DOCSLIB.ORG

Statistics 110

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Statistics 110 Cedar Crest College Statistics 110 Lecture Notes Author: E-Mail Address: James Hammer [email protected] June 23, 2015 Preface This is meant as a teaching aid. It can be freely distributed and edited in any way. For a copy of the LATEX document, please email the author. ii Contents 1 Nature of Probability and Statistics1 1.1 Descriptive and Inferential Statistics......................1 1.2 Variables and Types of Data...........................3 1.3 Data Collection and Sampling Techniques...................5 1.4 Experimental Design...............................6 2 Frequency Distribution and Graphs7 2.1 Organizing Data.................................7 2.2 Histograms, Frequency Polygons, and Ogives.................. 10 2.3 Other Types of Graphs.............................. 12 3 Data Description 19 3.1 Measures of Central Tendencies......................... 19 3.2 Measures of Variation............................... 26 3.3 Measures of Position............................... 33 3.4 Exploratory Data Analysis............................ 38 4 Probability and Counting Rules 41 4.1 Sample Spaces and Probability......................... 41 4.2 The Addition Rules for Probability....................... 47 4.3 The Multiplication Rules & Conditional Probability.............. 49 4.4 Counting Rules.................................. 56 4.5 Probability and Counting Rules......................... 62 5 Discrete Probability Distributions 65 5.1 Probability Distributions............................. 65 5.2 Mean, Variance, Standard Deviation, and Expectation............ 67 5.3 The Binomial Distribution............................ 72 6 The Normal Distribution 75 6.1 Normal Distributions............................... 75 6.2 Applications of the Normal Distribution.................... 77 6.3 The Central Limit Theorem........................... 80 iii 7 Confidence Intervals and Sample Size 83 7.1 Confidence Intervals for the Mean Shen σ is Known.............. 83 7.2 Confidence Intervals for the Mean When σ is Unknown............ 87 7.3 Confidence Intervals and Sample Size for Populations............. 89 8 Hypothesis Testing 93 8.1 Traditional Method................................ 93 8.2 z Test for a Mean................................. 95 8.3 t Test for a Mean................................. 98 8.4 z Test for a Proportion.............................. 100 10 Correlation and Regression 103 10.1 Scatter Plots and Correlation.......................... 103 iv Chapter 1 Nature of Probability and Statistics 1.1 Descriptive and Inferential Statistics Definition 1 (Statistics). Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data. Definition 2 (Variable). A variable is a characteristic of attribute that can assume different values. Definition 3 (Data). The data are the values (measurements or observations) that the variables can assume. Variables whose values are determined by chance are called random variables. Definition 4 (Population). A population consists of all subjects (human or otherwise) that are being studied. Definition 5 (Sample). A sample is a group of subjects selected from a population. Definition 6 (Descriptive Statistics). Descriptive statistics consists of the collection, organization, summarization, and presentation of data. Definition 7 (Inferential Statistics). Inferential statistics consists of generalizing from sample to populations, performing estimations and hypothesis tests, determining relationship among variables and making predictions. Definition 8 (Probability). Probability is the measure of the likeliness that an event will occur. Probability is quantified as a number between 0 and 1. The higher the probability of an event, the more likely the event will occur. Definition 9 (Hypothesis Testing). The area of inferential statistics called hypothesis testing is a decision-making process for evaluating claims about a population based on information obtained from samples. 1 Hammer 2 Statistics 110 Lecture Notes Problem 10. Determine whether descriptive statistics or inferential statistics were used. Explain why. a. The average jackpot for the top five lottery winners was $367:6 million. b. A study done by the American Academy of Neurology suggested that older people who had a high caloric diet more than doubled their risk of memory loss. c. Based on a survey of 9317 consumers done by the National Retail Federation, the average amount that consumers spent on Valentines Day in 2011 was $116. d. Scientists at the University of Oxford in England found that a good laugh significantly raises a person's pain level tolerance. Suggested Problems: 1-19. Cedar Crest College June 23, 2015 1.2. Variables and Types of Data Hammer 3 1.2 Variables and Types of Data Definition 11 (Qualitative Variables). Qualitative variables are variables that have dis- tinct categories according to some characteristic or attribute. Definition 12 (Quantitative Variables). Quantitative variables are variables that can be counted or measured. Definition 13 (Discrete Variables). Discrete variables assume values that can be counted. Definition 14 (Continuous Variables). Continuous variables can assume an infinite num- ber of values between any two specific values. They are obtained by measuring. They often include fractions and decimals. Problem 15. Classify each of the variables as a discrete variable or as a continuous variable. a. The highest wind speed of a hurricane. b. The weight of baggage on an airplane. c. The number of pages in these lecture notes. d. The amount of money a person spends per