Module 7: Hypothesis Testing I Statistics (OA3102)
|
|
- Hector Townsend
- 6 years ago
- Views:
Transcription
1 Module 7: Hypothesis Testing I Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter Revision:
2 Goals for this Module Introduction to testing hypotheses Steps in conducting a hypothesis test Types of errors Lots of terminology Large sample tests for the means and proportions Aka z-tests Type II error probability calculations Sample size calculations Revision:
3 A Simple Example You hypothesize a coin is fair To test, take a coin and start flipping it If it is fair, you expect that about half of the flips will be heads and half tails After a large number of flips, if you see either a large fraction of heads or a large fraction of tails you tend to disbelieve your hypothesis This is an informal hypothesis test! How to decide when the fraction is too large? Revision:
4 Is the Coin Fair? Back to the in-class exercise from Module 1: flip a coin 10 times and count the number of heads Assuming the coin is fair (and flips are independent), the number of heads has a binomial distribution with n=10 and p=0.5 So, if our assumption is true, the distribution of the number of heads is: And, if your assumption is not true, what do we expect to see? Either too few or too many heads Revision:
5 Setting Up a Test for the Coin Idea: Make a rule, based on the number of heads observed, from which we will conclude either that our assumption (p=0.5) is true or false Here s one rule: Assume p=0.5 if 3 < x < 7 Otherwise, conclude p 0.5 If p=0.5, what s our chance of making a mistake? ~11% If p=0.8, what s our chance of detecting the biased coin? ~68% Revision:
6 CIs vs. Hypothesis Testing Previously we would have answered this question with a confidence interval: Say we observed yn= 0.7 : we re 95% confident that the interval [0.5, 0.9] covers the true p Now we look to answer the question using a hypothesis test: If the true probability of heads is p = 0.5 (i.e., the coin is fair), how unlikely would it be to see 7 heads out of ten flips? If we see an outcome inconsistent with our hypothesis (the coin is fair), then we reject it Revision:
7 Why Do Hypothesis Tests? Confidence intervals provide more information than hypothesis tests But often we need to test a particular theory E.g., The mod decreases the mean down-time. Sometimes confidence intervals are hard/impossible When the theory you re testing has several dimensions E.g., regression slope and intercept When intervals don t make sense E.g., are two categorical variables independent? Revision:
8 Elements of a Statistical Test Null hypothesis Denoted H 0 We believe the null unless/until it is proven false Alternative hypothesis Denoted H a It s what we would like to prove Test statistic The test statistic is the empirical evidence from the data Rejection region If the test statistic falls in the rejection region, we say reject the null hypothesis or we ve proven the alternative Otherwise, we fail to reject the null Revision:
9 true situation Errors in Hypothesis Testing don t reject H 0 conclusion reject H 0 H 0 true no error type I error H a true type II error no error Revision:
10 Probability of a Type I Error a=pr(type I error) = Pr(H 0 is rejected when it is true) Also called the significance level or the level of the test Experimenter chooses prior to the test Conventions: a=0.10, 0.05, and 0.01 Can increase or decrease depending on particular problem Choice of a defines rejection region (or size of p-value ) that results in rejection of null Revision:
11 Probability of a Type II Error b=pr(type II error) = Pr(not rejecting H 0 when it is false) It is a function of the actual alternative distribution and the sample size 1-b called the power of a test It s Pr(rejecting H 0 when it is false) Ideal is both small a and b (i.e., high power), but for a fixed sample size they trade off By convention, we control a by choice and b with sample size (bigger sample, more power) Revision:
12 Choosing the Null and Alternative Based on the Severity of Error In hypothesis testing, we get to control the Type I error So, if one error is more severe than the other, set the test up so that it s the Type I error E.g., in the US, we consider sending an innocent person to jail more serious than letting a guilty person go free Hence, the burden of proof of guilt is placed on the prosecution at trial ( innocent until proven guilty ) I.e., the null hypothesis is a person is innocent, and the trial process controls the chance of sending an innocent person to jail Revision:
13 Example 10.1 Consider a political poll of n=15 people Want to test H 0 : p = 0.5 vs. H a : p < 0.5, where p is the proportion of the population favoring a candidate The test statistic is Y, the number of people in the sample favoring the candidate If the RR={y < 2}, find a Solution: Revision:
14 Example 10.1 (continued) Revision:
15 Example 10.2 Continuing with the previous problem, if p=0.3 what is the probability of a Type II error (b)? I.e., what s the probability of concluding that the candidate will win? Solution: Revision:
16 Example 10.2 (continued) Revision:
17 Example 10.3 Still continuing with the previous problem, if p=0.1 then what is the probability of a Type II error (b)? Solution: Revision:
18 Example 10.3 (continued) Revision:
19 Example 10.4 Still continuing the problem, if RR={y < 5}: What is the level of the test (a)? If p = 0.3, what is b? Solution: Revision:
20 Example 10.4 (continued) Revision:
21 Terminology Null and alternative hypotheses We will believe the null until it is proven false Acceptance vs. rejection region The null is proven false if the test statistic falls in the rejection region Type I vs. Type II error Type I: Rejecting the null hypothesis when it is really true Type II: Accepting the null hypothesis when it is really false Significance level or level of the test (a) Probability of a Type I error Pr(Type II error) = b and 1-b is called the power It s a function of the actual alternative distribution Revision:
22 Another Example As a program manager, you want to decrease mean down-time m for a type of equipment Manufacturer suggests modification Goal is to see if mod actually does decrease m Implement modification on a sample of equipment (n=25) and measure the downtime (say, per month) Currently, equipment down-time is 75 mins/month Standard deviation is s=9 mins Revision:
23 Intuition Behind the Test 1. We will assume the status quo is true a. In this case, that there is no change in the mean down-time 2. unless there is sufficient evidence to contradict it a. In this case, meaning we observe a much smaller sample mean than we would expect to see assuming m=75 s=9/5 X m=75 Revision:
24 Steps in Hypothesis Testing 1. Identify the parameter of interest 2. State null and alternative hypotheses 3. Determine form of test statistic 4. Calculate rejection region 5. Calculate test statistic 6. Determine test outcome by comparing test statistic to rejection region Revision:
