STATISTICS FOR ECONOMISTS: A BEGINNING - U of T

1 STATISTICS FOR ECONOMISTS: A BEGINNINGJohn E. FloydUniversity of TorontoJuly 2, 2010 PREFACEThe pages that follow contain the material presented in my introductoryquantitative methods in economics class at the University of Toronto. Theyare designed to be used along with any reasonable STATISTICS textbook. Themost recent textbook for the course was James T. McClave, P. George Ben-son and Terry Sincich, STATISTICS for Business and Economics, Eighth Edi-tion, Prentice Hall, 2001. The material draws upon earlier editions of thatbook as well as upon John Neter, William Wasserman and G. A. Whitmore,Applied STATISTICS , Fourth Edition, Allyn and Bacon, 1993, which was usedpreviously and is now out of print. It is also consistent with Gerald Kellerand Brian Warrack, STATISTICS for Management and Economics, Fifth Edi-tion, Duxbury, 2000, which is the textbook used recently on the St. GeorgeCampus of the University of Toronto.

2 The problems at the ends of the chap-ters are questions from mid-term and final exams at both the St. Georgeand Mississauga campuses of the University of Toronto. They were set byGordon Anderson, Lee Bailey, Greg Jump, Victor Yu and others manuscript should be useful for economics and business students en-rolled in basic courses in STATISTICS and, as well, for people who have studiedstatistics some time ago and need a review of what they are supposed to havelearned. Indeed, one could learn STATISTICS from scratch using this materialalone, although those trying to do so may find the presentation somewhatcompact, requiring slow and careful reading and thought as one goes would like to thank the above mentioned colleagues and, in addition, Ado-nis Yatchew, for helpful discussions over the years, and John Maheu forhelping me clarify a number of points. I would especially like to thank Gor-don Anderson, who I have bothered so frequently with questions that hedeserves the status of the original version of this manuscript was completed, I received somedetailed comments on Chapter 8 from Peter Westfall of Texas Tech Univer-sity, enabling me to correct a number of errors.

3 Such comments are E. FloydJuly 2, 2010c J. E. Floyd, University of TorontoiiiContents1 Introduction to STATISTICS , Data and Statistical What is STATISTICS ? .. The Use of STATISTICS in Economics and Other Social Sciences Descriptive and Inferential STATISTICS .. A Quick Glimpse at Statistical Inference .. Data Sets .. Numerical Measures of Position .. Numerical Measures of Variability .. Numerical Measures of Skewness .. Numerical Measures of Relative Position:Standardised Values .. Bivariate Data: Covariance and Correlation .. Exercises .. 312 Why Probability? .. Sample Spaces and Events .. Univariate, Bivariate and Multivariate Sample Spaces .. The Meaning of Probability .. Probability Assignment .. Probability Assignment in Bivariate Sample Spaces .. Conditional Probability .. Statistical Independence .. Bayes Theorem.

4 The AIDS Test .. Basic Probability Theorems .. Exercises .. 55iii3 Some Common Probability Random Variables .. Probability Distributions of Random Variables .. Expected Value and Variance .. Covariance and Correlation .. Linear Functions of Random Variables .. Sums and Differences of Random Variables .. Binomial Probability Distributions .. Poisson Probability Distributions .. Uniform Probability Distributions .. Normal Probability Distributions .. Exponential Probability Distributions .. Exercises .. 964 Statistical Sampling: Point and Interval Populations and Samples .. The Sampling Distribution of the SampleMean .. The Central Limit Theorem .. Point Estimation .. Properties of Good Point Estimators .. Unbiasedness .. Consistency .. Efficiency .. Confidence Intervals .. Confidence Intervals With Small Samples.

5 One-Sided Confidence Intervals .. Estimates of a Population Proportion .. The Planning of Sample Size .. Prediction Intervals .. Exercises .. Appendix: Maximum LikelihoodEstimators .. 1305 Tests of The Null and Alternative Hypotheses .. Statistical Decision Rules .. Application of Statistical Decision Rules .. Values .. Tests of Hypotheses about PopulationProportions .. Power of Test .. Planning the Sample Size to Control Both the and Risks Exercises .. 1516 Inferences Based on Two Comparison of Two Population Means .. Small Samples: Normal Populations With the Same Variance Paired Difference Experiments .. Comparison of Two Population Proportions .. Exercises .. 1647 Inferences About Population Variances and Tests of Good-ness of Fit and Inferences About a Population Variance .. Comparisons of Two Population Variances .. Chi-Square Tests of Goodness of Fit.

