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Jeff Racine

Overview 9/11/2013 5:37:00 AM Statistics: is a branch of mathematics dealing with the collection, presentation, and interpretation of data The field can be divided into two broad categories:  Descriptive statistics: which seeks to describe general characteristics of a set of data  Inferential statistics: which seeks to draw inferences about (unknown) features of a population based on a (known) sample drawn from a population Broadly speaking, statistics provide a rigors framework for describing, analyzing, and drawing on the basis of data Why do we need statistics?  Mainly due to uncertainty (we cannot know with certainity many things in life, even though they clearly exist) Variables and Distributions  In any statistical study, we analyze info/data obtained from individuals  Individuals can be people, animals, households, plants, and or objects of interest  A variable is any characteristic of an individual  A variable varies among individuals (as opposed to constants) o Ex. Age, income  The distribution of a variable tells us what values the variables takes and how often/likely it takes these values Population and Samples  Population: set of all possible observations on some characteristic (eg. Age, height, income)  Sample: subset of a population  Random sample: one obtained if every member of the population has an equal of being in the sample  Categorical population: a population whose characteristic is inherently nonnumerical (e.g. sex, race)  Quantitative population: a population whose characteristic is inherently numerical o For categorical population , we are typically interested in proportions (e.g. the proportion who are male) o For quantitative population, we are typically interested in proportions (e.g. the average, and how „spread-out‟ the variable is) Types of Categorical Data  Nominal data o Numbers merely label difference in kind o Cannot be manipulated arithmetically o Can only be counted o Ex. Indicator for a categorical variable (o=male, 1=female)  Ordinal data o Label difference in kind o Order or rank observation based on importance o No meaningful arithmetic analysis possible o Can only be counted and ranked o Ex. Ranking firms according to profit (e.g. HSBC) 9/11/2013 5:37:00 AM Presenting Data: Tables and Graphs  When presenting data either in the form of tables or graphs, it is often desirable of first split the data into groups/classes  How we split the data into groups/clases o Data types:  Categorical (nominal or ordinal)  Quantitative (discrete or continuous) o Regardless of the data type, data classes should be:  Collectively exhaustive (must exhaust all logical possibilities for classifying available data)  Mutually exclusive (must not overlap or have data in common)  One immediately confronts the issue of how many classes to create o Desirable class number:  Should fit data type  Often recommended: between 5 to 20  Sturgess‟s rule: desirable number of classes=K, an integer, where k is the integer closest to (use standard rules for rounding)  1+3.3log n10  where n is the sample size, and where log n 10 the power to which the base (10) is raised to yield n  ex. If n=1000 then log 1100=3, so 1+3.3 log10000=10.9, and therefore k=11 so by rule use 11 classes when n=1000 Data Classes  Class width= the difference between the lower and upper limits of a class  To achieve uniform class widths in a table, divide data set width by desirable class number  Approximate class width: o Large value= small value/ desirable class number Tabular Components  We first consider creating effective tabular summaries  An effective table includes: o Number (often based on chapter or page numbers) o Title (focus on what, where and when) o Caption (brief verbal summary) o Footnotes (e.g. size of sampling error, likely extent of systematic error, etc) o Decimals (consistent number of decimals) o Rounding (consistent rounding rules) o Class sums (sum of data pertaining to each class, crucial for open- ended classes) Frequency distributions  Frequency distribution are an effective way to summarize data  Absolute frequency distributions o Absolute class frequency:  Absolute number of observation that fall into a given class o Absolute frequency distribution:  Tabular summary of data set  Shows absolute numbers of observations that fall into several data classes o In R there is a function list() used for graphical representation Relative Frequency distributions:  Relative class frequency: o Ratio of a particular class‟s number of observation to the total number of observations made  Relative frequency