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ECON 221
Pirapagaran Tharmalingam

ECON221 – Course Notes Lecture 1  Learning Objectives o Define Statistics o Describe the uses of statistics o Distinguish Descriptive & Inferential Statistics o Define Population, Sample, Parameter, and Statistic o Define Quantitative and Qualitative Data o Define Random Sample  What is statistics? o Define: A methodology for collecting, classifying, summarizing, organizing, presenting, analyzing and interpreting numerical information  Uses of Statistics o Application Areas:  Economics  Forecasting, Demographics  Engineering  Construction & Materials  Sports  Individual and Team Performance  Business  Consumer preferences, financial trends o Uses of statistics in Econ  It’s the methodology that we use to confront theories  The theory of demand and and other testable propositions with the facts  It is the set of procedures and intellectual processes by which we decide whether or not to accept a theory as true  It is the process by which we decide what and what not to believe  Use of stats lead us to:  A wider belief in the ‘truth’ of a particular theory OR  To its rejection as inconsistent with the facts  Descriptive & Inferential Statistics o Descriptive Statistics  Description and presentation of data  Utilizes numerical and graphical methods  To find patterns in the data  To summarize the information it reveals  To present that information in a meaningful way  Descriptive statistics LACK measure of reliability o Inferential Statistics  Using the data to make inference (estimates, decisions, predictions) about features of the environment from which the data were selected or about the underlying mechanism that generated the data  Inferential statistics have a measure of reliability  E.g. an election says a poll is accurate to within 3 percentage points  Key Terms o Population  All items of interest o Sample  Portion of population o Parameter  Summary measure about population o Statistic  Summary measure about a sample  Components of Statistical Procedures o Population  Set of all objects or individuals of interest  E.g. all UW students, all econ majors, etc o Variable  A characteristic or property of the population unit that is of interest  E.g. height: from pop’n of uW students, GPA o Sample  A subset of the units of a population  E.g. 4 year students, 50 econ 401 students o Statistical inference  Estimation, prediction, or other generalization about a population based on information in a sample  Examples:  Possible inferences from a sample data: o The average gpa of all econ majors is 87% o 35% of all econ majors get less than 75% GPA o the median GPA for econ majors is 85%  Check text ex. 1.1, 1.2  Data Sets o Three kinds:  Cross sectional  Time-series  Panel Data o Within data sets there are two kinds of data  Quantitative  Qualitative  Types of Data o Quantitative Data  Measured on a naturally occurring scale  Equal intervals along scale (allows for meaningful mathematical calculations)  Ratio data  Data with absolute zero (zero means no value) o E.g. bank balance, grade  Interval data  Data with relative zero (zero has value) e.g. temperature o Qualitative Data  Measured by classification only  Non-numerical in nature  Ordinal data  Meaningfully ordered categories o E.g. best to worst ranking, age categories  Nominal data  Categories without a meaningful order o E.g. political affiliation, industry classification, ethnic/cultural groups  Random Sample o Every sample size n has an equal chance of selection  Collecting Data o Data sources  Published source – books, journals, abstracts  Primary vs. Secondary  Designed Experiment  Often used for gathering information about an intervention  Survey  Data gathered through questions from a sample of people  Observational Study  Data gathered through observation, no interaction with units Lecture 2  Learning Objectives o Describe Qualitative Data o Describe Quantitative Data o Explain Numerical Data Properties o Describe Summary Measures o Analyze Numerical Data Using Summary Measures  Methods for Describing Sets of Data o Describing Data using Graphs o Describing Data using Charts  Data Presentation o Qualitative Data  Summary Table  Bar graph, Pie Chart, Pareto Diagram o Quantitative Data  Stem & Leaf Display  Frequency Distribution  Histogram  How can Qualitative data be Described? o Since the qualitative data are non-numeric in nature they are best described using classes (also known as bins) o Class frequency is the number of data points which fall into a class o Class relative frequency is the number of data points in the class divided by total number of data points. Class percentage is class relative frequency multiplied by a 100. o How can the Results be Summarized in a Frequency Table  Frequency/Percent/Cumulative Percent o How are data displayed on a bar graph?  