CHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities:
|
|
- Eleanor Shields
- 5 years ago
- Views:
Transcription
1 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 begin with, the person is assumed innocent. The prosecutor presents evidence, trying to convince the jury to reject the original assumption of innocence, and conclude that the person is guilty. Parts of a Statistical Test The null hypothesis, H 0 The alternative hypothesis, H a The test statistic and its p-value The rejection region The conclusion The two competing hypotheses are the alternative hypothesis H a, generally the hypothesis that the researcher wishes to support, and the null hypothesis H 0, a contradiction of the alternative hypothesis. The researcher uses the sample data to Reject H 0 and conclude that H a is true. Accept (do not reject) H 0 as true. Test statistic: A single number calculated from the sample data. p-value: A probability calculated using the test statistic. Rejection region: One set, consisting of values that support the alternative hypothesis and lead to rejecting H 0. 1
2 Accepting region: One set, consisting of values that support the null hypothesis. Critical values: The value that separate the acceptance and rejection regions. A Type I error for a statistical test is the error of rejecting the null hypothesis when it is true. A level of significance (significance level α: for a statistical test of hypothesis is α = P (Type I error)=p (falsely rejecting H 0 )=P (rejecting H 0 when it is true) A Type II error for a statistical test is the error of accepting the null hypothesis when it is false. β = P (Type II error)=p (falsely accepting H 0 )=P (accepting H 0 when it is false) the power of a statistical test, given as 1 β = P (reject H 0 when H a is true) measures the ability of the test to perform as required. Large-Sample Statistical Test for µ 1. Null hypothesis: H 0 : µ = µ 0 One-Tailed Test H a : µ > µ 0 (or, H a : µ < µ 0 ) Two-Tailed Test H a : µ µ 0 3. Test statistic: z = x µ 0 σ/ n estimated as z = x µ 0 s/ n One-Tailed Test z > z α (or z < z α when the alternative hypothesis is H a : µ < µ 0 ) 2
3 Two-Tailed Test z > z α/2 or z < z α/2 Assumptions: The n observations in the sample are randomly selected from the population and n is large (n 30) p-value: The p-value or observed significant level of a statistical test is the smallest value of α for which H 0 can be rejected. It is the actual risk of committing a Type I error, if H 0 is rejected based on the observed value of the test statistic. The p-value measures the strength of the evidence against H 0. If the p-value is less than or equal to a preassigned significance level α, then the null hypothesis can be rejected, and you can report that the results are statistically significant at level α. Small-Sample Hypothesis Test for µ 1. Null hypothesis: H 0 : µ = µ 0 One-Tailed Test H a : µ > µ 0 (or, H a : µ < µ 0 ) Two-Tailed Test H a : µ µ 0 3. Test statistic: t = x µ 0 s/ n One-Tailed Test t > t α (or t < t α when the alternative hypothesis is H a : µ < µ 0 ) Two-Tailed Test t > t α/2 or t < t α/2 or when p-value< α 3
4 The critical values of t are based on (n 1) degrees of freedom. Large-Sample Statistical Test for p 1. Null hypothesis: H 0 : p = p 0 One-Tailed Test H a : p > p 0 (or, H a : p < p 0 ) Two-Tailed Test H a : p p 0 3. Test statistic: z = ˆp p 0 p0 q 0 n with ˆp = x n One-Tailed Test z > z α (or z < z α when the alternative hypothesis is H a : µ < µ 0 ) Two-Tailed Test z > z α/2 or z < z α/2 or when p-value< α Assumptions: The sampling satisfies the assumptions of a binomial experiment and n is large enough so that the sampling distribution of ˆp can be approximated by a normal distribution (np 0 > 5 and nq 0 > 5). Assumptions: The sample is randomly selected from a normally distributed population. - Examples: 1. Suppose a scheduled flight must average at least 60% occupancy in order to be profitable, and an examination of the occupancy rate for 120 flights from Atlanta to Dallas showed a mean occupancy per flight of 58% and a standard deviation of 11%. a. If µ is the mean occupancy per flight and if the company wishes to determine whether or 4