year for online purchases. June 23, 2015 Cedar Crest College Hammer 4 Statistics 110 Lecture Notes Definition 16 (Boundary). The boundary of a number is defined as a class in which a data value would be placed before the data values are rounded. Problem 17. Find the boundaries of each variable. a.8 :4 quarts b. 138 mmHg c. 137:63 mg/dL Definition 18 (Nominal Level of Measurement). The nominal level of measurement classifies data into mutually exclusive (nonoverlapping) categories in which no order or rank- ing can be imposed on the data. Definition 19 (Ordinal Level of Measurement). The ordinal level of measurement clas- sifies data into categories that can be ranked; however, precise differences between the ranks do not exist. Definition 20 (Interval Level of Measurement). The interval level of measurement ranks data, and precise differences between units of measure do exist; however, there is no meaningful zero. Definition 21 (Ratio Level of Measurement). The ratio level of measurement possesses all of the characteristics of interval measurement, and there exists a true zero. In addi- tion, true ratios exist when the same variable is measured on two different members of the population. Problem 22. What level of measurement would be used to measure each variable? a. The age of patients in a local hospital. b. The rating of movies released this month. c. Colors of athletic shirts sold by the Iron Pigs. d. Temperatures of hot tubs in local health clubs. Suggested Problems: 5 − 22 Cedar Crest College June 23, 2015 1.3. Data Collection and Sampling Techniques Hammer 5 1.3 Data Collection and Sampling Techniques Definition 23 (Random Sample). A random Sample is a sample in which all members of the population have an equal chance of being selected. Definition 24 (Systematic Sample). A systematic sample is a sample obtained by select- ing every kth member of the population. Definition 25 (Stratified Sample). A stratified sample is a sample obtained by dividing the population into subgroups or strata according to some characteristic relevant to the study. The subjects are selected from each subgroup. Definition 26 (Cluster Sample). A cluster sample is obtained by dividing the population into sections or clusters and then selecting one or more clusters and using all members in the cluster(s) as the members of the sample. Definition 27 (Sampling Error). Sampling error is the difference between the results obtained from a sample and the results obtained from the population from which the sample was selected. Definition 28 (Nonsampling Error). A nonsampling error occurs when the data are obtained erroneously or the sample is biased (non-representative). Problem 29. State which sampling method was used in the following: a. Out of 10 hospitals in the municipality, a researcher selects one and collects records for a 24-hour period on the types of emergencies that were treated here. b. A researcher divides a group of students according to gender; major field; and low, average, and high grade point average. Then she randomly selects six students from each group to answer questions in the survey. c. The subscribers to a magazine are numbered. Then a sample of these people is selected using random numbers. d. Every 10th bottle of Super-Duper Cola is selected, and the amount of liquid in the bottle is measured. The purpose is to see if the machines that fill the bottles are working properly. Suggested Problems: 11 − 16 June 23, 2015 Cedar Crest College Hammer 6 Statistics 110 Lecture Notes 1.4 Experimental Design Definition 30 (Observational Study). In an observational study, the researcher merely observes what is happening or what has happened in the past and tries to draw conclusions based on these observations. Definition 31 (Experimental Study). In an experimental study, the researcher manip- ulates one of the variables and tries to determine how the manipulation influences other variables. Definition 32 (Variable Types). The independent variable in an
Load more

Recommended publications
  • Programming in Stata
    Programming and Post-Estimation • Bootstrapping • Monte Carlo • Post-Estimation Simulation (Clarify) • Extending Clarify to Other Models – Censored Probit Example What is Bootstrapping? • A computer-simulated nonparametric technique for making inferences about a population parameter based on sample statistics. • If the sample is a good approximation of the population, the sampling distribution of interest can be estimated by generating a large number of new samples from the original. • Useful when no analytic formula for the sampling distribution is available. B 2 ˆ B B b How do I do it? SE B = s(x )- s(x )/ B /(B -1) å[ åb =1 ] b=1 1. Obtain a Sample from the population of interest. Call this x = (x1, x2, … , xn). 2. Re-sample based on x by randomly sampling with replacement from it. 3. Generate many such samples, x1, x2, …, xB – each of length n. 4. Estimate the desired parameter in each sample, s(x1), s(x2), …, s(xB). 5. For instance the bootstrap estimate of the standard error is the standard deviation of the bootstrap replications. Example: Standard Error of a Sample Mean Canned in Stata use "I:\general\PRISM Programming\auto.dta", clear Type: sum mpg Example: Standard Error of a Sample Mean Canned in Stata Type: bootstrap "summarize mpg" r(mean), reps(1000) » 21.2973±1.96*.6790806 x B - x Example: Difference of Medians Test Type: sum length, detail Type: return list Example: Difference of Medians Test Example: Difference of Medians Test Example: Difference of Medians Test Type: bootstrap "mymedian" r(diff), reps(1000) The medians are not very different.