Bar graphs are more suitable when the purpose is the comparison of categories o How are data displayed using a pareto diagram? o How are data displayed using a pie chart?  Pie charts are more suitable when the main objective is to investigate the portion of the whole that is in a particular category  How can Quantitative data be described? o After looking at data: o How does dot plot work?  Data points line up on top of each other depending on the frequencies o What is a histogram?  A histogram is a graph in which classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are placed adjacent to each other  Information given by a histogram  Has the following strengths o Provides visual indication of which class is more frequent (modal information) o Supplies an indication of the degree of spread, or variation, of the data o Displays the shape of the distribution  Has the following weaknesses: o Can obscure time differences between data sets o Can be manipulated to present the data in a fashion different from reality  How is a histogram constructed?  Identify the largest and smallest value in the data set (max and min)  Divide the interval between the max and the min into sub intervals (classes, bins)  Keep in mind that each data point must fall into one and only one class and no data point must be on the boundary  How many classes should the data be split into? There are two rules which can help answer that question (let n denote the sample size) o (2n) 1/3~ (2n) 0.3333 o Sturges’ Rule: 1 + log(n) / log(2)  To determine the width of the subintervals divide the range of data (max-min) by the number of classes  Determining the number of classes  Course text PAGE 58 o # of observations in a data set v. number of classes  less than 25 = 5-6 classes  25-50 = 7-14 classes  more than 50 = 15-20 classes  Newbold, P.Carlson W.L and Thorne B o Sample size v. Number of classes  Less than 50 = 5-6 classes  50-100 = 6-8 classes  More than 100 = 8-10 classes  Numerical Data Properties and Measures o Central Tendency  Mean, Median, Mode o Variation  Range, Interquartile Range, Variance, Standard Deviation o Relative Standing  Percentiles, Z-Scores  Notation that will be useful for this course: o Summation Notation: (Greek Letter sigma)  Lets say we have ‘n’ observations, denoted as X1, X2 … Xn. Each observation denoted with a subscript The Expression ∑ ni=1x is equal to “x1+ x 2 x 3 x +4x +5+ x ” n  and reads “sum of x ior i equals 1 to n” o What is central tendency?  Central tendency is the tendency of data center about certain numerical values  A measure of central tendency is a single value that summarizes a set of data. It locates the center of the values  As shown previously, the most common measures are MEAN (arithmetic average), median (positional center), and mode (most frequent value)  Calculating Mean  The arithmetic mean (which is usually referred to as, simply the mean) is the sum of data values divided by the number of observations  Population mean would be denoted as μ (Greek letter ‘mu’). This is a parameter.  Sample mean would be denoted as x (read as ‘x-bar’). This is a statistic.  Calculating mean for ungrouped items: Sample mean x n x x1x 2x 3x n i1 i x  n n Population mean  (value typically not known) N = population size N  xi i1  N  Calculating the Mean for grouped data o Sometimes data you encounter is divided into classes and all you can observe are the class frequencies (or relative frequencies) and means of observation in the classes o In this case the mean of the data set would be the weighted mean, with weights being the frequency of the respective classes o Here there are k classes, fi denotes the class frequency, and Xi-bars indicated the class means o This formula also applies in cases when Xi-bars are unobservable but the frequency of each class is still known o In this case class means would be subsistuted for mid-points  Main disadvantage of the mean? o The mean is easily influenced by the extreme values (for example, outliers, values which fall afar from the main cluster of data) o Lets demonstrate it with an example suppose we have the following 2,2,2,3,3,3,6 and then replace one fo the 3’s with an outlier 17  The Role of the Mean in “Balancing” the data … o A histogram ‘balances’ when supported by the mean (in this case 140.6)  What is Median?  