25 1. Identify Parameter of Interest In hypothesis tests, we are testing the parameter of a distribution E.g., m or s for a normal distribution E.g., p for a binomial distribution E.g., a and b for a gamma distribution So, the first step is to identify the parameter of interest Often we ll be testing the mean m of a normal, since the CLT often applies to the sample mean E.g., in the equipment down-time example, we re interested in testing the mean down-time For the coin and election examples, we re testing p Revision:
26 2. State Null and Alternative Hypotheses H 0 : The null hypothesis is a specific theory about the population that we wish to disprove We will believe the null until it is disproved Example: Mean down-time is equal to 75 In notation, H : m = 75 0 H a : The alternative hypothesis is what we want to prove What we will believe if the null is rejected Example: Mean down-time is less than 75 In notation, H : 75 a m Revision:
27 Null and Alternative Hypotheses are Fundamentally Different The null hypothesis is what you have assumed Generally, it s the status quo or less desirable test outcome Failing to prove the alternative does not mean the null is true, only that you don t have enough evidence to reject it The alternative is proven based on empirical evidence It s the desired test outcome and/or the outcome upon which the burden of proof rests The significance level (a) is set so that the chance of incorrectly proving the alternative is tolerably low Having proved the alternative is a much stronger outcome than failing to reject the null Thus, structure the test so the alternative is what needs proving Revision:
28 In the Example In the example, we want to test is whether the mod decreases mean down-time So, the null hypothesis is the status quo and the alternative carries the burden of proof to show a decrease We write this out as H : 75 0 m = H : 75 a m H0 The other possibilities are : m = 75 H : 75 and : 0 m = 75 Ha m H : 75 a m What would you be testing with these hypotheses? Revision:
29 Expressing the Null as an Equality We will express the null hypothesis as an equality and the alternative as an inequality E.g., H0 : m = 75 versus H a : m 75 In reality, the hypotheses divide the real line into two separate regions E.g., H0 : m 75 versus H a : m 75 However, the most powerful test occurs when the null hypothesis is at the boundary of its region Hence, we write the null as an equality Revision:
30 3. Determine Test Statistic (and its Sampling Distribution) The test statistic is (a function of) the sample statistic corresponding to population parameter you are testing Population and sample statistic examples: Population mean Sample mean Difference of two population means Difference of two sample means It is sometimes a function of the sample statistic as we may rescale the sample statistic We use the sampling distribution to determine whether the observed statistic is unusual Revision:
31 In the Example In the example, we are testing the mean m so the obvious choice for the sample statistic is X In this case, it s easier to make the test statistic the rescaled sample statistic, X m0 Z =, s / n where m 0 is the null hypothesis mean Why? Because, assuming the null hypothesis is true, we know the sampling distribution of Z: Z~N(0,1) This is exactly true if X has a normal distribution and approximately true via the CLT for large sample sizes Revision:
32 4. Calculate Rejection Region Rejection region depends on the alternative hypothesis Set the significance level a so that the Pr(fall in rejection region null hyp. is true) = a Means you will have to determine the appropriate quantile or quantiles of the sampling distribution Revision:
33 The Way to Think About It (for a two-sided test) Rejection region unlikely under the null (i.e., probability a) If test statistic falls in this region, reject the null Acceptance region likely under the null hypothesis If test statistic falls in this region, fail to reject null Revision:
34 In the Example In our example, the test statistic In addition, we know H : a m 75, so this is a one-tailed or one-sided test So, we need to find z a so that Pr(Z < z a ) = a Choosing a=0.05: From Table 4 we get Pr(Z < ) = 0.05 Using Excel: =NORMSINV(0.05) And in R: qnorm(0.05) Z~N(0,1) Revision:
35 Picturing the Rejection Region For the rescaled test statistic, the rejection region is in red Probability of falling in that region, assuming the null is true, is a=0.05 Z~N(0,1) pdf Here s the equivalent test without rescaling Probability of falling in that region, assuming the null is true, is still a= x 9/5 = X~N(75,81/25) pdf Revision:
36 5. Calculate the Test Statistic Now, plug the necessary quantities into the formula and calculate the test statistic E.g., in the example, imagine we tested 25 pieces of equipment for a month, measuring the total down-time for each: Calculating the sample mean gives X = 68.6 Thus, the test statistic is X m z = = = 3.8 s / n 9 / 25 Revision:
37 6. Determine the Outcome Now, compare the test statistic with the rejection region If it falls within the rejection region you have rejected the null hypothesis Equivalently, proven the alternative If it falls in the acceptance region, you have failed to reject the null hypothesis Equivalently, failed to prove the alternative Revision:
38 In the Example We observed z = 3.8 rejection region The picture: which falls in the Very unusual to see this, if the null is true z = Thus, conclude that the mod is effective at reducing mean down-time for the equipment Revision:
39 One Way to Think About It The test statistic tells us our observation was 3.8 standard errors way from the hypothesized mean The calculation How likely is this? X m z = = = 3.8 s / n 9 / 25 If the null were true, it would be very unlikely Pr( Z 3.8 H is true) = 3.8 = Another way to think about it is about one chance in 14,000 to see something like this or more extreme (i.e., 1/ ) Pretty convincing evidence that the mod decreases mean down-time Revision:
40 Logic of Hypothesis Testing It s proof by contradiction: Suppose I am a general/flag officer People would salute me on Tuesdays No one ever salutes me I must not be a general/flag officer Null Hypothesis What I expect if null hypothesis is true What I see is inconsistent with what I expect My initial assumption must be wrong Revision:
41 Now, for the Example Assume m is 75 Sample mean (n=25) normally distributed w/ mean 75, s.e. 9/ 25 = 1.8 Observe a sample mean of 68.6 that is 3.8 s.e.s below assumed mean Real mean m must be less than 75 Null Hypothesis What you expect to see if null is true Not very likely to happen if null is true (p = ) Null must be false Revision:
42 Large-Sample or z-tests The statistic is (approximately) normally distributed ˆ 0 The rescaled test statistic is Z = The null hypothesis is Three possible alternative hypotheses and tests: Alternative Hypothesis H a : H a : H a : H : = 0 0 s ˆ Rejection Region for Level a Test z z z a z a z z or z z a/ 2 a/ 2 (upper-tailed test) (lower-tailed test) (two-tailed test) Revision:
43 Picturing z-tests Upper-tailed test Lower-tailed test Two-tailed test Revision: * Figure from Probability and Statistics for Engineering and the Sciences, 7 th ed., Duxbury Press, 2008.