5 not this scheduled flight is unprofitable, give the alternative and the null hypotheses for the test. b. Does the alternative hypothesis in part a imply a one or two-tailed test? c. Do the occupancy data for the 120 flights suggest that this scheduled flight is unprofitable? 2. A random sample of 120 observations was selected from a binomial population, and 72 successes were observed. Do the data provide sufficient evidence to indicate that p is greater than 0.5? 3. The following n = 10 observations are a sample from a normal population: 7.4, 7.1, 6.5, 7.5, 7.6, 6.3, 6.9, 7.7, 6.5, 7.0 a. Find a 99% upper one-sided confidence bound for the population mean µ. b. Test H 0 : µ = 7.5 versus H a : µ < 7.5. Use α = c. Do the results of part a support your conclusion in part b? Large-Sample Statistical Test for (µ 1 µ 2 ) 1. Null hypothesis: H 0 : (µ 1 µ 2 ) = D 0, where D 0 is some specific difference that you wish to tests. One-Tailed Test H a : (µ 1 µ 2 ) > D 0 or (µ 1 µ 2 ) < D 0 Two-Tailed Test (µ 1 µ 2 ) D 0 3. Test statistic: z = ( x 1 x 2 ) D 0 SE = ( x 1 x 2 ) D 0 s s2 2 n 2 One-Tailed Test z > z α or z < z α when (µ 1 µ 2 ) < D 0 Two-Tailed Test z > z α/2 or z < z α/2 or when p-value< α 5
6 Assumptions: The samples are randomly and independently selected from the two populations and 30 and n Test of Hypothesis Concerning the Difference Between Two Means: Independent Random Small Samples 1. Null hypothesis: H 0 : (µ 1 µ 2 ) = D 0, where D 0 is some specific difference that you wish to tests. One-Tailed Test H a : (µ 1 µ 2 ) > D 0 or (µ 1 µ 2 ) < D 0 Two-Tailed Test H a : (µ 1 µ 2 ) D 0 3. Test statistic: t = ( x 1 x 2 ) D 0 ( ) s n 2 where s 2 = ( 1)s 2 1 +(n 2 1)s 2 2 +n 2 2 One-Tailed Test t > t α or t < t α when (µ 1 µ 2 ) < D 0 Two-Tailed Test t > t α/2 or t < t α/2 or when p-value< α The critical values of t are based on ( + n 2 2) df. Assumptions: The samples are randomly and independently selected from normally distributed populations. The variances of the populations σ 2 1 and σ 2 2 are equal. Examples: 1. Random samples of 50 recent college graduates in each major were selected and the following information was obtained: 6
7 Major Education Social science Mean SD a. Do the data provide sufficient evidence to indicate a difference in average starting salaries for college graduates who majored in education and the social sciences? Test using α = b. Find a 95% confidence interval for difference between means for the two groups in the general population. Compare your result with part a. 2. A geologist collected the titanium contents of the samples, found using two different methods: Method 1: 0.011, 0.013, 0.013, 0.015, 0.014, 0.013, 0.010, 0.013, 0.011, Method 2: 0.011, 0.016, 0.013, 0.012, 0.015, 0.012, 0.017, 0.013, 0.014, a. Use an appropriate method to test for a significant difference in the average titanium contents using the two different methods. b. Determine a 95% confidence interval estimate for (µ 1 µ 2 ). Does your interval estimate support your conclusion in part a? Large-Sample Statistical Test for (p 1 p 2 ) 1. Null hypothesis: H 0 : (p 1 p 2 ) = 0, or alternatively H 0 : p 1 = p 2. One-Tailed Test H a : (p 1 p 2 ) > 0 or (p 1 p 2 ) < 0 Two-Tailed Test (p 1 p 2 ) 0 3. Test statistic: z = (ˆp 1 ˆp 2 ) 0 SE = (ˆp 1 ˆp 2 ) p1 q 1 + p 2q 2 n 2 = (ˆp 1 ˆp 2 ) pq + pq n 2 where ˆp 1 = x 1 / and ˆp 2 = x 2 /n 2. Since the common value of p 1 = p 2 = p (used in the standard error) is unknown,it is estimated by and the test statistic is z = (ˆp 1 ˆp 2 ) 0 ˆpˆq + ˆpˆq ˆp = x 1 + x 2 + n 2 n 2 or z = 7 (ˆp 1 ˆp 2 ) ( ) 1 ˆpˆq + 1 n 2
8 One-Tailed Test z > z α or z < z α when (p 1 p 2 ) < 0 Two-Tailed Test z > z α/2 or z < z α/2 or when p-value< α Assumptions: Samples are selected in a random and independent manner from two binomial populations and and n 2 are large enough, that is ˆp 1, ˆq 1, n 2ˆp 2 and n 2ˆq 2 should all be greater than 5. - Example: Independent random samples of 280 and 350 observations were selected from binomial populations 1 and 2 respectively. Sample 1 had 132 successes, and sample 2 had 178 successes. Do the data present sufficient evidence to indicate that the proportion of successes in populatio is smaller than the proportion in population 2? Suggested Exercises: 9.7, 9.11, 9.15, 9.17, 9.21, 9.23, 9.27, 9.31, 9.33, 9.35, 9.37, 9.41, 9.45, 9.51, 9.57, 9.61, 9.69, 9.75, 10.7, 10.11, 10.15, 10.21, 10.23, 10.27, 10.31,
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 informationCHAPTER 8. Test Procedures is a rule, based on sample data, for deciding whether to reject H 0 and contains:
CHAPTER 8 Test of Hypotheses Based on a Single Sample Hypothesis testing is the method that decide which of two contradictory claims about the parameter is correct. Here the parameters of interest are
More informationStatistics 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 informationPreliminary 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 informationMATH 240. Chapter 8 Outlines of Hypothesis Tests