    [Show full text]
  • The Cascade Bayesian Approach: Prior Transformation for a Controlled Integration of Internal Data, External Data and Scenarios
    risks Article The Cascade Bayesian Approach: Prior Transformation for a Controlled Integration of Internal Data, External Data and Scenarios Bertrand K. Hassani 1,2,3,* and Alexis Renaudin 4 1 University College London Computer Science, 66-72 Gower Street, London WC1E 6EA, UK 2 LabEx ReFi, Université Paris 1 Panthéon-Sorbonne, CESCES, 106 bd de l’Hôpital, 75013 Paris, France 3 Capegemini Consulting, Tour Europlaza, 92400 Paris-La Défense, France 4 Aon Benfield, The Aon Centre, 122 Leadenhall Street, London EC3V 4AN, UK; [email protected] * Correspondence: [email protected]; Tel.: +44-(0)753-016-7299 Received: 14 March 2018; Accepted: 23 April 2018; Published: 27 April 2018 Abstract: According to the last proposals of the Basel Committee on Banking Supervision, banks or insurance companies under the advanced measurement approach (AMA) must use four different sources of information to assess their operational risk capital requirement. The fourth includes ’business environment and internal control factors’, i.e., qualitative criteria, whereas the three main quantitative sources available to banks for building the loss distribution are internal loss data, external loss data and scenario analysis. This paper proposes an innovative methodology to bring together these three different sources in the loss distribution approach (LDA) framework through a Bayesian strategy. The integration of the different elements is performed in two different steps to ensure an internal data-driven model is obtained. In the first step, scenarios are used to inform the prior distributions and external data inform the likelihood component of the posterior function. In the second step, the initial posterior function is used as the prior distribution and the internal loss data inform the likelihood component of the second posterior function.
    [Show full text]
  • What Is Bayesian Inference? Bayesian Inference Is at the Core of the Bayesian Approach, Which Is an Approach That Allows Us to Represent Uncertainty As a Probability
    Learn to Use Bayesian Inference in SPSS With Data From the National Child Measurement Programme (2016–2017) © 2019 SAGE Publications Ltd. All Rights Reserved. This PDF has been generated from SAGE Research Methods Datasets. SAGE SAGE Research Methods Datasets Part 2019 SAGE Publications, Ltd. All Rights Reserved. 2 Learn to Use Bayesian Inference in SPSS With Data From the National Child Measurement Programme (2016–2017) Student Guide Introduction This example dataset introduces Bayesian Inference. Bayesian statistics (the general name for all Bayesian-related topics, including inference) has become increasingly popular in recent years, due predominantly to the growth of evermore powerful and sophisticated statistical software. However, Bayesian statistics grew from the ideas of an English mathematician, Thomas Bayes, who lived and worked in the first half of the 18th century and have been refined and adapted by statisticians and mathematicians ever since. Despite its longevity, the Bayesian approach did not become mainstream: the Frequentist approach was and remains the dominant means to conduct statistical analysis. However, there is a renewed interest in Bayesian statistics, part prompted by software development and part by a growing critique of the limitations of the null hypothesis significance testing which dominates the Frequentist approach. This renewed interest can be seen in the incorporation of Bayesian analysis into mainstream statistical software, such as, IBM® SPSS® and in many major statistics text books. Bayesian Inference is at the heart of Bayesian statistics and is different from Frequentist approaches due to how it views probability. In the Frequentist approach, probability is the product of the frequency of random events occurring Page 2 of 19 Learn to Use Bayesian Inference in SPSS With Data From the National Child Measurement Programme (2016–2017) SAGE SAGE Research Methods Datasets Part 2019 SAGE Publications, Ltd.