44 Example 10.5 VP claims mean contacts/week is less than15 Data collected on random sample of n=36 people Given y =17 and s 2 =9, is there evidence to refute the claim at a significance level of a=0.05? Specify the hypotheses to be tested: Revision:
45 Example 10.5 (continued) Specify the test statistic and the rejection region Revision:
46 Example 10.5 (continued) Conduct the test and state the conclusion Revision:
47 Large Sample Tests for Population Proportion (p) If we have a large sample, then via the CLT, has an approximate normal distribution For the null hypothesis H 0 : p = p 0 there are three possible alternative hypotheses Alternative Hypothesis H p p a : H p p a : H p p a : where the test statistic is Rejection Region for Level a Test z z Revision: z a z a z z or z z a/ 2 a/ 2 z = pˆ (upper-tailed test) p (lower-tailed test) 0 p (1 p ) / n 0 0 pˆ = y / n (two-tailed test)
48 What s a Large Sample? Remember that the y in indicator variables pˆ = y / n is a sum of If you sum enough, then the CLT kicks in and you can assume ˆp has an approximately normal distribution So use the same rule we used back in the CLT lecture: n 9 max( p0, q0) min( p, ) 0 q0 Revision:
49 Example 10.6 Machine must be repaired is produces more than 10% defectives A random sample of n=100 items has 15 defectives Is there evidence to support the claim that the machine needs repairing at a significance level of a=0.01? Specify the hypotheses to be tested: Revision:
50 Example 10.6 (continued) Specify the test statistic and the rejection region Revision:
51 Example 10.6 (continued) Conduct the test and state the conclusion Revision:
52 Example 10.7 Reaction time study on men and women conducted Data on independent random samples of 50 men and 50 women collected giving 2 2 y = 3.6, s = 0.18, y = 3.8, and s = 0.14 men men women women Is there evidence to suggest a difference in the true mean reaction times between men and women at the a=0.01 level? Specify the hypotheses to be tested: Revision:
53 Example 10.7 (continued) Specify the test statistic and the rejection region Revision:
54 Example 10.7 (continued) Conduct the test and state the conclusion Revision:
55 Calculating Type II Error Probabilities The probability of a Type II error (b) is the probability a test fails to reject the null when the alternative hypothesis is true Note that it depends on a particular alternative hypothesis We can write it mathematically as Pr(reject H 0 H a is true with = a ) To determine, first figure out the rejection region (a function of the null hypothesis), then calculate the probability of falling in the acceptance region when = a Revision:
56 Calculating Type II Error Probabilities, continued For example, consider the test versus H a : 0 H : = 0 0 Then the rejection region is of the form RR = ˆ : ˆ k for some value of k So, the probability of a Type II error is Pr ˆ not in RR when true with = a = Ha a b = Pr ˆ k = ˆ a k a = Pr s s a ˆ ˆ Revision:
57 Example 10.8 Returning to Example 10.5, find b if m a =16 Remember that m 0 =15, a=0.05, n=36, y =17 and s 2 =9 Pictorially, we have: Revision:
58 Example 10.8 (continued) Now solving: Revision:
59 Example 10.8 (continued) An alternative way to solve: Revision:
60 Example 10.8 (continued) Revision:
61 Alternatively, the Power of a Test Remember, that s the power of a test is the probability a test will reject the null for a particular alternative hypothesis It s just 1-b Why is this important? Prior to doing a test, natural problem is that you want to make sure you have sufficient power to prove interesting alternatives Sometimes after a test results in a null result, you might want to know the probability of rejecting at the observed level Revision:
62 Sample Size Calculations The sample size n for which a level a test also has b at the alternative value m a is n 2 s za zb ma m0 = s z a/2 zb ma m0 2 for a one-tailed test for a two-tailed test (approximate sol'n) Here z a and z b are the quantiles of the normal distribution for a and b Revision:
63 Example 10.9 Returning to Exercise 10.5, assuming s 2 =9, what sample size n is required to test vs. H m = with a=b=0.05? a : 16 Solution: H : m = 15 0 Revision:
64 Another Example Consider a test: H 0 : m =30,000 vs H a : m 30,000, where The desired significance level is a=0.01 The population has a normal distribution with s =1,500 Find the required sample size so that at m a =31,000, the probability of a Type II error is 0.1: s za 2 z b n = = ma m 0 2 Revision:
65 Relationship Between Hypothesis Tests and Confidence Intervals The large sample hypothesis test (z-test) is based on the statistic ˆ 0 Z = s where the acceptance region is ˆ 0 RR = za za s ˆ which we can write as Revision: ˆ 2 2 RR = ˆ z s ˆ z s a 2 ˆ 0 a 2 ˆ
66 Relationship Between Hypothesis Tests and Confidence Intervals Now, remember the general form for a twosided large sample confidence interval: a % CI = ˆ z s ˆ 2 ˆ, z s a a 2 ˆ Note the similarities to the acceptance region: RR = ˆ z s ˆ z s a 2 ˆ 0 a 2 So, if 0 ˆ z s ˆ 2 ˆ, a z s a 2 ˆ reject the hypothesis test we would fail to Can interpret 100(1-a)% CI as the set of all values for 0 for which H 0 : = 0 is acceptable at level a Revision: ˆ