distributions: o Tabular summary of data set o Shows portions of all observations that fall into each of several data classes Cumulative Frequency Distributions  Cumulative class frequency o The sum of all class frequency up to and including the class in question LE distribution (typical, often called „cumulative; distribution)  LE: less then or equal to upper class limit  Moves from lesser to greater classes ME distribution (less used, often classed „survivors‟ distribution)  ME: more then or equal to lower class limit  Moves from greater to lesser classes Cross Tabulations  Cross tabulation are tabular summaries for two variables  One variable represented by row heads  The other variable represented by column heads  Information for both variables entered in table cells  In R we use the xtabs() functions Sept.10.2013 Graph Types  Core Graphs o Two dimensional graphs o Ex. Histograms, scatter, diagrams, etc.  Specialty graphs o Combine elements from core graphs to display data in unique ways o Ex. Bar graphs, pie, charts. Etc.  3G graphs o display 3 elements; width, height, depth Frequency Histograms  Horizontal axis o Identifies upper and lower limits of data classes  Vertical axis o Shows number of observations in each class (absolute freq) o Shows ratio of class freq to class width (relative freq)  Rectangles o Represents corresponding class frequencies by area or by height (if all class intervals are alike)  In R we use the hist() function (hist(.prob=TRUE) for relative freq divided by the class width Frequency Polygon and Density Estimations  Frequency polygons o Portray frequency distributions as many-sided figures o Class mark= avg of two class limits o Final points on horizontal axis lie ½ length below the lowest (or above the highest) class limit  Density estimator o Portray freq distributions as smooth curves o Remove irregularities from histograms depicting info gathered is sample surveys o Estimate how histograms would appear in census info were graphed with many tiny classes o In R we use the plot() function Ways to chart categorical data  Unlike quantitative, when a variable is categorical the data in the graph can be ordered any way we want (alphabetical, by increasing value, by personal prefence)  Bar Graph: o Each category is represented by a bar o In R, see the barplot() function  Pie charts: o The slices must represent the pieces o In R pie() Summary Statistics: Symbolic Expression  Population parameters (typically unknown) o Summary statistics based on population data o Designated by Greek letter (eg. Mu, sigma)  Sample Statistics (computed from sample) o Summary statistics based on sample data o Designated by roman letter (eg. X(with bar above),s)  Observed values o Traditionally represented by x or X o Different value indicated by subscripts 1,2,3,…etc.  Eg. X1, X2  Observation totals o Population total= N o Sample total=n Sept. 12.2013 Summary Statistics: Summation notation  In statistics, we often need to accumulate or sum some variable of interest  For example, A might denote, say, income, with A1 being the income of the first person in our sample, A2 that for the second and son on  It is supremely convenient to avoid having to write things like o A1+A2+A3+…….OR o (A1-B)+(A2-B)+(A3-B)+…… or even o (A1-B)^2+(A2-B)^2+(A3-B)^2+……  One convenient operator that we shall use heavily is the summation operator  In R use the sum() function  The sum of a variable X can be written as o Summation(from i=1 to n) Xi= X1+X2+….+Xn o Note Summation(from i=1 to n) and Summation (from i*) and Summation are often interchanged o Care must be taken when expanding sums. For instance 2 2 n  Summation(from i=1 to n) X i=X  square the summation before and after are differnetnt  Some useful manipulations involving the summation operator follow from the definition of the arithmetic mean, o X(BAR) = 1/n Summation(from i=1 to n)Xi o Common ones you may find useful are as follows:  Summation(from i=1 to n)Xi= nX(bar)  Summation(from i=1 to n)(Xi- Xbar) =0  Summation(from i=1 to n)(Xi-Xbar)^2= Summation(from i=1 to n) X i-nXbar^2 Summary Measure Types  Central Tendency Measures o Locate a data set‟s „center‟  Dispersion measure o Focus on the spread of data around a set‟s center  Shape Measure o Describe the symmetry (or asymmetry) of a frequency/density curve, as well as how flat or peaked it is  Proportions o Used for categorical data o Describe the frequency of observations in a category as fraction of all observations made Central Tendency Measures  The Arithmetic Mean o Point of balance o Beware- the mean can be strongly affected by the presence of „extreme‟ value (potential „outliers‟) o Calculation let Xi, denote an observation , i=1,2,….  