MATH 4 Chapter 8 Outlines of Hypothesis Tests Test for Population Proportion p Specify the null and alternative hypotheses, ie, choose one of the three, where p is some specified number: () H : p H : p
More informationPopulation 1 Population 2
Two Population Case Testing the Difference Between Two Population Means Sample of Size n _ Sample mean = x Sample s.d.=s x Sample of Size m _ Sample mean = y Sample s.d.=s y Pop n mean=μ x Pop n s.d.=
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 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 informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 9-1 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population
More information1 Statistical inference for a population mean
1 Statistical inference for a population mean 1. Inference for a large sample, known variance Suppose X 1,..., X n represents a large random sample of data from a population with unknown mean µ and known
More informationTopic 17 Simple Hypotheses
Topic 17 Simple Hypotheses Terminology and the Neyman-Pearson Lemma 1 / 11 Outline Overview Terminology The Neyman-Pearson Lemma 2 / 11 Overview Statistical hypothesis testing is designed to address the
More informationSmoking Habits. Moderate Smokers Heavy Smokers Total. Hypertension No Hypertension Total
Math 3070. Treibergs Final Exam Name: December 7, 00. In an experiment to see how hypertension is related to smoking habits, the following data was taken on individuals. Test the hypothesis that the proportions
More informationChapter Six: Two Independent Samples Methods 1/51
Chapter Six: Two Independent Samples Methods 1/51 6.3 Methods Related To Differences Between Proportions 2/51 Test For A Difference Between Proportions:Introduction Suppose a sampling distribution were
More information8.1-4 Test of Hypotheses Based on a Single Sample
8.1-4 Test of Hypotheses Based on a Single Sample Example 1 (Example 8.6, p. 312) A manufacturer of sprinkler systems used for fire protection in office buildings claims that the true average system-activation
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 8. Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis
Chapter 8 Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 1 Content 1. Identifying the Target Parameter 2.
More informationLECTURE 5. Introduction to Econometrics. Hypothesis testing
LECTURE 5 Introduction to Econometrics Hypothesis testing October 18, 2016 1 / 26 ON TODAY S LECTURE We are going to discuss how hypotheses about coefficients can be tested in regression models We will
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 informationThe 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 informationT test for two Independent Samples. Raja, BSc.N, DCHN, RN Nursing Instructor Acknowledgement: Ms. Saima Hirani June 07, 2016
T test for two Independent Samples Raja, BSc.N, DCHN, RN Nursing Instructor Acknowledgement: Ms. Saima Hirani June 07, 2016 Q1. The mean serum creatinine level is measured in 36 patients after they received
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 informationSingle Sample Means. SOCY601 Alan Neustadtl
Single Sample Means SOCY601 Alan Neustadtl The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size
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 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 informationEstimating the accuracy of a hypothesis Setting. Assume a binary classification setting
Estimating the accuracy of a hypothesis Setting Assume a binary classification setting Assume input/output pairs (x, y) are sampled from an unknown probability distribution D = p(x, y) Train a binary classifier
More information10/4/2013. Hypothesis Testing & z-test. Hypothesis Testing. Hypothesis Testing
& z-test Lecture Set 11 We have a coin and are trying to determine if it is biased or unbiased What should we assume? Why? Flip coin n = 100 times E(Heads) = 50 Why? Assume we count 53 Heads... What could
More informationStudy Ch. 9.3, #47 53 (45 51), 55 61, (55 59)
GOALS: 1. Understand that 2 approaches of hypothesis testing exist: classical or critical value, and p value. We will use the p value approach. 2. Understand the critical value for the classical approach
More informationStatistical Process Control (contd... )
Statistical Process Control (contd... ) ME522: Quality Engineering Vivek Kumar Mehta November 11, 2016 Note: This lecture is prepared with the help of material available online at https://onlinecourses.science.psu.edu/
More informationFirst we look at some terms to be used in this section.