67 What We Covered in this Module Introduced hypothesis testing Steps in conducting a hypothesis test Types of errors Lots of terminology Large sample tests for the means and proportions Aka z-tests Type II error probability calculations Sample size calculations Revision:
68 Homework WM&S chapter 10 Required: 2, 3, 20, 21, 27, 34, 38, 41, 42 Extra credit: None Useful hints: Always first write out the null and alternative hypotheses. I also recommend drawing the null distribution and then highlighting the rejection region. This can be particularly helpful when calculating b... Exercises 21 and 27: We didn t do these types of problems in class, but just use what you learned with two-sample confidence intervals The relevant hypotheses are H0: m1 m2 = 0 H0: p1 p2 = 0 : 0 and Ha m1 m2 H : 0 a p1 p2 Revision:
STAT 220 Midterm Exam, Friday, Feb. 24
STAT 220 Midterm Exam, Friday, Feb. 24 Name Please show all of your work on the exam itself. If you need more space, use the back of the page. Remember that partial credit will be awarded when appropriate.
More informationManagerial Decision Making
Course Business Managerial Decision Making Session 4 Conditional Probability & Bayesian Updating Surveys in the future... attempt to participate is the important thing Work-load goals Average 6-7 hours,
More informationThe Evolution of Random Phenomena
The Evolution of Random Phenomena A Look at Markov Chains Glen Wang glenw@uchicago.edu Splash! Chicago: Winter Cascade 2012 Lecture 1: What is Randomness? What is randomness? Can you think of some examples
More information4-3 Basic Skills and Concepts
4-3 Basic Skills and Concepts Identifying Binomial Distributions. In Exercises 1 8, determine whether the given procedure results in a binomial distribution. For those that are not binomial, identify at
More informationProbability and Statistics Curriculum Pacing Guide
Unit 1 Terms PS.SPMJ.3 PS.SPMJ.5 Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods
More informationVisit us at:
White Paper Integrating Six Sigma and Software Testing Process for Removal of Wastage & Optimizing Resource Utilization 24 October 2013 With resources working for extended hours and in a pressurized environment,
More informationSTA 225: Introductory Statistics (CT)
Marshall University College of Science Mathematics Department STA 225: Introductory Statistics (CT) Course catalog description A critical thinking course in applied statistical reasoning covering basic
More informationLecture 1: Machine Learning Basics
1/69 Lecture 1: Machine Learning Basics Ali Harakeh University of Waterloo WAVE Lab ali.harakeh@uwaterloo.ca May 1, 2017 2/69 Overview 1 Learning Algorithms 2 Capacity, Overfitting, and Underfitting 3
More informationAGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS
AGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS 1 CALIFORNIA CONTENT STANDARDS: Chapter 1 ALGEBRA AND WHOLE NUMBERS Algebra and Functions 1.4 Students use algebraic
More informationDiagnostic Test. Middle School Mathematics
Diagnostic Test Middle School Mathematics Copyright 2010 XAMonline, Inc. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by
More informationEdexcel GCSE. Statistics 1389 Paper 1H. June Mark Scheme. Statistics Edexcel GCSE
Edexcel GCSE Statistics 1389 Paper 1H June 2007 Mark Scheme Edexcel GCSE Statistics 1389 NOTES ON MARKING PRINCIPLES 1 Types of mark M marks: method marks A marks: accuracy marks B marks: unconditional
More informationEvaluating Statements About Probability
CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Evaluating Statements About Probability Mathematics Assessment Resource Service University of Nottingham
More informationStopping rules for sequential trials in high-dimensional data
Stopping rules for sequential trials in high-dimensional data Sonja Zehetmayer, Alexandra Graf, and Martin Posch Center for Medical Statistics, Informatics and Intelligent Systems Medical University of
More informationExtending Place Value with Whole Numbers to 1,000,000
Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Recognize that a digit
More informationCal s Dinner Card Deals
Cal s Dinner Card Deals Overview: In this lesson students compare three linear functions in the context of Dinner Card Deals. Students are required to interpret a graph for each Dinner Card Deal to help
More informationQuantitative analysis with statistics (and ponies) (Some slides, pony-based examples from Blase Ur)
Quantitative analysis with statistics (and ponies) (Some slides, pony-based examples from Blase Ur) 1 Interviews, diary studies Start stats Thursday: Ethics/IRB Tuesday: More stats New homework is available
More informationA Game-based Assessment of Children s Choices to Seek Feedback and to Revise
A Game-based Assessment of Children s Choices to Seek Feedback and to Revise Maria Cutumisu, Kristen P. Blair, Daniel L. Schwartz, Doris B. Chin Stanford Graduate School of Education Please address all
More information12- A whirlwind tour of statistics
CyLab HT 05-436 / 05-836 / 08-534 / 08-734 / 19-534 / 19-734 Usable Privacy and Security TP :// C DU February 22, 2016 y & Secu rivac rity P le ratory bo La Lujo Bauer, Nicolas Christin, and Abby Marsh
More informationMathematics (JUN14MS0401) General Certificate of Education Advanced Level Examination June Unit Statistics TOTAL.
Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Statistics 4 Tuesday 24 June 2014 General Certificate of Education Advanced
More informationAlgebra 1, Quarter 3, Unit 3.1. Line of Best Fit. Overview
Algebra 1, Quarter 3, Unit 3.1 Line of Best Fit Overview Number of instructional days 6 (1 day assessment) (1 day = 45 minutes) Content to be learned Analyze scatter plots and construct the line of best
More informationSoftware Maintenance
1 What is Software Maintenance? Software Maintenance is a very broad activity that includes error corrections, enhancements of capabilities, deletion of obsolete capabilities, and optimization. 2 Categories
More informationProof Theory for Syntacticians
Department of Linguistics Ohio State University Syntax 2 (Linguistics 602.02) January 5, 2012 Logics for Linguistics Many different kinds of logic are directly applicable to formalizing theories in syntax
More informationPage 1 of 11. Curriculum Map: Grade 4 Math Course: Math 4 Sub-topic: General. Grade(s): None specified
Curriculum Map: Grade 4 Math Course: Math 4 Sub-topic: General Grade(s): None specified Unit: Creating a Community of Mathematical Thinkers Timeline: Week 1 The purpose of the Establishing a Community
More informationMeasures of the Location of the Data
OpenStax-CNX module m46930 1 Measures of the Location of the Data OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 The common measures
More informationTU-E2090 Research Assignment in Operations Management and Services
Aalto University School of Science Operations and Service Management TU-E2090 Research Assignment in Operations Management and Services Version 2016-08-29 COURSE INSTRUCTOR: OFFICE HOURS: CONTACT: Saara
More informationCAN PICTORIAL REPRESENTATIONS SUPPORT PROPORTIONAL REASONING? THE CASE OF A MIXING PAINT PROBLEM
CAN PICTORIAL REPRESENTATIONS SUPPORT PROPORTIONAL REASONING? THE CASE OF A MIXING PAINT PROBLEM Christina Misailidou and Julian Williams University of Manchester Abstract In this paper we report on the
More informationGreen Belt Curriculum (This workshop can also be conducted on-site, subject to price change and number of participants)
Green Belt Curriculum (This workshop can also be conducted on-site, subject to price change and number of participants) Notes: 1. We use Mini-Tab in this workshop. Mini-tab is available for free trail
More informationMathematics Success Grade 7
T894 Mathematics Success Grade 7 [OBJECTIVE] The student will find probabilities of compound events using organized lists, tables, tree diagrams, and simulations. [PREREQUISITE SKILLS] Simple probability,
More informationBuild on students informal understanding of sharing and proportionality to develop initial fraction concepts.
Recommendation 1 Build on students informal understanding of sharing and proportionality to develop initial fraction concepts. Students come to kindergarten with a rudimentary understanding of basic fraction
More informationCritical Thinking in the Workplace. for City of Tallahassee Gabrielle K. Gabrielli, Ph.D.
Critical Thinking in the Workplace for City of Tallahassee Gabrielle K. Gabrielli, Ph.D. Purpose The purpose of this training is to provide: Tools and information to help you become better critical thinkers
More informationTHE INFORMATION SYSTEMS ANALYST EXAM AS A PROGRAM ASSESSMENT TOOL: PRE-POST TESTS AND COMPARISON TO THE MAJOR FIELD TEST
THE INFORMATION SYSTEMS ANALYST EXAM AS A PROGRAM ASSESSMENT TOOL: PRE-POST TESTS AND COMPARISON TO THE MAJOR FIELD TEST Donald A. Carpenter, Mesa State College, dcarpent@mesastate.edu Morgan K. Bridge,
More informationEDEXCEL FUNCTIONAL SKILLS PILOT. Maths Level 2. Chapter 7. Working with probability
Working with probability 7 EDEXCEL FUNCTIONAL SKILLS PILOT Maths Level 2 Chapter 7 Working with probability SECTION K 1 Measuring probability 109 2 Experimental probability 111 3 Using tables to find the
More informationGetting Started with Deliberate Practice
Getting Started with Deliberate Practice Most of the implementation guides so far in Learning on Steroids have focused on conceptual skills. Things like being able to form mental images, remembering facts
More informationIntroduction to Causal Inference. Problem Set 1. Required Problems
Introduction to Causal Inference Problem Set 1 Professor: Teppei Yamamoto Due Friday, July 15 (at beginning of class) Only the required problems are due on the above date. The optional problems will not
More informationMath 96: Intermediate Algebra in Context
: Intermediate Algebra in Context Syllabus Spring Quarter 2016 Daily, 9:20 10:30am Instructor: Lauri Lindberg Office Hours@ tutoring: Tutoring Center (CAS-504) 8 9am & 1 2pm daily STEM (Math) Center (RAI-338)
More informationGrade 2: Using a Number Line to Order and Compare Numbers Place Value Horizontal Content Strand
Grade 2: Using a Number Line to Order and Compare Numbers Place Value Horizontal Content Strand Texas Essential Knowledge and Skills (TEKS): (2.1) Number, operation, and quantitative reasoning. The student
More informationThe Indices Investigations Teacher s Notes
The Indices Investigations Teacher s Notes These activities are for students to use independently of the teacher to practise and develop number and algebra properties.. Number Framework domain and stage:
More informationThe lab is designed to remind you how to work with scientific data (including dealing with uncertainty) and to review experimental design.