Arithmetic Mean:  Population: mu= Summation(from i=1 to n)Xi/N  Sample: Xbar= Summation(from i=1 to n)Xi/n o In R we use the mean() function  The Median o Divides data into equal halves  Odd number of observations:  Median= middle observation in the ordered array  Even number of observations:  Median- average of the two middle in the ordered array o Calculation: After sorting your data in ascending order  Median (n odd) median=X (n+1)/2  Median( n even) median= (X n/2+ X (n/2)+1)2  The mode: the most frequently occurring value in a data set o Can be found either extreme of data set (and thus be atypical of the majority of observations) o May not appear in a data set at all (because no single observation occurs more than once) o May occur more than once in a data set  Multimodal frequency distribution: data set has 2 more or more values with the highest freq (eg. A data set with 2 modes is termed „bimodal;) o The mode could also be obtained from a density curve, and is that values(s) of X lying directly below the peak(s) of the density curve Dispersion Measure  The Variance (Average (Squared) Deviation) o Most common measure of dispersion o Measure the typical squared deviation about the mean o Calculated by averaging the squares of individual deviation from the mean  Calculation  Variance: o Population: mu = (Summation(from i=1 to n)(Xi- mu)^2)/N 2 o Sample: s =(Summation(from i=1 to n) (Xi- Xbar)^2)/n-1  In R we use the var() function (note this computes the sample variance) Standard Deviation  The positive square root of the variance  Falls in the same range of magnitude (and appears in the same units) as the observation themselves  Calculation: o Population: sigma= square root(sigma^2) o Sample: square root (s^2) o In R we use the sd() function Dispersion Measures  Overall Range o Equals the difference between the largest and smallest observation in a data set( in R we could use the range() function)  range(X)  Interfractile Ranges o Measures difference between 2 valus in the ordered array (called „fractiles‟)  Quartiles: divide the array into 4 quarters  Interquartile Range: difference between 3 rdand 1 st quartiles (contain middle 50% of data)  In R we use the IQR() and quantile() functions Shape Measures: Skewnes  A frequency distribution‟s degree of distortions from horizontal symmetry  Pearson‟s (first) coefficient of skew is (we want to know is it +/- ?): o Skewness= mean-mode/ standard deviation o An alternative (more popular, default in R) moment-based measure is given by  Skewness= Summation (i=1 to n)( (X1-Xbar)^3)/n/( Summation (i=1 to n)( (X1-Xbar)^2)/n)^3/2 o Skewness=0 fro symmetric distributions o For right-skewed distributions the mean is bigger than the median which is bigger than the mode (for left-skewed just the opposite holds) Shape Measures: Kurtosis  A frequency curve‟s degee of peakness  Coefficient of kurtosis  Kurtosis= = Summation (i=1 to n)( (X1-Xbar)^4)/n / ( Summation (i=1 to n)( (X1-Xbar)^2)/n)^2  Kurtosis= 3 for the normal distribution that we introduce shortly( handy reference distribution)  In order to compute certain higher-order “moments” (e.g. skewness and kurtosis) of a quantitive variable we need to install an optional R PACKAGE  In RStudio we first install the moments package via the Install packages icon in the packages tab (lower right pane by default)  The function kurtosis() and skewness() can then be accessed after we oad the moments package via library (moments)  The five-Number Summmary  The five-number summary of set of observations consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written on order from smallest to largest Minimum Q1 Median Q3 Maximum  In R the function is fivenum() Suspected Outliers  Outliers are troublesome data points, and it is important to be able to identify them  One way to raise the flag for suspected outlier is to compare the distance from the suspicious data point to the nearest quartile (Q1 or Q3)  We then compare this distance to the interquartile range (distance between Q1 and Q3)  We call an observation a suspected outlier if it falls more than 1.5 times the size of the interquartile range (IQR) below the first quartile or above the third quartile  This is called the “1.5 IQR rule for outliers” Boxplots  A box-and-whisker plot (sometimes called simply a „box plot‟) is a histogram-like method of displaying data, invented by John Tukey o Draw a box with ends at the quartile Q1 and Q3 o Draw the median as a horizontal line in the box o Now extend the “whiskers” to the farthest points that are