    [Show full text]
  • Basic Econometrics / Statistics Statistical Distributions: Normal, T, Chi-Sq, & F
    Basic Econometrics / Statistics Statistical Distributions: Normal, T, Chi-Sq, & F Course : Basic Econometrics : HC43 / Statistics B.A. Hons Economics, Semester IV/ Semester III Delhi University Course Instructor: Siddharth Rathore Assistant Professor Economics Department, Gargi College Siddharth Rathore guj75845_appC.qxd 4/16/09 12:41 PM Page 461 APPENDIX C SOME IMPORTANT PROBABILITY DISTRIBUTIONS In Appendix B we noted that a random variable (r.v.) can be described by a few characteristics, or moments, of its probability function (PDF or PMF), such as the expected value and variance. This, however, presumes that we know the PDF of that r.v., which is a tall order since there are all kinds of random variables. In practice, however, some random variables occur so frequently that statisticians have determined their PDFs and documented their properties. For our purpose, we will consider only those PDFs that are of direct interest to us. But keep in mind that there are several other PDFs that statisticians have studied which can be found in any standard statistics textbook. In this appendix we will discuss the following four probability distributions: 1. The normal distribution 2. The t distribution 3. The chi-square (␹2 ) distribution 4. The F distribution These probability distributions are important in their own right, but for our purposes they are especially important because they help us to find out the probability distributions of estimators (or statistics), such as the sample mean and sample variance. Recall that estimators are random variables. Equipped with that knowledge, we will be able to draw inferences about their true population values.
    [Show full text]
  • Weighted Means and Grouped Data on the Graphing Calculator
    Weighted Means and Grouped Data on the Graphing Calculator Weighted Means. Suppose that, in a recent month, a bank customer had $2500 in his account for 5 days, $1900 for 2 days, $1650 for 3 days, $1375 for 9 days, $1200 for 1 day, $900 for 6 days, and $675 for 4 days. To calculate the average balance for that month, you would use a weighted mean: ∑ ̅ ∑ To do this automatically on a calculator, enter the account balances in L1 and the number of days (weight) in L2: As before, go to STAT CALC and 1:1-Var Stats. This time, type L1, L2 after 1-Var Stats and ENTER: The weighted (sample) mean is ̅ (so that the average balance for the month is $1430.83.) We can also see that the weighted sample standard deviation is . Estimating the Mean and Standard Deviation from a Frequency Distribution. If your data is organized into a frequency distribution, you can still estimate the mean and standard deviation. For example, suppose that we are given only a frequency distribution of the heights of the 30 males instead of the list of individual heights: Height (in.) Frequency (f) 62 - 64 3 65- 67 7 68 - 70 9 71 - 73 8 74 - 76 3 ∑ n = 30 We can just calculate the midpoint of each height class and use that midpoint to represent the class. We then find the (weighted) mean and standard deviation for the distribution of midpoints with the given frequencies as in the example above: Height (in.) Height Class Frequency (f) Midpoint 62 - 64 (62 + 64)/2 = 63 3 65- 67 (65 + 67)/2 = 66 7 68 - 70 69 9 71 - 73 72 8 74 - 76 75 3 ∑ n = 30 The approximate sample mean of the distribution is ̅ , and the approximate sample standard deviation of the distribution is .
    [Show full text]
  • Practical Meta-Analysis -- Lipsey & Wilson Overview
    Practical Meta-Analysis -- Lipsey & Wilson The Great Debate • 1952: Hans J. Eysenck concluded that there were no favorable Practical Meta-Analysis effects of psychotherapy, starting a raging debate • 20 years of evaluation research and hundreds of studies failed David B. Wilson to resolve the debate American Evaluation Association • 1978: To proved Eysenck wrong, Gene V. Glass statistically Orlando, Florida, October 3, 1999 aggregate the findings of 375 psychotherapy outcome studies • Glass (and colleague Smith) concluded that psychotherapy did indeed work • Glass called his method “meta-analysis” 1 2 The Emergence of Meta-Analysis The Logic of Meta-Analysis • Ideas behind meta-analysis predate Glass’ work by several • Traditional methods of review focus on statistical significance decades testing – R. A. Fisher (1944) • Significance testing is not well suited to this task • “When a number of quite independent tests of significance have been made, it sometimes happens that although few or none can be – highly dependent on sample size claimed individually as significant, yet the aggregate gives an – null finding does not carry to same “weight” as a significant impression that the probabilities are on the whole lower than would often have been obtained by chance” (p. 99). finding • Source of the idea of cumulating probability values • Meta-analysis changes the focus to the direction and – W. G. Cochran (1953) magnitude of the effects across studies • Discusses a method of averaging means across independent studies – Isn’t this what we are