The Median is the middle observation of a set of observations that are arranged in increasing or decreasing order  IF the number of observations in a sample is odd then the median is the middle observations.  How does one locate the median? o The median would be located in the position number 0.5(n+1) if the sample has an oddnumber of observations o The first number used in calculating the median  What is the mode?ated at 0.5n and the second one at 0.5n+1  The mode, if one exists, is the most frequently occurring value  The data can be: uni-modal, bi-modal, and multi-modal  Data displayed in a histogram can have a modal class (the clas with highest frequency) o Which of these measures are unique  Mean is a unique measure, the data set has only one mean  Median is a unique measure even the data set with even number of observations has one median (even though it is calculated with the two observations)  Mode may not exist and if it existst may not necessarily be unique o How much information is taken into account by each of these measures?  Mean uses all of the information  Median uses less information  Mode uses the least  What is skewness? o The data is symmetric if the distribution has the same shape on either side of the center o Otherwise the data is Skwed. The distribution extends more to one side than to the other o This is caused by extreme values “dragging” the mean to their side o How can skewness be measured? o Skewness can be measured by the AVERAGE CUBED DEVIATION from the sample mean (for example): if the large deviations are skewed) NOT ON final (Below formula)positive and data will positively n 3  (x ix) 3 i1 m   n1  Why does one need numeric measures of variability o The mean (and other central tendencies) does not provide sufficient or complete description of data o Variability indicates how spread the data is over all possible values o Measures of variability are numbers which describe how spread the o Most commonly used measures of variability include: range, variance, and standard deviation  How is Range Calculated o Range is the difference between the largest and the smallest observation o The larger value of this measure of variability indicates a greater o Range is vulnerable to extreme values o Range, also, loses snesitivy when the number of observations is large o What other types of Range Measures exist  Quartiles divide a set of observations into four equal parts  The interquartile Range measures the spread of the middle 50% of the data. It is calculated as follows: IQR = Q3 – Q1  Semi-interquartile range calculated as: (Q3-Q1)/2  Whao The sample variance is the sume of squared deviations from the mean dvided by (n-1). o We are using squared deviations, such as the sum of deviations from the mean is equal to zero n 2 (x  x)  i 2 s  i1 n1 o o mean divided by N which denotes the population sizetions from the o How is Variance from a Frequency Distribution Calculated?  Can be approximated using thi class means (x)  And the sample mean (x) n 2  fi(x i x) 2 s  i1  n1  Note That n n  f  i  i1  What is Standard Deviation o Smeanful measure of data variability.uare root of variance. It is a o Sample Standard deviation (denoted s) would be calculated according to the following formula: n 2  (x ix) s  i1  s 2 n1 o  Whao Population standard deviation will be denoted as “sigma” o In most cases the population standard deviation is not observable since the population variance is unobservable  Some properties of Standard deviation o Standard deviation is always greater or equal to zero. This value would equal zero when all the observations are the same. o Larger values of standard deviation indicate greater spread of data o Helps us determine the likely size of chance of error in measurement  How can we interpret standard deviation? o number of standard deviations. obserations fits within a certain o These are two rules which describe that amount … o 1) Empirical Rule  Sometimes is referred to as the Normal rule.  