not suspected to outliers (i.e. that are within 1.5 times the interquartile range (Q3-Q1) from the then of the box) o Then, for every point more than 1.5 times the interquartile range from the end of a box, draw a dot  In R the function is boxplot() Normal („Gaussian‟) Frequency distribution  The normal distribution is a special type of population whose relative frequency distribution is characterized by: o Single peak with mean, median, and mode coinciding at the center of distribution o Perfect symmetry indefinitely in both directions from center (approaching, but never touching, the horizontal axis) o Predictable percentages of all population values the lie within +/-1, +/-2 and +/-3 standard deviation from the mean (roughly, 68%, 95%, and 100%) Chebyshev‟s Theorem  Chebyshev’s Theorem is an empirical rule that applies to all distributions, not only the Normal distribution  Regardless of he shape of a population frequency distribution, the proportion of observation falling within k standard devivations of the mean is at least (given that k quals 1 or more): o Proportional falling within +/- kσ of μ>=1 - 1/k^2   Coefficient of Variation o Used to compare degree of dispersion among data sets o Ratio of the standard deviation to the arithmetic mean o In R ad()/mean() 9/11/2013 5:37:00 AM Origins of Data  Data can originate in a number of ways  Internal Data o Created as by-products of regular activities o Ex. Customer, employee, production records, government records  External Data (typical source fro this course) o Created by entities other than the person, firm, or government that wants to use the data o Ex. Print sources, CD-ROMs, web sites  In the past, users were limited to print data (e,,g, print and CDROM) o Frequently expensive o Useful for transferring to computer applications  The internet is an exceptionally efficient source of data Important Data Websites  Canadian government SOURCES  Foreign Government sources  Fortune 500 company sources Sampling Versus Census Taking  A census is a complete survey of every member in the population  A sample is a partial survey in which data is collected for only a subset of the population  Reasons for sampling versus conduction a census o Expense can be prohibitive o Speed of response o Impossible. For example we may have infinite population (I,e, observation occur from an infinite or recurring process) o Destructive sampling (e.g., lifetime of light bulb, safety of automobile) o Accuracy can, oddly enough, be better (e.g. higher quality information results via hiring fewer persons who can be better trained, say) Types of Samples: Non probability  Nonprobability sample: occurs when a sample is taken from an existing population in a haphazard fashion without the use of some randomizing device assigning each member a known (positive) probability of selection o Voluntary response sample (e.g., phone survey (self selection issue) o Convenience sample (eg. Most easily selected persons) o Judgment sample ( based on data collects personal judgment (experience) ) o All of the above will likely not be representative of the population as a whole, and are perhaps the worst types of samples to model Types of Samples: probability  Probability sample: occurs when a sample is taken with the help of a randomizing device that assures each member a known (positive, not necessarily equal( probability of selection o Simple random a sample: obtained if every member of the population has an equal chance of being in the sample (main type) st o Systematic random sample: randomly select 1 element, then include every kth element thereafter till sample complete o Stratified random sample: take random samples from every stratum (clearly distinguishable subgroups) in a population o Clustered random sample: population naturally subdivided into geographically distinct clusters, and samples are created by taking censuses in randomly chosen clusters Errors in Survey Data  We expect that there will exist random error when we conduct a survey (also known as chance error or sampling error)  Arises only in sample survey (I,e, this will be zero in a census)  Equals difference between o Value obtained from single random sample o Value obtained by taking a census  Size of error (and frequency) often estimated and reported with observed data (e.g., “A census would reveal a number within 2% of the survey value”) Bias in Survey Data  Sadly, random errors are the least of our problems, and even if we had a census, we might still encounter serious problems  Systematic errors( a.k.a, bias, nonsampling error) equals the difference between) o Value obtained by