    [Show full text]
  • Lecture 3: Measure of Central Tendency
    Lecture 3: Measure of Central Tendency Donglei Du ([email protected]) Faculty of Business Administration, University of New Brunswick, NB Canada Fredericton E3B 9Y2 Donglei Du (UNB) ADM 2623: Business Statistics 1 / 53 Table of contents 1 Measure of central tendency: location parameter Introduction Arithmetic Mean Weighted Mean (WM) Median Mode Geometric Mean Mean for grouped data The Median for Grouped Data The Mode for Grouped Data 2 Dicussion: How to lie with averges? Or how to defend yourselves from those lying with averages? Donglei Du (UNB) ADM 2623: Business Statistics 2 / 53 Section 1 Measure of central tendency: location parameter Donglei Du (UNB) ADM 2623: Business Statistics 3 / 53 Subsection 1 Introduction Donglei Du (UNB) ADM 2623: Business Statistics 4 / 53 Introduction Characterize the average or typical behavior of the data. There are many types of central tendency measures: Arithmetic mean Weighted arithmetic mean Geometric mean Median Mode Donglei Du (UNB) ADM 2623: Business Statistics 5 / 53 Subsection 2 Arithmetic Mean Donglei Du (UNB) ADM 2623: Business Statistics 6 / 53 Arithmetic Mean The Arithmetic Mean of a set of n numbers x + ::: + x AM = 1 n n Arithmetic Mean for population and sample N P xi µ = i=1 N n P xi x¯ = i=1 n Donglei Du (UNB) ADM 2623: Business Statistics 7 / 53 Example Example: A sample of five executives received the following bonuses last year ($000): 14.0 15.0 17.0 16.0 15.0 Problem: Determine the average bonus given last year. Solution: 14 + 15 + 17 + 16 + 15 77 x¯ = = = 15:4: 5 5 Donglei Du (UNB) ADM 2623: Business Statistics 8 / 53 Example Example: the weight example (weight.csv) The R code: weight <- read.csv("weight.csv") sec_01A<-weight$Weight.01A.2013Fall # Mean mean(sec_01A) ## [1] 155.8548 Donglei Du (UNB) ADM 2623: Business Statistics 9 / 53 Will Rogers phenomenon Consider two sets of IQ scores of famous people.
    [Show full text]
  • Statistical Power in Meta-Analysis Jin Liu University of South Carolina - Columbia
    University of South Carolina Scholar Commons Theses and Dissertations 12-14-2015 Statistical Power in Meta-Analysis Jin Liu University of South Carolina - Columbia Follow this and additional works at: https://scholarcommons.sc.edu/etd Part of the Educational Psychology Commons Recommended Citation Liu, J.(2015). Statistical Power in Meta-Analysis. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/3221 This Open Access Dissertation is brought to you by Scholar Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. STATISTICAL POWER IN META-ANALYSIS by Jin Liu Bachelor of Arts Chongqing University of Arts and Sciences, 2009 Master of Education University of South Carolina, 2012 Submitted in Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy in Educational Psychology and Research College of Education University of South Carolina 2015 Accepted by: Xiaofeng Liu, Major Professor Christine DiStefano, Committee Member Robert Johnson, Committee Member Brian Habing, Committee Member Lacy Ford, Senior Vice Provost and Dean of Graduate Studies © Copyright by Jin Liu, 2015 All Rights Reserved. ii ACKNOWLEDGEMENTS I would like to express my thanks to all committee members who continuously supported me in the dissertation work. My dissertation chair, Xiaofeng Liu, always inspired and trusted me during this study. I could not finish my dissertation at this time without his support. Brian Habing helped me with R coding and research design. I could not finish the coding process so quickly without his instruction. I appreciate the encouragement from Christine DiStefano and Robert Johnson.
    [Show full text]
  • Descriptive Statistics Frequency Distributions and Their Graphs
    Chapter 2 Descriptive Statistics § 2.1 Frequency Distributions and Their Graphs Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number in each class. The frequency f of a class is the number of data points in the class. Class Frequency, f 1 – 4 4 LowerUpper Class 5 – 8 5 Limits 9 – 12 3 Frequencies 13 – 16 4 17 – 20 2 Larson & Farber, Elementary Statistics: Picturing the World , 3e 3 1 Frequency Distributions The class width is the distance between lower (or upper) limits of consecutive classes. Class Frequency, f 1 – 4 4 5 – 1 = 4 5 – 8 5 9 – 5 = 4 9 – 12 3 13 – 9 = 4 13 – 16 4 17 – 13 = 4 17 – 20 2 The class width is 4. The range is the difference between the maximum and minimum data entries. Larson & Farber, Elementary Statistics: Picturing the World , 3e 4 Constructing a Frequency Distribution Guidelines 1. Decide on the number of classes to include. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns. 2. Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number. 3. Find the class limits. You can use the minimum entry as the lower limit of the first class. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper class limits. 4.