6 One-Dimensional Count Data: The Multinomial Distribution Contingency Tables: Tests of Independence .. Exercises .. 1888 Simple Linear The Simple Linear Regression Model .. Point Estimation of the RegressionParameters .. The Properties of the Residuals .. The Variance of the Error Term .. The Coefficient of Determination .. The Correlation Coefficient BetweenXandY.. Confidence Interval for the PredictedValue ofY.. Predictions About the Level ofY.. Inferences Concerning the Slope andIntercept Parameters .. Evaluation of the Aptness of the Model .. Randomness of the Independent Variable .. An Example .. Exercises .. 218v9 Multiple The Basic Model .. Estimation of the Model .. Confidence Intervals and Statistical Tests .. Testing for Significance of the Regression .. Dummy Variables .. Left-Out Variables .. Multicollinearity .. Serially Correlated Residuals.

7 Non-Linear and Interaction Models .. Prediction Outside the Experimental Region: Forecasting .. Exercises .. 25510 Analysis of Regression Results in an ANOVA Framework .. Single-Factor Analysis of Variance .. Two-factor Analysis of Variance .. Exercises .. 280viChapter 1 Introduction to STATISTICS ,Data and What is STATISTICS ?In common usage people think of STATISTICS as numerical data the unem-ployment rate last month, total government expenditure last year, the num-ber of impaired drivers charged during the recent holiday season, the crime-rates of cities, and so forth. Although there is nothing wrong with viewingstatistics in this way, we are going to take a deeper approach. We will viewstatistics the way professional statisticians view it as a methodology forcollecting, classifying, summarizing, organizing, presenting, analyzing andinterpreting numerical The Use of STATISTICS in Economics and OtherSocial SciencesBusinesses use statistical methodology and thinking to make decisions aboutwhich products to produce, how much to spend advertising them, how toevaluate their employees, how often to service their machinery and equip-ment, how large their inventories should be, and nearly every aspect ofrunning their operations.

8 The motivation for using STATISTICS in the studyof economics and other social sciences is somewhat different. The objectof the social sciences and of economics in particular is to understand how12 INTRODUCTIONthe social and economic system functions. While our approach to statisticswill concentrate on its uses in the study of economics, you will also learnbusiness uses of STATISTICS because many of the exercises in your textbook,and some of the ones used here, will focus on business and understandings of how things work are calledtheories. Eco-nomic theories are descriptions and interpretations of how the economic sys-tem functions. They are composed of two parts a logical structure whichis tautological (that is, true by definition), and a set of parameters in thatlogical structure which gives the theory empirical content (that is, an abilityto be consistent or inconsistent with facts or data). The logical structure,being true by definition, is uninteresting except insofar as it enables us toconstruct testable propositions about how the economic system works.

9 Ifthe facts turn out to be consistent with the testable implications of the the-ory, then we accept the theory as true until new evidence inconsistent withit is uncovered. A theory is valuable if it is logically consistent both withinitself and with other theories established as true and is capable of beingrejected by but nevertheless consistent with available evidence. Its logicalstructure is judged on two grounds internal consistency and usefulness asa framework for generating empirically testable illustrate this, consider the statement: People maximize utility. This statement is true by definition behaviour is defined as what peopledo (including nothing) and utility is defined as what people maximize whenthey choose to do one thing rather than something else. These definitionsand the associated utility maximizing approach form a useful logical struc-ture for generating empirically testable propositions.

10 One can choose theparameters in this tautological utility maximization structure so that themarginal utility of a good declines relative to the marginal utility of othergoods as the quantity of that good consumed increases relative to the quan-tities of other goods consumed. Downward sloping demand curves emerge,leading to the empirically testable statement: Demand curves slope down-ward. Thistheory of demand(which consists of both the utility maxi-mization structure and the proposition about how the individual s marginalutilities behave) can then be either supported or falsified by examining dataon prices and quantities and incomes for groups of individuals and commodi-ties. The set of tautologies derived using the concept of utility maximizationare valuable because they are internally consistent and generate empiricallytestable propositions such as those represented by the theory of demand. If itdidn t yield testable propositions about the real world, the logical structureof utility maximization would be of little , consider the statement: Canada is a wonderful country.