8 Hypothesis Testing 8.1 Introduction MATH1015 Biostatistics Week 8 In Chapter 7, we ve studied the estimation of parameters, point or interval estimates. The construction of CI relies on the sampling
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 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 informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
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 informationSTAT Chapter 8: Hypothesis Tests
STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular
More informationMathematical 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 informationTUTORIAL 8 SOLUTIONS #
TUTORIAL 8 SOLUTIONS #9.11.21 Suppose that a single observation X is taken from a uniform density on [0,θ], and consider testing H 0 : θ = 1 versus H 1 : θ =2. (a) Find a test that has significance level
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 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 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 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 information1 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 informationUnit 19 Formulating Hypotheses and Making Decisions
Unit 19 Formulating Hypotheses and Making Decisions Objectives: To formulate a null hypothesis and an alternative hypothesis, and to choose a significance level To identify the Type I error and the Type
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationMath 101: Elementary Statistics Tests of Hypothesis
Tests of Hypothesis Department of Mathematics and Computer Science University of the Philippines Baguio November 15, 2018 Basic Concepts of Statistical Hypothesis Testing A statistical hypothesis is an
More informationChapter 8 of Devore , H 1 :
Chapter 8 of Devore TESTING A STATISTICAL HYPOTHESIS Maghsoodloo A statistical hypothesis is an assumption about the frequency function(s) (i.e., PDF or pdf) of one or more random variables. Stated in
More informationChapter 9. Hypothesis testing. 9.1 Introduction
Chapter 9 Hypothesis testing 9.1 Introduction Confidence intervals are one of the two most common types of statistical inference. Use them when our goal is to estimate a population parameter. The second
More informationChapter 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 informationECO220Y Hypothesis Testing: Type I and Type II Errors and Power Readings: Chapter 12,
ECO220Y Hypothesis Testing: Type I and Type II Errors and Power Readings: Chapter 12, 12.7-12.9 Winter 2012 Lecture 15 (Winter 2011) Estimation Lecture 15 1 / 25 Linking Two Approaches to Hypothesis Testing
More information1; (f) H 0 : = 55 db, H 1 : < 55.
Reference: Chapter 8 of J. L. Devore s 8 th Edition By S. Maghsoodloo TESTING a STATISTICAL HYPOTHESIS A statistical hypothesis is an assumption about the frequency function(s) (i.e., pmf or pdf) of one
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 informationTable of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).
Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X.04) =.8508. For z < 0 subtract the value from,
More informationMathematical statistics
October 20 th, 2018 Lecture 17: Tests of Hypotheses Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation
More informationBIO5312 Biostatistics Lecture 6: Statistical hypothesis testings
BIO5312 Biostatistics Lecture 6: Statistical hypothesis testings Yujin Chung October 4th, 2016 Fall 2016 Yujin Chung Lec6: Statistical hypothesis testings Fall 2016 1/30 Previous Two types of statistical
More informationSampling Distributions: Central Limit Theorem
Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)
More information16.400/453J Human Factors Engineering. Design of Experiments II
J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential
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 informationExam 2 (KEY) July 20, 2009
STAT 2300 Business Statistics/Summer 2009, Section 002 Exam 2 (KEY) July 20, 2009 Name: USU A#: Score: /225 Directions: This exam consists of six (6) questions, assessing material learned within Modules
More informationBINF702 SPRING 2015 Chapter 7 Hypothesis Testing: One-Sample Inference
BINF702 SPRING 2015 Chapter 7 Hypothesis Testing: One-Sample Inference BINF702 SPRING 2014 Chapter 7 Hypothesis Testing 1 Section 7.9 One-Sample c 2 Test for the Variance of a Normal Distribution Eq. 7.40
More informationPower and the computation of sample size
9 Power and the computation of sample size A statistical test will not be able to detect a true difference if the sample size is too small compared with the magnitude of the difference. When designing
More informationA proportion is the fraction of individuals having a particular attribute. Can range from 0 to 1!