    [Show full text]
  • Flood Frequency Analysis Using Copula with Mixed Marginal Distributions
    Flood frequency analysis using copula with mixed marginal distributions Project Report, Aug. 2007 Project Report August 2007 - 1 - Flood frequency analysis using copula with mixed marginal distributions Project Report, Aug. 2007 Prepared by and - 2 - Flood frequency analysis using copula with mixed marginal distributions Project Report, Aug. 2007 Abstract...................................................................................................................7 I. Introduction......................................................................................................8 II. Nonparametric method of estimating marginal distribution.................................14 II.1 Univariate kernel density estimation...........................................................14 II.2 Orthonormal series method.......................................................................17 III. The copula function.........................................................................................19 IV. Multivariate flood frequency analysis................................................................23 IV.1 Details of the case study............................................................................25 IV.2 Associations among flood characteristics.....................................................26 IV.3 Marginal distributions of flood characteristics..............................................29 IV.3.1 Parametric estimation.........................................................................30 IV.3.2 Nonparametric estimation...................................................................32
    [Show full text]
  • Formulas Used by the “Practical Meta-Analysis Effect Size Calculator”
    Formulas Used by the “Practical Meta-Analysis Effect Size Calculator” David B. Wilson George Mason University April 27, 2017 1 Introduction This document provides the equations used by the on-line Practical Meta- Analysis Effect Size Calculator, available at: http://www.campbellcollaboration.org/resources/effect_size _input.php and http://cebcp.org/practical-meta-analysis-effect-size-calculator The calculator is a companion to the book I co-authored with Mark Lipsey, ti- tled “Practical Meta-Analysis,” published by Sage in 2001 (ISBN-10: 9780761921684). This calculator computes four effect size types: 1. Standardized mean difference (d) 2. Correlation coefficient (r) 3. Odds ratio (OR) 4. Risk ratio (RR) In addition to computing effect sizes, the calculator computes the associated variance and 95% confidence interval. The variance can be used to generate the w = 1=v weight for inverse-variancep weighted meta-analysis ( ) or the standard error (se = v). 2 Notation Notation commonly used through this document is presented below. Other no- tation will be defined as needed. 1 Formulas for “Practical Meta–Analysis Online Calculator” d Standardized mean difference effect size n1 Sample size for group 1 n2 Sample size for group 2 N Total sample size s1 Standard deviation for group 1 s2 Standard deviation for group 2 s Full sample standard deviation se1 Standard error for the mean of group 1 se2 Standard error for the mean of group 2 sgain Gain score standard deviation X1 Mean for group 1 X2 Mean for group 2 Xij Mean for subgroup i of group j ∆1
    [Show full text]
  • Categorical Data Analysis
    Categorical Data Analysis Related topics/headings: Categorical data analysis; or, Nonparametric statistics; or, chi-square tests for the analysis of categorical data. OVERVIEW For our hypothesis testing so far, we have been using parametric statistical methods. Parametric methods (1) assume some knowledge about the characteristics of the parent population (e.g. normality) (2) require measurement equivalent to at least an interval scale (calculating a mean or a variance makes no sense otherwise). Frequently, however, there are research problems in which one wants to make direct inferences about two or more distributions, either by asking if a population distribution has some particular specifiable form, or by asking if two or more population distributions are identical. These questions occur most often when variables are qualitative in nature, making it impossible to carry out the usual inferences in terms of means or variances. For such problems, we use nonparametric methods. Nonparametric methods (1) do not depend on any assumptions about the parameters of the parent population (2) generally assume data are only measured at the nominal or ordinal level. There are two common types of hypothesis-testing problems that are addressed with nonparametric methods: (1) How well does a sample distribution correspond with a hypothetical population distribution? As you might guess, the best evidence one has about a population distribution is the sample distribution. The greater the discrepancy between the sample and theoretical distributions, the more we question the “goodness” of the theory. EX: Suppose we wanted to see whether the distribution of educational achievement had changed over the last 25 years. We might take as our null hypothesis that the distribution of educational achievement had not changed, and see how well our modern-day sample supported that theory.
    [Show full text]