taking a census o The (unknown) rue value)  Can be hard to detect  Size of error cannot be estimated  Bias can creep into a survey in a number of ways o Can be built into the survey‟s design (pre survey) o Can occur during the survey‟s execution (during survey) o Can transpire during final procession (post survey) Selection bias:  Selection bias: systematic tendency to include elementary units with particular characteristics (while excluding those units with other (opposite) characteristic)  Source of selection bias o Nonrandom samples  (e.g. surveying all employees who leave work at 5pm. o Faulty design of random samples  E.g. selecting every 12th month to survey monthly stock prices o Faulty execution of perfect sampling plans  E.g. interviewers substituting other persons fro those randomly selected) Nonresponse bias:  Systematic tendency for elementary units with particular characteristics not to contribute data( while other units do) Sources of Nonresponse Bias o Contact problems  (e.g., potential respondent tears up mailed questionnaire, or won‟t answer calls of telephone surveyor, or refuses to cooperate with surveyor at the door o Badly designed questionaires  Eg. Unattractive design; hard-toread print. Questions that are borinh, intrusive, unclearm excessive in number, badly sequenced. Multiple-choce questions that over lap Response Bias:  Tendency for answers to survey questions to be wrong in some systematic way  Sources for response bias o Failing to  Pretest questionnaires  Motivate truthful answers  Define terms clearly o Asking leading questions o Inappropriate behavior of interviews and/or respondents o Processing errors Surveys versus Experiments  There are addition shortcomings of surveys (a.k.a “observational studies”) o Can only reveal association (correlation) o Data collected by merely recording information about selected characteristics (while ignoring other characteristics)  One recent development in economics is the use of experimental data to circumvent the shortcomings associated with observational studies  Experiments have a number of useful features including: o Can establish causation o Data is collected by exposing elementary units to some kind of change or treatment (while leaving all else unchanged) Association versus Causation  Association o Can show that A varies with b, but cannot prove the A causes b o May or may not indicate causation  Ex. Opening umberallas us associated with rainfall..which causes which? The answer cannot be settled purely on the basis of the data, rather ot must come from outside the data.  Causation o A has a casual relationship with B(i.e. if A then B) o Can be established (or refuted) by well-designed experiment  Ex. 1. Have participants randomly open and close umberalla, observe frequency of rainfall Experimental limits  With a well designed experiment, we introduce a change into a carefully controlled setting then observe the consequences, if any: if they do occur, we know the cause because we introduce it  However, experimentation is not feasible if: o Physically impossible  Ex. Resolution of issue requires sex or race be assigned to otherwise identical groups o Practically impossible  E.g. execution requires participation of large numbers of people unwilling to cooperate o Ethically unacceptable  E,g, creating a control group by forbidding head start enrollment for some disadvantaged children theory of probability 9/11/2013 5:37:00 AM Basic Probability Concepts  We now lay the foundation for the filed of statistical inference by studying the theory of probability Statistical inference  The set of techniques used to turn sample evidence into valid conclusions about statistical population of interest The theory of Probability  Calculus of the likelihood of specific occurrences  Gives rise to entire field of inferential statistics  Points to likely truth when we cannot know it with certainty Basic Probability Concepts  Experiments: any repeatable process from which an outcome measurement, or result is obtained  The Random Experiment o An experiment whose outcome cannot be predicted with certainty o Any activity that results in one and only one of several clearly defined possible outcomes, but that does not allow us to tell in advance which of these will prevail in any particular instance The Sample Space o A listing of all the basic outcomes of random experiment, also known sometimes as the  Outcome space  Probability space  EX. Tossing a single coin , rolling a single die The Nature of Random Events  A random Event o Any subset of the sample space  A simple Event o Any single basic outcome from a