Exam 1 Study Guide
Exam 1 Study Guide GES 255
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This 11 page Study Guide was uploaded by Nora Salmon on Thursday October 8, 2015. The Study Guide belongs to GES 255 at University of Alabama - Tuscaloosa taught by Robert Batson in Fall 2015. Since its upload, it has received 131 views.
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Date Created: 10/08/15
EXAM 1 STUDY GUIDE FALL 2015 GES 255002 Chapter One Overview and Descriptive Statistics 0 11 Populations Samples and Processes I A population is a welldefined collection of objects I A sample is a subset of a population I A variable is any characteristic whose value may change from one element to the other in a population I Data sets Univariateobservations on a single variable 0 Ex variable9transmission on a car9automatic or manual Multivariateobservations made on more than one variable 0 Ex 2 variables9systolic blood pressure AND diastolic blood pressure I Dr Batson compare and contrast 0 Enumerative studies use a fixed finite sampling frame 0 Analytic studies use a changing sampling frame I Analytic studies are studies aimed at improving a product for example 0 12 Pictorial and tabular Methods in Descriptive Statistics pictures provided at the end of study guide I Constructing a stemandleaf display Select one or more leading digits for the stem values a display with 520 stems is typical List stem values in a vertical column Order the leaves for each stem from least to greatest and record each leaf or observation beside the corresponding stem value Indicate the units for stems and leaves somewhere on the display I Constructing a dot plot Create a measurement scale horizontal Draw a dot representing each observation above the corresponding location on the measurement scale Stack repeated values as dots one on top of the other I Histograms Two different ways of constructing histograms depending on the type of variable 0 A variable is discrete if the possible values are finite or countable I The relative frequency of a value is the fraction or proportion of times the value occurs number of times xi occurs o number of ob sewations in data set 1 FALL 2015 GES 255002 EXAM 1 STUDY GUIDE A histogram for discrete data requires determining the frequency and relative frequency of each x value Next draw a rectangle whose height is the relative frequency or alternatively the frequency of that value and whose width is 1 o Centered over the x value on its axis 0 A variable is continuous if the possible values are an interval a ray or a real number Identify the max and min data points Determine the number of classes categories class width that covers the whole range of data Make a frequency table and assign each data point to its class Draw a histogram whose horizontal axis is comprised of the classes Above each class interval draw a rectangle whose height is the relative frequency of that class Histogram Shapes o A unimodal histogram is one that rises to a single peak and then declines o A bimodal histogram has two different peaks 0 A histogram is symmetric if the left half is a mirror image of the right half 0 A unimodal histogram is positively skewed if the right or upper tail is stretched out compared with the left lower tail 0 13 Measures of Location The sample mean bar x is the average of a given set of numbers The sample median of a data set is the middlemost value once the observations have been ordered from smallest to largest Dr Batson the median is the value of x that has 50 of the population on either side it divides the range into two parts of equal probability A trimmed mean is a mean computed by eliminating some percentage of both the smallest and largest numbers in a data set this way the mean will be affected less by the presence of outliers It s negatively skewed if the left tail is stretched out compared with the right tail EXAM 1 STUDY GUIDE 3 FALL 2015 GES 255002 Dr Batson this is a compromise between the mean and the median o 14 Measures of Variability I In a sample of size n The range is the maximum value the minimum value An observation s deviation from the mean is its value subtracted from the sample mean The sample variance is the square of the standard deviation and is given by Eve if i1 n 1 The standard deviation is the square root of the variance S The upper fourth is the median of the upper half of the data lower fourth is the median of the larger half of the data 0 Fourth spreadupper fourthlower fourth O I In a population The first quartile is the value of x that has 25 of population to left 75 to right The third quartile is the value of x that has 75 of population to left 25 to right I Constructing a boxplot of a data set Compute the minimum x lower fourth median upper fourth and maximum x values Draw a horizontal measurement scale including the min and max Draw a box enclosing the lower fourth and upper fourth Draw a line at the median location to divide the box in two won t always be exactly in the middle of the box Draw line segments from the edges of the box out to the respective extreme x values I Boxplots with outliers Complete steps 