Name: Partner(s): Lab #1 The Scientific Method Due 6/25 Objective The lab is designed to remind you how to work with scientific data (including dealing with uncertainty) and to review experimental design.
More informationReduce the Failure Rate of the Screwing Process with Six Sigma Approach
Proceedings of the 2014 International Conference on Industrial Engineering and Operations Management Bali, Indonesia, January 7 9, 2014 Reduce the Failure Rate of the Screwing Process with Six Sigma Approach
More informationInstructor: Mario D. Garrett, Ph.D. Phone: Office: Hepner Hall (HH) 100
San Diego State University School of Social Work 610 COMPUTER APPLICATIONS FOR SOCIAL WORK PRACTICE Statistical Package for the Social Sciences Office: Hepner Hall (HH) 100 Instructor: Mario D. Garrett,
More informationCS Machine Learning
CS 478 - Machine Learning Projects Data Representation Basic testing and evaluation schemes CS 478 Data and Testing 1 Programming Issues l Program in any platform you want l Realize that you will be doing
More informationAP Statistics Summer Assignment 17-18
AP Statistics Summer Assignment 17-18 Welcome to AP Statistics. This course will be unlike any other math class you have ever taken before! Before taking this course you will need to be competent in basic
More informationStatewide Framework Document for:
Statewide Framework Document for: 270301 Standards may be added to this document prior to submission, but may not be removed from the framework to meet state credit equivalency requirements. Performance
More informationProbability Therefore (25) (1.33)
Probability We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment,
More informationGrade 6: Correlated to AGS Basic Math Skills
Grade 6: Correlated to AGS Basic Math Skills Grade 6: Standard 1 Number Sense Students compare and order positive and negative integers, decimals, fractions, and mixed numbers. They find multiples and
More informationHow to make your research useful and trustworthy the three U s and the CRITIC
How to make your research useful and trustworthy the three U s and the CRITIC Michael Wood University of Portsmouth Business School http://woodm.myweb.port.ac.uk/sl/researchmethods.htm August 2015 Introduction...
More informationJulia Smith. Effective Classroom Approaches to.
Julia Smith @tessmaths Effective Classroom Approaches to GCSE Maths resits julia.smith@writtle.ac.uk Agenda The context of GCSE resit in a post-16 setting An overview of the new GCSE Key features of a
More informationAn overview of risk-adjusted charts
J. R. Statist. Soc. A (2004) 167, Part 3, pp. 523 539 An overview of risk-adjusted charts O. Grigg and V. Farewell Medical Research Council Biostatistics Unit, Cambridge, UK [Received February 2003. Revised
More informationAn Introduction to Simio for Beginners
An Introduction to Simio for Beginners C. Dennis Pegden, Ph.D. This white paper is intended to introduce Simio to a user new to simulation. It is intended for the manufacturing engineer, hospital quality
More informationEvidence-based Practice: A Workshop for Training Adult Basic Education, TANF and One Stop Practitioners and Program Administrators
Evidence-based Practice: A Workshop for Training Adult Basic Education, TANF and One Stop Practitioners and Program Administrators May 2007 Developed by Cristine Smith, Beth Bingman, Lennox McLendon and
More informationNote: Principal version Modification Amendment Modification Amendment Modification Complete version from 1 October 2014
Note: The following curriculum is a consolidated version. It is legally non-binding and for informational purposes only. The legally binding versions are found in the University of Innsbruck Bulletins
More informationLaw Professor's Proposal for Reporting Sexual Violence Funded in Virginia, The Hatchet
Law Professor John Banzhaf s Novel Approach for Investigating and Adjudicating Allegations of Rapes and Other Sexual Assaults at Colleges About to be Tested in Virginia Law Professor's Proposal for Reporting
More informationOn-the-Fly Customization of Automated Essay Scoring
Research Report On-the-Fly Customization of Automated Essay Scoring Yigal Attali Research & Development December 2007 RR-07-42 On-the-Fly Customization of Automated Essay Scoring Yigal Attali ETS, Princeton,
More informationActivity 2 Multiplying Fractions Math 33. Is it important to have common denominators when we multiply fraction? Why or why not?
Activity Multiplying Fractions Math Your Name: Partners Names:.. (.) Essential Question: Think about the question, but don t answer it. You will have an opportunity to answer this question at the end of
More informationUnraveling symbolic number processing and the implications for its association with mathematics. Delphine Sasanguie
Unraveling symbolic number processing and the implications for its association with mathematics Delphine Sasanguie 1. Introduction Mapping hypothesis Innate approximate representation of number (ANS) Symbols
More informationClassifying combinations: Do students distinguish between different types of combination problems?