random experiment o Univariate, bivariate, multivariate  A composite Event o Any combination of two or more basic outcomes How Random Events Relate  Mutually Exclusive Events o Random events that have no outcome in common o Often also called disjoint or incompatible events  Collectively Exhaustive Events o Random events that contain all basic outcomes in the sample space o When the appropriate random experiment is conducted, one of these events is bound to occur  Complementary Events o Two random events for which all basic outcomes not contained in one event are contained in the other event (denoted A and Ā, the bar denoting „complemetnary‟) o Such events ar noth mutally exclusive and collectively exhaustive at the same time  Union of Events (Logical operator U (or)) o All basic outcome contained in one or the other event  Interesection of Events (logical operation [and]) o All basic outcomes contained in one and the other event ___________ General Addition Law for Events which are not Mutually Exclusive If there are two events, A and B, which are not mutually exclusive, then: P(A or B) = P(A) + P(B) – P(A and B) 
 Law of probability  Unconditional probability p(A) o The likelihood that a particular event will occur, regardless of whether another event occurs  Joint Probability p(A inter B) o The likehood that two or more vents will occur simultaneously  Conditional probability o The likelihood that a particular event will occur, given the fact that another event has already occurred or is certain to occur  Unconditional from Joint Probability Rule o To obtain an unconditional probability from joint probability, we sum the joint probability over all possible occurrences of the other event(s) o p(A)= Sum p(A intersection B)  ex. Sum o (face inter suit)  =p(facr inter heart) + p(face inter club) + p(face inter spade) + p(face inter diamond)  =3/52 +3/52 +3/52 +3/52 =12/52 Joint Probability Tables  a joint probability table shows frequencies or relative frequencies for joint events   passed  failed  Marginal prob Male  3800  4700  8500  Female  1600  2400  400  Marginal  total=12500 prob Joint prob passed failed Marginal prob Male 0.304 0.376 0.680 Female 0.128 Marginal prob 1  o if we divide each entry by the total number of students we obtain a se of relative frequencies that can be used to approximate the probability of occurrence of the relevant outcome, which we present on the next slide Laws of Probability: Multiplication  the general multiplication law (probability of A and B) o the joint probability of two events happening at the same time equals the unconditional probability of one event times the conditional probability of the other event, given that the first event has already occurred  p(A inter B) = p(A) * p(B|A)= p(B) * p(A|B)  the events A and B are dependent when the probability of occurrence of A is affected by the occurrence of B, hence p(A) is not equal to p(A|B). Independent events are those for which p(A)=p(A|B) o the special multiplication law (multiplication law for independent events)  p(A inter B) =p(A) * p(B) Sept.26.2013 Bayes’ Theorem (Reverend Thomas Bayes [1702-1761])  How do we estimate or revise probabilities when new information is obtained? By using Bayes‟ Theorem!  Consider a random experiment having several possible mutually exclusive outcomes E1…..En. Suppose that the probabilities of these outcomes, p(E1) have been obtained  We call these Prior Probabilities since they are obtained before new information is taken into account  A revised probability is called a Posterior Probability  Posterior probability are conditional probabilities, the conditioning event being new information  By using Bayes‟ Theorem, a prior probability is modified  The new probability is a posterior (conditional) probability  Let A be the new information. By the definition of conditional probability, the posterior probability of the outcome Ei, is given by o P(Ei|A)= p(Ei inter A)/ p(A)  Recall from the general multiplicative rule of probability that o P(Ei inter A) = p(Ei) * p(A|Ei)  Substituting (2)into (1) Background: Discrete Probability Distributions  A probability distribution is the theoretical (i.e., analytical) counterpart of the frequency distribution discussed earlier  By applying the laws of probability we can determine the probability that a variable takes on a specific value or range of values  We can always convert a categorical variable into a quantitative one by assigning it a numerical value, and it is often more convenient notationally speaking to do so o The simplest example is a male/female sample where males are assigned the value 1 