Proportions A proportion is the fraction of individuals having a particular attribute. It is also the probability that an individual randomly sampled from the population will have that attribute Can range
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 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 informationHypothesis Testing. ) the hypothesis that suggests no change from previous experience
Hypothesis Testing Definitions Hypothesis a claim about something Null hypothesis ( H 0 ) the hypothesis that suggests no change from previous experience Alternative hypothesis ( H 1 ) the hypothesis that
More informationSummary: 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 informationSolution: First note that the power function of the test is given as follows,
Problem 4.5.8: Assume the life of a tire given by X is distributed N(θ, 5000 ) Past experience indicates that θ = 30000. The manufacturere claims the tires made by a new process have mean θ > 30000. Is
More informationOHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd Basic Statistics Sample size?
ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Basic Statistics Sample size? Sample size determination: text section 2-4-2 Page 41 section 3-7 Page 107 Website::http://www.stat.uiowa.edu/~rlenth/Power/
More informationCH.9 Tests of Hypotheses for a Single Sample
CH.9 Tests of Hypotheses for a Single Sample Hypotheses testing Tests on the mean of a normal distributionvariance known Tests on the mean of a normal distributionvariance unknown Tests on the variance
More informationPerformance 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 informationMAT 212 Introduction to Business Statistics II Lecture Notes
MAT 212 Introduction to Business Statistics II Lecture Notes Muhammad El-Taha Department of Mathematics and Statistics University of Southern Maine 96 Falmouth Street Portland, ME 04104-9300 MAT 212, Spring
More informationTopic 17: Simple Hypotheses
Topic 17: November, 2011 1 Overview and Terminology Statistical hypothesis testing is designed to address the question: Do the data provide sufficient evidence to conclude that we must depart from our
More information20 Hypothesis Testing, Part I
20 Hypothesis Testing, Part I Bob has told Alice that the average hourly rate for a lawyer in Virginia is $200 with a standard deviation of $50, but Alice wants to test this claim. If Bob is right, she
More informationPOLI 443 Applied Political Research
POLI 443 Applied Political Research Session 6: Tests of Hypotheses Contingency Analysis Lecturer: Prof. A. Essuman-Johnson, Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College
More information8: Hypothesis Testing
Some definitions 8: Hypothesis Testing. Simple, compound, null and alternative hypotheses In test theory one distinguishes between simple hypotheses and compound hypotheses. A simple hypothesis Examples:
More informationHypothesis for Means and Proportions
November 14, 2012 Hypothesis Tests - Basic Ideas Often we are interested not in estimating an unknown parameter but in testing some claim or hypothesis concerning a population. For example we may wish
More informationLECTURE 12 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING
LECTURE 1 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING INTERVAL ESTIMATION Point estimation of : The inference is a guess of a single value as the value of. No accuracy associated with it. Interval estimation
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 informationTwo Sample Problems. Two sample problems
Two Sample Problems Two sample problems The goal of inference is to compare the responses in two groups. Each group is a sample from a different population. The responses in each group are independent
More informationPHP2510: Principles of Biostatistics & Data Analysis. Lecture X: Hypothesis testing. PHP 2510 Lec 10: Hypothesis testing 1
PHP2510: Principles of Biostatistics & Data Analysis Lecture X: Hypothesis testing PHP 2510 Lec 10: Hypothesis testing 1 In previous lectures we have encountered problems of estimating an unknown population
More informationFormulas and Tables. for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. ˆp E p ˆp E Proportion
Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. Ch. 3: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx)
More informationHypothesis Testing The basic ingredients of a hypothesis test are
Hypothesis Testing The basic ingredients of a hypothesis test are 1 the null hypothesis, denoted as H o 2 the alternative hypothesis, denoted as H a 3 the test statistic 4 the data 5 the conclusion. The
More informationHypothesis tests
6.1 6.4 Hypothesis tests Prof. Tesler Math 186 February 26, 2014 Prof. Tesler 6.1 6.4 Hypothesis tests Math 186 / February 26, 2014 1 / 41 6.1 6.2 Intro to hypothesis tests and decision rules Hypothesis
More informationSection 9.1 (Part 2) (pp ) Type I and Type II Errors
Section 9.1 (Part 2) (pp. 547-551) Type I and Type II Errors Because we are basing our conclusion in a significance test on sample data, there is always a chance that our conclusions will be in error.