For a symmetrical, bell shaped frequency distribution approximately 68 percent of the observations will lie within plus and minus one standard deviation of the mean; about 95 percent of the observations will lie within plus and minus two standard deviations of the mean; and practically all (99.7 percent) will lie within o 2) plus and minus three standard deviations of the mean Chebyshev  unknown distributions (0,3/4,8/9) Empirical  known, almost all inferential statistics is based on empirical rule (68/95, 99.7) Box Plot (Q3 + (IQR *1.5) Q3 Q1 Q1 – (1.5*IQR) Lecture Three  Relative Standing Measures o Descriptive measures of a relationship of a measurement to the rest of its data o Common Measures  Percentile ranking/score  Percentile rankings make use of the pth percentile  For any p, the pth percentile o Has p% of the measures lying below it and o (100-p)% above it  The median is the 50 percentile o 50% observations above and below  Percentile given a score o Percentile of score x = # of scores less than x/total number of scores * 100  Finding the score given a percentile o X = p/100 * n  Z-score  X subtracted by mean divided by standard deviation  Z scores follow empirical rule for mounded distributions  Anything above 2 or -2 = unusual values and outside -3 or 3 are outliers  Outliers o Outlier  an observation that is unusually large/small relative to data being described o Can have dramatic affect on mean, standard deviation, and scale of histogram o Causes: invalid measurement, misclassified measurement, a rare (chance) event o 2 detection methods  Box Plots  Lower Quartile, Middle Quartile, and Upper Quartile necessary, IQR  Reveals the: center, spread, distribution, presence of outliers  Excellent for comparing two or more data sets  Z-Scores  Graphing Bivariate Relationships o Relationship between two quantitative variables can tell if positive/negative/no relationship o Time Series Plot  Data produced over time (time on horizontal)  Points connected by straight lines  Distorting Truth o Errors in presenting data  Using ‘chart junk’  No relative basis in comparing data batches  Compressing the vertical axis  No zero point on the vertical axis o Lecture Four Learning Objectives  Define Experiment, Outcome, Sample Point, Sample Space, Event & Probability o Events, Sample Spaces and Probability  Experiment  Process of observation that leads to a single outcome with no predictive certainty (tossing 2 coins)  Sample point/simple event  Most basic outcome of an experiment or event that cannot be broken own into simpler components  Sample Space  A listing of all sample points for an experiment  Sample Point probability  Relative frequency of the occurrence of the sample point e.g. HT/out of all sample points o Sample Space Properties  Mutually Exclusive  2 outcomes can not occur at the same time o Male and Female in same person  Collectively Exhaustive  One outcome in sample space must occur o Male or Female o Sample Space Examples  Observe Gender  male, female  Play a football game  Win, Lose, Tie  Select 1 card, note color (Red black) o Events  Any collection of sample points  Simple Event  Outcome with one characteristic  Compound Event  Collection of outcomes or simple events  Two or more characteristics  Joint Event is a special case o Two events occurring simultaneously  Use of Venn Diagram, Two-Way Table, or Tree Diagram to find Probabilities o Compound Event (atleast one tail) inside the venn diagram o Outcome HH outside o S  indicating sample space o  Describe and Use Probability Rules o What is Probability  Numerical measure of the likelihood that event will occur  P(Event)  P(A)  Prob(A)  Lies between 0 & 1  Sum of sample points is 1 o Probability  P(Event) = X/T  X = number of event outcomes  T = Total number of sample points in Sample Space  Each of T sample points is equally likely – P (sample point) = 1/T  Approaches to Probability  Relative Frequency Approximation o Conduct (or observe) an experiment a large number of times and count the number of times event A actually occurs, then an estimate of P(A) is  P(A) = # of times A occurred/# of times trial was repeated  Note  approximation of the actual probability  The Classical Approach o Requires equally likely outcomes o If a procedure has n different simple events, each with an equal chance of occurring, and s is the number of ways event A can occur, then o P(A) = s/n = # of ways A can occur/# of different simple events o This is actual probability  The Subjective Probabilities Approach o P(A), the probability of A is found