety of Asymptotic Experiences calculus sin frontera Section.x: The Variety of Asymptotic Experiences We talked in class about the function y = /x when x is large. Whether you do it with a table x-value y = /x 0 0. 00.0 000.00 or with

More information

Conceptual Explanations: Simultaneous Equations Distance, rate, and time

Conceptual Explanations: Simultaneous Equations Distance, rate, and time Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.

More information

Lecture 10: Powers of Matrices, Difference Equations

Lecture 10: Powers of Matrices, Difference Equations Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each

More information

ECON Introductory Econometrics. Lecture 2: Review of Statistics

ECON Introductory Econometrics. Lecture 2: Review of Statistics ECON415 - Introductory Econometrics Lecture 2: Review of Statistics Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 2-3 Lecture outline 2 Simple random sampling Distribution of the sample

More information

The Components of a Statistical Hypothesis Testing Problem

The Components of a Statistical Hypothesis Testing Problem Statistical Inference: Recall from chapter 5 that statistical inference is the use of a subset of a population (the sample) to draw conclusions about the entire population. In chapter 5 we studied one

More information

Slides for Data Mining by I. H. Witten and E. Frank

Slides 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 information

Performance Evaluation and Comparison

Performance Evaluation and Comparison Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Cross Validation and Resampling 3 Interval Estimation

More information

STAT 515 fa 2016 Lec Statistical inference - hypothesis testing

STAT 515 fa 2016 Lec Statistical inference - hypothesis testing STAT 515 fa 2016 Lec 20-21 Statistical inference - hypothesis testing Karl B. Gregory Wednesday, Oct 12th Contents 1 Statistical inference 1 1.1 Forms of the null and alternate hypothesis for µ and p....................

More information

Parameter Estimation, Sampling Distributions & Hypothesis Testing

Parameter Estimation, Sampling Distributions & Hypothesis Testing Parameter Estimation, Sampling Distributions & Hypothesis Testing Parameter Estimation & Hypothesis Testing In doing research, we are usually interested in some feature of a population distribution (which

More information

Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing

Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire

More information

Multiple Linear Regression for the Salary Data

Multiple Linear Regression for the Salary Data Multiple Linear Regression for the Salary Data 5 10 15 20 10000 15000 20000 25000 Experience Salary HS BS BS+ 5 10 15 20 10000 15000 20000 25000 Experience Salary No Yes Problem & Data Overview Primary

More information

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.

Review: 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 information

Physics 6720 Introduction to Statistics April 4, 2017

Physics 6720 Introduction to Statistics April 4, 2017 Physics 6720 Introduction to Statistics April 4, 2017 1 Statistics of Counting Often an experiment yields a result that can be classified according to a set of discrete events, giving rise to an integer

More information

Bias Variance Trade-off

Bias Variance Trade-off Bias Variance Trade-off The mean squared error of an estimator MSE(ˆθ) = E([ˆθ θ] 2 ) Can be re-expressed MSE(ˆθ) = Var(ˆθ) + (B(ˆθ) 2 ) MSE = VAR + BIAS 2 Proof MSE(ˆθ) = E((ˆθ θ) 2 ) = E(([ˆθ E(ˆθ)]

More information

Practice Problems Section Problems

Practice Problems Section Problems Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,

More information