and females the value 0 o In this case we talk about random variable X denoting a person‟s sex with XE{0,1}  By way of example, consider tossing a fair coin 3 times. There are 8 possible outcomes, each outcome is equally likely, and has a probability 1/8  The outcomes are a1=(h,h,h), a2=(h,h,t) a3=(h,t,h), a4=(t,h,h) a5=(h,t,t) a6=(t,h,t) a7-(t,t,h) a8=(t,t,t)  Letting X denote the number of heads obtained in 3 tosses, find p(X=2)  We see that p(X-2)=p(a2)_p(a3)+p(a4)= 3/8 Basic Concepts  Random variable: any quantitative variable the numerical value of which is determined by a random experiment, and thus by chance  Probability distribution: a table, graph, or formula that associates each possible value, x, of a random variable, X, with its probability of occurrence, p(X=x)  Summary measures for the probability distribution are often desired, such as the mean and standard deviation  If the variable is a discrete random variable, the distribution is a discrete probability distribution and can be represented in: o Tabular form o Graphical form o Symbolic form (e.g, p(X=x))  No probability can be negative (p(X=x) >=0)  The sum of the probabilities of all values of the discrete random variable must equal 1  Cumulative distribution function: Let X be discrete random variable and let x be any real number, The CDF of X is the function F(x)=p(X<=x)  F(x) denotes the probability that the random variable X assumes a value less than or equal to x o For ant value b,0 <=F(b) <=1 o If a0.5  Let X be the number of successes in n trials. Then Y=n-X is the number of failures in n trials. For example, if n=10 then for any specific Binomial Summary measures  For a binomially distributed random variable, we can obtain the following results: o Arithmetic Mean or Expected value: o variance o Standard Deviation Normal Approximation to the Binomial  What is the shape of the binomial distribution? If pi=0.5, then the binomial distribution is symmetric  It can be shown, by means of the central limit theorem, that a bell shaped approximation works fairly well provided that n*pi>=5 and n(1-pi)>=5  Because of this the normal distribution is sometimes used to approximate the binominal distribution due to its ease of use The Sample Proportion of Success P=X/n  Often we are interested in the proportion of successes in a sample rather than the number of successes  Recall that the sample proportion is denoted as P(hat)= X/n. The random variable must take on the values 0,1…n, so the sample proportion must take on the value 0/n,1/n..1  It follows that the probability for X=3 is the same as the probability for P(har)=3/n  Therefore, the probability distribution for the random variable P(hat) can be derived from the probability distribution for the random variable X  The mean of P(hat) is pi, and the variance is pi(1-pi)/n Continuous Probability Density Functions  Continuous random variables o Unlike their discrete counterparts, continuous random variables are „uncountable‟ and can assume any value on the real number line o Rather than looking at p(X=x) (i.e the probability that a continuous random variable lies in an interval)  The probability density function o Commonly shows probabilities with ranges of values along the continuum of possible values that the random variable might take on o Let some smooth curve how the probability of a continuous random variable X is distributed. If the smooth curve can be represented by a formula f(x), then the function f(x) is called the probability density function  Areas will be seen to play an important role in determining probabilities  Since there exist an infinite number of points in an interval, we cannot assign a non-zero probability to each point and still have their probabilities sum to one, therefore, the probability of any specific occurrence of a continuous random variable is defined to be zero  The total are under the probability density function (just as under the relative frequency histogram) must therefore equal 1  We shall study two common distributions o The uniform probability distribution o The normal probability distribution Basic Rules for probability density functions  If a smooth curve is to represent a probability density function, then the following two requirements must be met: o The total area between the curve and the horizontal axis must be unity i.e. Integral (-infinity to +infinity)F(x)dx=1 o The curve must never fall below the horizontal axis i.e, f(x)>=0  Note that the probability of obtaining any one value is zero  The probability that X falls between 2 values a and b, p(a
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