Midterm Review Econ 240A 1 The Big Picture 2 The Classical Statistical Trail Rates Inferential Statistics Descriptive Statistics Probability Discrete Random Proportions Application Binomial Variables Power 4 4 Discrete Probability Distributions Moments Descriptive Statistics Power One Lab One Concepts central tendency mode median mean dispersion range inter quartile range standard deviation variance Are central tendency and dispersion enough descriptors 4 Concepts Normal Distribution Central tendency mean or average Dispersion standard deviation Non normal distributions Density Function for the Standardized Normal Variate Draw a Histogram 0 45 0 4 0 35 Density 0 3 0 25 0 2 0 15 0 1 0 05 5 4 3 2 1 0 0 1 2 3 4 5 Standard Deviations 5 The Classical Statistical Trail Rates Descriptive Statistics Inferential Statistics Classicall Proportions Application Modern Probability Discrete Random Binomial Variables Power 4 4 Discrete Probability Distributions Moments Exploratory Data Analysis Stem and Leaf Diagrams Box and Whiskers Plots 7 Weight Data Males 140 145 160 190 155 165 150 190 195 138 160 155 153 145 170 175 175 170 180 135 170 157 130 185 190 155 170 155 215 150 145 155 155 150 155 150 180 160 135 160 130 155 150 148 155 150 140 180 190 145 150 164 140 142 136 123 155 Females 140 120 130 138 121 125 116 145 150 112 125 130 120 130 131 120 118 125 135 125 118 122 115 102 8 115 150 110 116 108 95 125 133 110 150 108 9 Box Diagram median First or lowest quartile 25 of observations below Upper or highest quartile 25 of observations above 10 3rd Quartile 1 5 IQR 156 46 5 202 5 1st value below 195 11 The Classical Statistical Trail Rates Inferential Statistics Descriptive Statistics Probability Discrete Random Proportions Application Binomial Variables Power 4 4 Discrete Probability Distributions Moments Power Three Lab Two Probability 13 Operations on events The event A and the event B both occur A B Either the event A or the event B occurs or both do A B The event A does not occur i e not A A 14 Probability statements Probability of either event A or event B p A B p A p B p A B if the events are mutually exclusive then p A B 0 probability of event B p B 1 p B 15 Conditional Probability Example in rolling two dice what is the probability of getting a red one given that you rolled a white one P R1 W1 16 In rolling two dice what is the probability of getting a red one given that you rolled a white one 17 Conditional Probability Example in rolling two dice what is the probability of getting a red one given that you rolled a white one P R1 W1 p R1 W 1 p R1 W 1 p W 1 1 36 1 6 18 Independence of two events p A B p A i e if event A is not conditional on event B then p A B p A p B 19 The Classical Statistical Trail Rates Inferential Statistics Descriptive Statistics Probability Discrete Random Proportions Application Binomial Variables Power 4 4 Discrete Probability Distributions Moments Power 4 Lab Two 21 Three flips of a coin 8 elementary outcomes p p p H H 1 p H H T p 1 p T T H H T 1 p p T H 1 p T T 3 heads 2 heads 2 heads 1 head 2 heads 1 head 1 head 0 heads 22 The Probability of Getting k Heads The probability of getting k heads along a given branch in n trials is p k 1p n k The number of branches with k heads in n trials is given by Cn k So the probability of k heads in n trials is Prob k Cn k pk 1 p n k This is the discrete binomial distribution where k can only take on discrete values of 0 1 k 23 Expected Value of a discrete random variable E x n x i p x i i 0 the expected value of a discrete random variable is the weighted average of the observations where the weight is the frequency of that observation 24 Variance of a discrete random variable VAR xi n 2 x i E x i p x i i 0 the variance of a discrete random variable is the weighted sum of each observation minus its expected value squared where the weight is the frequency of that observation 25 Lab Two The Binomial Distribution Numbers Plots Coin flips one two ten Die Throws one ten twenty The Normal Approximation to the Binomial As n p k N np np 1 p Sample fraction of successes p k n E p np n p Var p np 1 p n 2 p N p p 1 p n 26 Density Function for the Standardized Normal Variate Lab Three and Power 5 6 1 2 z 0 1 2 f z 1 2 e Z N 0 1 Prob 1 96 z 1 96 0 95 0 45 0 4 0 35 Density 0 3 0 25 0 2 0 15 2 5 z p E p p 2 5 0 1 0 05 5 4 3 2 1 0 0 1 2 3 1 96Standard Deviations 1 96 4 5 p p 1 p n prob 1 96 p E p p 1 96 0 95 prob 1 96 p p p 1 96 p 0 95 prob 1 96 p p p 1 96 p 0 95 prob p 1 96 p p p 1 96 p 0 95 27 Hypothesis Testing Rates Proportions One tailed test Step 1 hypotheses H0 p f Ha p f One tailed test Step 2 test statistic z p E p p p f p p p 1 p n Density Function for the Standardized Normal Variate One tailed test Step 3 choose e g 5 0 4 Z 1 645 0 35 0 3 Density Step 4 this determines The rejection region for H0 0 45 Reject if 0 25 0 2 p f 1 645 0 15 5 0 1 0 05 5 4 3 2 1 0 0 1 Standard Deviations 2 3 4 28 5 Remaining Topics Interval estimation and hypothesis testing for population means using sample means Decision theory Regression Estimators OLS Maximum lilelihood Method of moments ANOVA 29 Midterm Review Cont Econ 240A 30 Last Time 31 The Classical Statistical Trail Rates Inferential Statistics Descriptive Statistics Probability Discrete Random Proportions Application Binomial Variables Power 4 4 Discrete Probability Distributions Moments Remaining Topics Interval estimation and hypothesis testing for population means using sample means Decision theory Regression Estimators OLS Maximum lilelihood Method of moments ANOVA 33 Lab Three Power 7 Population Random variable x Distribution f 2 f Pop Sample Sample Statistic 2 x N Sample Statistic n s 2 xi x 2 n 1 i 1 34 f x in this example is Uniform X U 0 5 1 12 E x 0 5 Var x 1 12 f x 0 …
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