Classifying combinations: Do students distinguish between different types of combination problems? Elise Lockwood Oregon State University Nicholas H. Wasserman Teachers College, Columbia University William
More informationSimple Random Sample (SRS) & Voluntary Response Sample: Examples: A Voluntary Response Sample: Examples: Systematic Sample Best Used When
Simple Random Sample (SRS) & Voluntary Response Sample: In statistics, a simple random sample is a group of people who have been chosen at random from the general population. A simple random sample is
More informationCorpus Linguistics (L615)
(L615) Basics of Markus Dickinson Department of, Indiana University Spring 2013 1 / 23 : the extent to which a sample includes the full range of variability in a population distinguishes corpora from archives
More informationEffect of Cognitive Apprenticeship Instructional Method on Auto-Mechanics Students
Effect of Cognitive Apprenticeship Instructional Method on Auto-Mechanics Students Abubakar Mohammed Idris Department of Industrial and Technology Education School of Science and Science Education, Federal
More informationUsing Proportions to Solve Percentage Problems I
RP7-1 Using Proportions to Solve Percentage Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by
More informationLevel 1 Mathematics and Statistics, 2015
91037 910370 1SUPERVISOR S Level 1 Mathematics and Statistics, 2015 91037 Demonstrate understanding of chance and data 9.30 a.m. Monday 9 November 2015 Credits: Four Achievement Achievement with Merit
More informationRote rehearsal and spacing effects in the free recall of pure and mixed lists. By: Peter P.J.L. Verkoeijen and Peter F. Delaney
Rote rehearsal and spacing effects in the free recall of pure and mixed lists By: Peter P.J.L. Verkoeijen and Peter F. Delaney Verkoeijen, P. P. J. L, & Delaney, P. F. (2008). Rote rehearsal and spacing
More informationBusiness Analytics and Information Tech COURSE NUMBER: 33:136:494 COURSE TITLE: Data Mining and Business Intelligence
Business Analytics and Information Tech COURSE NUMBER: 33:136:494 COURSE TITLE: Data Mining and Business Intelligence COURSE DESCRIPTION This course presents computing tools and concepts for all stages
More informationDeveloping a concrete-pictorial-abstract model for negative number arithmetic
Developing a concrete-pictorial-abstract model for negative number arithmetic Jai Sharma and Doreen Connor Nottingham Trent University Research findings and assessment results persistently identify negative
More informationSuccessfully Flipping a Mathematics Classroom
2014 Hawaii University International Conferences Science, Technology, Engineering, Math & Education June 16, 17, & 18 2014 Ala Moana Hotel, Honolulu, Hawaii Successfully Flipping a Mathematics Classroom
More informationMissouri Mathematics Grade-Level Expectations
A Correlation of to the Grades K - 6 G/M-223 Introduction This document demonstrates the high degree of success students will achieve when using Scott Foresman Addison Wesley Mathematics in meeting the
More informationGrade 5 + DIGITAL. EL Strategies. DOK 1-4 RTI Tiers 1-3. Flexible Supplemental K-8 ELA & Math Online & Print
Standards PLUS Flexible Supplemental K-8 ELA & Math Online & Print Grade 5 SAMPLER Mathematics EL Strategies DOK 1-4 RTI Tiers 1-3 15-20 Minute Lessons Assessments Consistent with CA Testing Technology
More informationCritical Thinking in Everyday Life: 9 Strategies
Critical Thinking in Everyday Life: 9 Strategies Most of us are not what we could be. We are less. We have great capacity. But most of it is dormant; most is undeveloped. Improvement in thinking is like
More informationMathematics subject curriculum
Mathematics subject curriculum Dette er ei omsetjing av den fastsette læreplanteksten. Læreplanen er fastsett på Nynorsk Established as a Regulation by the Ministry of Education and Research on 24 June
More informationRule-based Expert Systems
Rule-based Expert Systems What is knowledge? is a theoretical or practical understanding of a subject or a domain. is also the sim of what is currently known, and apparently knowledge is power. Those who
More informationAn Empirical Analysis of the Effects of Mexican American Studies Participation on Student Achievement within Tucson Unified School District
An Empirical Analysis of the Effects of Mexican American Studies Participation on Student Achievement within Tucson Unified School District Report Submitted June 20, 2012, to Willis D. Hawley, Ph.D., Special
More informationVOL. 3, NO. 5, May 2012 ISSN Journal of Emerging Trends in Computing and Information Sciences CIS Journal. All rights reserved.
Exploratory Study on Factors that Impact / Influence Success and failure of Students in the Foundation Computer Studies Course at the National University of Samoa 1 2 Elisapeta Mauai, Edna Temese 1 Computing
More informationARSENAL OF DEMOCRACY
ARSENAL OF DEMOCRACY Preview of Main Idea Between 1910 and 1930, Detroit became a major industrial center of the United States, indeed, the world. The ability of the automobile industry to produce an extraordinarily
More informationACTION LEARNING: AN INTRODUCTION AND SOME METHODS INTRODUCTION TO ACTION LEARNING
ACTION LEARNING: AN INTRODUCTION AND SOME METHODS INTRODUCTION TO ACTION LEARNING Action learning is a development process. Over several months people working in a small group, tackle important organisational
More informationIndividual Differences & Item Effects: How to test them, & how to test them well
Individual Differences & Item Effects: How to test them, & how to test them well Individual Differences & Item Effects Properties of subjects Cognitive abilities (WM task scores, inhibition) Gender Age
More informationActivities, Exercises, Assignments Copyright 2009 Cem Kaner 1
Patterns of activities, iti exercises and assignments Workshop on Teaching Software Testing January 31, 2009 Cem Kaner, J.D., Ph.D. kaner@kaner.com Professor of Software Engineering Florida Institute of
More informationState University of New York at Buffalo INTRODUCTION TO STATISTICS PSC 408 Fall 2015 M,W,F 1-1:50 NSC 210
1 State University of New York at Buffalo INTRODUCTION TO STATISTICS PSC 408 Fall 2015 M,W,F 1-1:50 NSC 210 Dr. Michelle Benson mbenson2@buffalo.edu Office: 513 Park Hall Office Hours: Mon & Fri 10:30-12:30
More informationPhysics 270: Experimental Physics
2017 edition Lab Manual Physics 270 3 Physics 270: Experimental Physics Lecture: Lab: Instructor: Office: Email: Tuesdays, 2 3:50 PM Thursdays, 2 4:50 PM Dr. Uttam Manna 313C Moulton Hall umanna@ilstu.edu
More informationMaximizing Learning Through Course Alignment and Experience with Different Types of Knowledge
Innov High Educ (2009) 34:93 103 DOI 10.1007/s10755-009-9095-2 Maximizing Learning Through Course Alignment and Experience with Different Types of Knowledge Phyllis Blumberg Published online: 3 February
More informationCurriculum Design Project with Virtual Manipulatives. Gwenanne Salkind. George Mason University EDCI 856. Dr. Patricia Moyer-Packenham
Curriculum Design Project with Virtual Manipulatives Gwenanne Salkind George Mason University EDCI 856 Dr. Patricia Moyer-Packenham Spring 2006 Curriculum Design Project with Virtual Manipulatives Table
More informationSoftware Security: Integrating Secure Software Engineering in Graduate Computer Science Curriculum
Software Security: Integrating Secure Software Engineering in Graduate Computer Science Curriculum Stephen S. Yau, Fellow, IEEE, and Zhaoji Chen Arizona State University, Tempe, AZ 85287-8809 {yau, zhaoji.chen@asu.edu}
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch 2 Test Remediation Work Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) High temperatures in a certain
More informationArtificial Neural Networks written examination
1 (8) Institutionen för informationsteknologi Olle Gällmo Universitetsadjunkt Adress: Lägerhyddsvägen 2 Box 337 751 05 Uppsala Artificial Neural Networks written examination Monday, May 15, 2006 9 00-14
More informationProbability estimates in a scenario tree
101 Chapter 11 Probability estimates in a scenario tree An expert is a person who has made all the mistakes that can be made in a very narrow field. Niels Bohr (1885 1962) Scenario trees require many numbers.