14 in the box plot instructions above Find the outliers o A mild outlier is 15fourth spread away from the upper or lower fourth respectively I Solid dots on the box plot 0 An extreme outlier is 3fourth spread away from the upper or lower fourth respectively I Open circles on the box plot Chapter Two Probability FALL 2015 GES 255002 EXAM 1 STUDY GUIDE 0 21 Sample Spaces and Events A random experiment is a sequence of trials or actions with a set of possible outcomes you can distinguish one outcome from another when a trial is complete and the outcome of each next trial is uncertain The sample space is all possible outcomes of a specific experiment An event is a collection or subset of outcomes in sample space 0 Dr Batson Simple event throwing a dice vs compound event multiple die I Events A and B are mutually exclusive if A and B are disjoint sets A n B Q I Dr Batson recommends studying Figure 21 on p 54 Relations from Set Theory The complement of A denoted by A is the set of all outcomes in S that are not contained in A The union of two events A and B denoted by A U B is the set of all outcomes that are either in A or in B or in both events The intersection of two events A and B denoted by Am B is the set of all outcomes that are in both A and B o 22 Axioms and Interpretations of Probability Probability is a function P whose range is 01 Where do probabilities come from Dr Batson PA is the quotlimiting relative frequency of an even A based on observing an infinite sequence of trials fpw If n When5491M as n infinity o 23 Counting Techniques Let N objects be the sample space 9 PANAN Ordered pairs If an event consists of ordered pairs of objects A B then the number of possible outcomes is the number of ways A can be selected times the number of ways B can be selected You can use permutations for an ordered sequence that is larger than an ordered pair EXAM 1 STUDY GUIDE 5 FALL 2015 GES 255002 For example the number k of possible first second and third place winers in a race of n100 people n Pk nn 1n k 1 n k I For an unordered subset of n objects you can calculate the amount of possible outcomes k using a combination n n k kn k 0 24 Conditional Probability I PAI B refers to the conditional probability of A given that event B has occurred Here B is the quotconditioning event Ckn P A B PAB 1f PBgtO PB Dr Batson recommends studying examples 225229 p 74 77 I The law of total probability Dr Batson A partition of p is a set of events such that A1U A2 U A3 U Ak The total prior probability of event B is o PB PBA1PA1 PBA2PA2 PBIAkPAk Bayes Theorem states that if a set is a partition and B is an arbitrary event in the partition then PB ADP A1 PB ADP A1 PB k PB AiPAI 0 PAjl 3 quot o 25 Independence I Defined by Dr Batson two events in a set are independent if the probability of one event occurring is not affected by the occurrence or nonoccurrence of the other n mathematical terms PAI BPA IfA and B are independent then A and B A and B and A and B are independent I Multiplication rule for PAn B A and B are independent if and only if PAn BPAPB For more than two events the events are independent if 0 PAn Bn cn ZPAPBPCPZ Chapter Three Discrete Random Variables and Probability Distributions FALL 2015 GES 255002 EXAM 1 STUDY GUIDE 0 31 Random Variables I A random variable is a function whose domain is the sample space 9 and whose range is a set of real numbers Dr Batson quotAn rv is a function X from 9 into R Mp 9 R 0 quotOutcomes in X p are always counts or measurements associated with one or more outcomes in the set 50 0 An event A in p is equivalent to the event B in Xp and vice versa if AX391B I Probability Example Let y sum of faces up roll of two fair dice Let event AY211 9 quotWhat is the probability of the faces adding up to 11 or greater Then PyA PY11 PY12 236136336 o PY11236 because 56 65 0 PY12136 because 66 I Two Types of Random Variables Dr Batson quotA discrete rv has range space that is a finite set of numbers or countable infinite set of numbers 0 Use counting principles summation Dr Batson quotA continuous rv is one whose range space is an interval a series of disjoint union of intervals a ray or a real line 0 Use derivative integrals o 32 Probability Distributions for Discrete Random Variables I Dr Batson quotLet X be a discrete random variable The probability distribution or mass function px for X describes how probability is distributed for every number xexm The pdf denotes a probability for each possible outcome x in the set 9 of X In math terms px PXx Ps 6p Xsx I A Bernoulli Random Variable is an rv whose range space is 01 Dr Batson quotEach value of 0c chosen specifies a particular instance of the Bernoulli so there is in fact a family of Bernoulli distributions with constant on referred to as the parameter of the distribution to which it belongs o P010l o P1 0L 0 Px0 otherwise I The Cumulative Distribution Function cdf The cdf Fx of a discrete rv X with pmf px is defined for each 19y x by FxPsz ysxgm EXAM 1 STUDY GUIDE 7 FALL 2015 GES 255002 Dr Batson quotThe cdf is a step function has a jump at every possible value of X and will be flat between possible values