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
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 informationClass 24. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 4 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 9. and 9.3 Lecture Chapter 10.1-10.3 Review Exam 6 Problem Solving
More informationIntroduction to Statistics
MTH4106 Introduction to Statistics Notes 15 Spring 2013 Testing hypotheses about the mean Earlier, we saw how to test hypotheses about a proportion, using properties of the Binomial distribution It is
More informationSoc3811 Second Midterm Exam
Soc38 Second Midterm Exam SEMI-OPE OTE: One sheet of paper, signed & turned in with exam booklet Bring our Own Pencil with Eraser and a Hand Calculator! Standardized Scores & Probability If we know the
More informationLECTURE NOTES. INTSTA2 Introductory Statistics 2. Francis Joseph H. Campeña, De La Salle University Manila
LECTURE NOTES INTSTA Introductory Statistics Francis Joseph H. Campeña, De La Salle University Manila Contents 1 Normal Distribution 1.1 Normal Distribution....................... Sampling and Sampling
More informationFormulas and Tables by Mario F. Triola
Copyright 010 Pearson Education, Inc. Ch. 3: Descriptive Statistics x f # x x f Mean 1x - x s - 1 n 1 x - 1 x s 1n - 1 s B variance s Ch. 4: Probability Mean (frequency table) Standard deviation P1A or
More informationStatistics Handbook. All statistical tables were computed by the author.
Statistics Handbook Contents Page Wilcoxon rank-sum test (Mann-Whitney equivalent) Wilcoxon matched-pairs test 3 Normal Distribution 4 Z-test Related samples t-test 5 Unrelated samples t-test 6 Variance
More informationFormulas and Tables. for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. ˆp E p ˆp E Proportion.
Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Ch. 2: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx) 2 s Å n(n 2 1) Mean (frequency
More informationThe Purpose of Hypothesis Testing
Section 8 1A:! An Introduction to Hypothesis Testing The Purpose of Hypothesis Testing See s Candy states that a box of it s candy weighs 16 oz. They do not mean that every single box weights exactly 16
More informationContent by Week Week of October 14 27
Content by Week Week of October 14 27 Learning objectives By the end of this week, you should be able to: Understand the purpose and interpretation of confidence intervals for the mean, Calculate confidence
More informationMATH 728 Homework 3. Oleksandr Pavlenko
MATH 78 Homewor 3 Olesandr Pavleno 4.5.8 Let us say the life of a tire in miles, say X, is normally distributed with mean θ and standard deviation 5000. Past experience indicates that θ = 30000. The manufacturer
More informationInferences for Correlation
Inferences for Correlation Quantitative Methods II Plan for Today Recall: correlation coefficient Bivariate normal distributions Hypotheses testing for population correlation Confidence intervals for population
More informationWe need to define some concepts that are used in experiments.
Chapter 0 Analysis of Variance (a.k.a. Designing and Analysing Experiments) Section 0. Introduction In Chapter we mentioned some different ways in which we could get data: Surveys, Observational Studies,
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 informationLecture 14. Analysis of Variance * Correlation and Regression. The McGraw-Hill Companies, Inc., 2000
Lecture 14 Analysis of Variance * Correlation and Regression Outline Analysis of Variance (ANOVA) 11-1 Introduction 11-2 Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationLecture 14. Outline. Outline. Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA)
Outline Lecture 14 Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA) 11-1 Introduction 11- Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationReview 6. n 1 = 85 n 2 = 75 x 1 = x 2 = s 1 = 38.7 s 2 = 39.2
Review 6 Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected ) A researcher finds that of,000 people who said that
More information