by simply guessing or estimating its value based on knowledge of the relevant circumstances o So the individual assigns probabilities based on personal experience, anecdotal evidence, etc. o E.g. probability of conservatives winning elleciton is .7 Lecture Five  Compound Events o Union  Outcomes in either events A or B or Both  ‘OR’ statement  U symbol (i.e., A U B) o Intersection  Outcomes in both events A and B  ‘AND’ statement  intersection symbol o Compound Event Probability  Numerical measure of likelihood that compound event will occur  Can often use a two-way table  Two variables only  Formula Methods  Additive Rule  Conditional Probability Rule  Multiplicative Rule o Complementary  The event that A does not occur  All events not in A: Ac  P(A) + P(Ac) = 1  Mutually Exclusive Events o Events do not occur simultaneously o A intersect B does not contain any sample points  Drawing spades + hearts  Additive Rule o Used to get compound probabilities for union vents o P (A or B) = P (A U B) = P (A) + P(B) – P(AintersectB) o For Mutually Exclusive Events: P (A OR B) = P (A) + P (B)  Conditional Probability o Event Probability GIVEN that another event occurred o Revise sample space to account for new information  Eliminated certain outcomes o P(A|B) = P (A intersect B)/ P(B)  Statistical Independence o Event occurrence does not affect probability of another event  Toss one coun twice o Causality not implied o Tests for independence  P (A|B) = P(A)  P (A intersect B) = P (A) * P(B)  Multiplicative Rule o Used to get compound probabilities for intersection of events (joint events) o P (A and B) = P (A intersect b) = P(A) * P(A) = p(B)*P(A|B) o For Independent Events:  P (A and B) = P (A intersect B) = P(A) * P(B)  Bayes’s Rule o Allows computation of an unknown conditional probability, P(B|A) by converting it to a known conditional probability P(A|B) o For k mutually exclusive events o P(B1|A) = Counting  Factorial: o The factorial is a non-negative integer where n greater than or equal to zero, is the product of all integers 1,2, …n o Factorial Notation:  n! = 1x2x..xn, for n > 0 and n! = 1 for n=0 o Recursive Formula:  n! = (n-1)! x n o Approximation formula:  Stirling’s formula: n! ~ square root of 2pien (n/e)^n)  Where pie = 3.1415927 and e = 2.718218  Methods for Counting Outcomes o Use:  We need to be able to count the number of simple events in a compound event and possible events in the sample space o Basic Rules  Addition Rule  Suppose event A can happen in p ways and event B in q ways  Then either event a OR event B but not both can happen in p+q ways  Multiplication Rule  Suppose event A can happen in p ways and an unrelated event B in q ways  Then both event A AND B can happen in p x q ways  W.o Replacement (1/5->1/4->1/3, etc.) o Ordering Matters (PERMUTATIONS)  Suppose that n distinct objects are to be ‘drawn’ sequentially or ordered from left to right in a row  The number of ways to arrange n distinct objects in a row is n!  Explanation: we can fi the first position in n ways, the next is n- 1 ways, etc.  The number of ways to select r objects from n distinct outcomes is n(n-1)(n-2)…(n-r+1)  By the rth pick (r-1) objects have already been used  Described as n taken to r terms  Therefore:  N(r) = n! / (n-r)! o Order does not matter (COMBINATIONS)  Suppose that n distinct objects are to be ‘drawn’ without replacement  The number of ways to choose r objects from n is denoted by (n over r) called “n choose r”  Combinations is also used when the number of sample points is too large to enumerate  Formula:  N over r = n(r) /r! = n!/r!(n-r)! or nCr  Proof  The nmber of ways to choose r objects from n and arrange them from left to right is n  Any choice of r objects can be arranged in r! ways  Therefore: o # of was to choose r objects from n = n (r) o # of ways to choose r objects from n = n(r)/r!  