More informationIBM Software Group. Mastering Requirements Management with Use Cases Module 6: Define the System
IBM Software Group Mastering Requirements Management with Use Cases Module 6: Define the System 1 Objectives Define a product feature. Refine the Vision document. Write product position statement. Identify
More informationWhat is related to student retention in STEM for STEM majors? Abstract:
What is related to student retention in STEM for STEM majors? Abstract: The purpose of this study was look at the impact of English and math courses and grades on retention in the STEM major after one
More informationThe KAM project: Mathematics in vocational subjects*
The KAM project: Mathematics in vocational subjects* Leif Maerker The KAM project is a project which used interdisciplinary teams in an integrated approach which attempted to connect the mathematical learning
More informationLab 1 - The Scientific Method
Lab 1 - The Scientific Method As Biologists we are interested in learning more about life. Through observations of the living world we often develop questions about various phenomena occurring around us.
More informationPolitics and Society Curriculum Specification
Leaving Certificate Politics and Society Curriculum Specification Ordinary and Higher Level 1 September 2015 2 Contents Senior cycle 5 The experience of senior cycle 6 Politics and Society 9 Introduction
More informationStatistical Analysis of Climate Change, Renewable Energies, and Sustainability An Independent Investigation for Introduction to Statistics
5/22/2012 Statistical Analysis of Climate Change, Renewable Energies, and Sustainability An Independent Investigation for Introduction to Statistics College of Menominee Nation & University of Wisconsin
More informationMathematics. Mathematics
Mathematics Program Description Successful completion of this major will assure competence in mathematics through differential and integral calculus, providing an adequate background for employment in
More informationPreliminary Chapter survey experiment an observational study that is not a survey
1 Preliminary Chapter P.1 Getting data from Jamie and her friends is convenient, but it does not provide a good snapshot of the opinions held by all young people. In short, Jamie and her friends are not
More informationSector Differences in Student Learning: Differences in Achievement Gains Across School Years and During the Summer
Catholic Education: A Journal of Inquiry and Practice Volume 7 Issue 2 Article 6 July 213 Sector Differences in Student Learning: Differences in Achievement Gains Across School Years and During the Summer
More informationAn Empirical and Computational Test of Linguistic Relativity
An Empirical and Computational Test of Linguistic Relativity Kathleen M. Eberhard* (eberhard.1@nd.edu) Matthias Scheutz** (mscheutz@cse.nd.edu) Michael Heilman** (mheilman@nd.edu) *Department of Psychology,
More informationSan José State University Department of Marketing and Decision Sciences BUS 90-06/ Business Statistics Spring 2017 January 26 to May 16, 2017
San José State University Department of Marketing and Decision Sciences BUS 90-06/30174- Business Statistics Spring 2017 January 26 to May 16, 2017 Course and Contact Information Instructor: Office Location:
More informationSan José State University Department of Psychology PSYC , Human Learning, Spring 2017
San José State University Department of Psychology PSYC 155-03, Human Learning, Spring 2017 Instructor: Valerie Carr Office Location: Dudley Moorhead Hall (DMH), Room 318 Telephone: (408) 924-5630 Email:
More informationDublin City Schools Mathematics Graded Course of Study GRADE 4
I. Content Standard: Number, Number Sense and Operations Standard Students demonstrate number sense, including an understanding of number systems and reasonable estimates using paper and pencil, technology-supported
More informationMachine Learning and Data Mining. Ensembles of Learners. Prof. Alexander Ihler
Machine Learning and Data Mining Ensembles of Learners Prof. Alexander Ihler Ensemble methods Why learn one classifier when you can learn many? Ensemble: combine many predictors (Weighted) combina
More informationAGENDA LEARNING THEORIES LEARNING THEORIES. Advanced Learning Theories 2/22/2016
AGENDA Advanced Learning Theories Alejandra J. Magana, Ph.D. admagana@purdue.edu Introduction to Learning Theories Role of Learning Theories and Frameworks Learning Design Research Design Dual Coding Theory
More information