I Dr Batson On the relationship between px and Fx for discrete rv s Let the range space of X be Xp x1 x2 x3 in ascending order 0 PXiFXi39FXi1Ii22 O PX1FX1 For any two real numbers a and b with a s b o PasxsbFbFa39 I Where a39 represents the largest xi strictly less than a o 33 Expected Values I Let X be a discrete rv with pdf px The expected or mean value of X is EX 2 xi 170939 O xEXS o This value is also called the population mean and is denoted by the Greek letter mu W I Expected value of a function X Sometimes interest will focus on the expected value of some function hX rather than on just X 0 You can find the expected value of a function of X by plugging it into the same formula for mean value I For YhX EYEhX 00 2 We gtpltxigt I y I The hX function is quite often a linear function aXb Dr Batson Let a b be real numbers 0 EaXbaEXb o EaXaEX o EXbEXb o Ebb I In plain English the expected value of a linear function equals the linear function evaluated at the expected value EX so you just compound the functions I The variance of X We use the variance of a discrete rv X to assess the amount of variability in the distribution of X FALL 2015 GES 255002 EXAM 1 STUDY GUIDE 0 If X has pmf Px and expected value p then the variance of X VX is 2 xi 1102 39Pxi VX xiEXS Some notes from Dr Batson VX is EXu2 o VX will always be greater than or equal to 0 and if VX0 then X is a constant 0 VX shortcut VXEX2EX2 I The standard deviation of X The standard deviation of X is simply the square root of VX 2 O 0X 0X o 34 The Binomial Probability Distribution I A binomial experiment is one that satisfies the following The experiment consists of a sequence of smaller experiments called trials where the number of tiralbs is fixed in advance Each trial can result in one of the same two possible outcomes AKA it s a Bernoulli trial The trials are independent The probability of success PS or simply p is constant from trial to trial I The binomial random variable X associated with a binomial experiment is Xthe number of 5 s successes among the trials Dr Batson Therefore the number of successes X in n trials has pdf n px px1 p 39x x O12n x O 0 NOTE Dr Batson substitutes px for the text s use of bx n p different notation same meaning I Mean variance of a binomial rv X EXnpM VXno1IonIool02 0sqrtnpq I Dr Batson s notes me bx n p whereas Bx n pcdf of X There are four approaches to compute the cdf O O EXAM 1 STUDY GUIDE 9 FALL 2015 GES 255002 Use the binomial tables such as Appendix Table A1 when n is small Calculate the respective by n p for y01 x and add the terms Use a computer program Approximation see 36 Chapter 4 when n is large 0 35 Hypergeometric and Negative Binomial Distributions I The hypergeometric distribution is the exact probability model for the number of 5 s in a sample I The binomial rv X is the number of 5 s when the number of trials is fixed beforehand whereas the negative binomial distribution arises when you fix the number of 5 s desired and let the number of trials be random I Dr Batson the situation of a hypergeometric distribution A finite population of N objects is to be samples from Each object chosen can be characterized as Success 5 or Failure F and there are M successes and NM failures A sample of n objects is chosen from the population without replacement 0 This is important because binomial distributions USUALLY INVOLVE REPLACEMENT so events can be independent Let Xnumber of objects among the n objects that are S s I The pdf of X is called the hypergeometric distribution and is given by M N M X n X x p N n M EX n The mean expected value ofX is EX N o EXnp The variance of X is VX M M VX n 1 N n N N N l o VXnp1p I Dr Batson recommends looking at Example 336 here 0 36 The Poisson Probability Distribution FALL 2015 GES 255002 EXAM 1 STUDY GUIDE I Dr Batson The Poisson process is a counting process for the number of events that have occurred up to a particular time Let X represent a count of events like arrivals errors etc per unit of time or length or area ex 10 screws per yard of wood etc Let krate of events per unit 0 Then X has a Poisson distribution if its pdf is e 11x px 1 x I for x012 o Parameter EXuVXo2 o osq rtt I Note from textbook in any binomial experiment in which n is large and p is small bx39 n p is approximately equal to px39 u where unp Example of a StemandLeaf Plot 64 35 64 33 70 Stem Thousands and hundreds digits 65 26 27 06 83 Leaf Tens and ones digits 66 05 94 14 67 9O 7O 00 98 7O 45 13 68 9O 7O 73 50 69 OO 27 36 O4 70 51 05 11 4O 50 22 71 31 69 68 05 13 65 72 80 09 2007 Thomson Higher Education Example of a Dot Plot Temperature Ranges for ORings K I I Temperature l l l l l l l l 30 40 50 60 7O 80 2007 Thomson Hiqher Education 3 o o o l l EXAM 1 STUDY GUIDE 11 FALL 2015 GES 255002 Example of a Histogram Relative frequency Hits game A I0 05 0 i l l i i i i i 0 IO 20 2007 Thomson Higher Education Skew I l M M M a Negative skew b Symmetric 0 Positive skew 2007 Thomson Higher Education Modified Box Plot with Outliers I I l 400 I l l I I I I I I I I 600 800 1 000 1 200 Daily nitrogen load x 7 load 1600
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