With replacement o Never considered in this course Statistical Independence = lack of correlation Mutually exclusive = no sharing of elements Lecture 6 Learning Objectives  Distinguish between the Two Types of Random Variables o A random variable is a numerical-valued function defined on the outcomes of an experiment o A variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure o A variable that assumes numerical values associated with random outcomes of an experiment o Two Types  Discrete Random Variable  Has either a finite number of values or a countable number of distinct possible values o Where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process o Test: for any given value of the random variable you can designate the next largest or next smallest value of the random variable o Poisson random variable is exception o Ex. #Sales, #Correct,  Continuous Random Variable  Random variable that has an infinite number of distinct possible values o Values can be associated with measurements on a continuous scale with no gaps or interruptions o So the variable can take on all possible values in an interval of numbers o Test: given a particular value of the random variable, you cannot designate the next largest or next smalles vaue o Ex. Weight, Hours  Discrete random variables “count”  Continuous random variables “measure” (Length, width, height, etc)  Another distinction: quantitative order  Discrete random variable o Sample points can be enumerated or listed in order  Continuous o Not possible to list sample points in order  Describe Discrete Probability Distributions o Discrete Probability Distribtion  List of all possible [x, p(x)] pairs  X = value of random variable (outcome)  P(x) = probability associated with value  Mutually exclusive (no overlap)  Collectively exhaustive (nothing left out)  0 < p(x) < 1 for all x  Sum of p(x) = 1 o Summary Measures  Expected Value (mean of probability distribution)  Weighted average of all possible values  Mean = E(X) = Sumofxp(x)  Variance   Standard Deviation  Square root of variance o Interpretation  E(x) is NOT the value of the random variable x that you “expect” to observe if you perform experiment once but rather a “long run” average o Variance of Discrete Random Variables  Sum of (x-mean)^2p(x)  Describe the Binomial and Poisson Distributions o Binomial Distribution  Number of ‘successes’ in a sample ofn observations (trials)  Number of reds in 15 spins of a roulette wheel  Number of correct ona 33 question exam  Binomial Distribution Properties  Two different sampling methods o Infinite population without replacement o Finite population with replacement  Sequence of n identical trials  Each trial has 2 outcomes o ‘success’ desired outcome or ‘failure  Constant trial probability  Trials are independent  Binomial Probability Distribution Function   x n x n! x n x p(x)   p q  p (1  p) x x!(n  x)!    p(x) = probability of x successes, n = sample size, p = probability of success, x= # of successes in a sample o Poisson Distribution  Discrete probability distribution that applies to occurrence of some event over a specific interval (integer value from zero to infinity)  The value gives the number of occurrences of the circumstance of interest during period  Events PER UNIT  Time , length, area, space  Possion Process  Constant event probability  average of 60/hour is 1/min for 60 1 minute intervals  One event per interval (don’t arrive together)  Independent events o Arrival of 1 person does not affect another’s arrival  Function  p(x) = probability of x given expected (mean) # of ‘successes’  P(x) = Ex. Mean^# of success/unit e^-ex.mean/ #ofsuccess!  Ex value/Mean = variance  Standard deviation  Describe the Uniform and Normal Distributions o Uniform Distribution  The Uniform Distribution  Characteristics o Uniform probability distributions result:  When the probability of all occurrences in the same space are the same OR  When a continuous random variable is evenly distributed over a particular interval o Are these probability distributions discrete or continuous?  They may be either discrete or continuous  Discrete Uniform Distribution o Consider a random number generator that cranks out random numbers between 0 and 9 o By construction of the computer program, the probability that any one of the 10 numbers will turn 1/10 or .1 o Therefore probability distribution X = .1 every time o The discrete probability function is:  P(x) = 1/s where P(x) = P(X=x); and x = a; a+1; a+2,,,, a+s(-1)  A denotes smallest outcome and s denotes the number of distinct outcomes  Eyeball the mean (median)  Mean: E(x) = a + (s-1)/2  Variance: = s^2 -1 / 12 o Probability Distributions for Continuous Random Variables  Continuous Probability Density Function  Mathetmatical Formula  Shows all values x, and frequencies f(x) o F(x) is NOT probability  Properties f(x)dx = 1 (area under cruve)  F(x) > 0, a a)  denotes the probability that z score is greater than a  P (z
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