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4 LECTURE PUBP704: Statistical Method for Policy Analysis Instructor: Naoru Koizumi, Ph.D. Office: Arlington 254 Office hours: Thursdays 3:00-4:00 and by appointment Phone: 703-993-8380 E-mail: nkoizumi@gmu.edu (for primary communication) Course Website: http://classweb.gmu.edu/nkoizumi/PUBP704/syllabus.htm Teaching Assistant: Huaqun Li (hli5@gmu.edu) NEXT WEEK READINGS: [2nd Edition] Ch. 11 / [1st Edition] Ch. 5 Announcements and Today s topic Announcements PS1 has been uploaded (download both PS1 and 2 datasets Due next Thursday (Feb. 21st). Encourage teaming-up with your classmates. Feel free to send me/TA an email (or drop by my office) for questions. Past project examples have been uploaded. Today s Topic Overview: Random Variable and Distributions Law of Large Numbers / Central Limit Theorem Introduction to Hypothesis Testing 1 Review: Basic Statistics Typical Value (Central Tendency Measure) Mean: average Not a good statistic if there is an outlier, but cardinal Median: the value at 50% Good if there is an outlier, but ordinal Variability (Dispersion Measure) Variance: sum of squared distance from the mean /(n-1) Standard deviation: Square root of variance, more intuitive Standard score / Z-score: Could help you find out how far each obs is from the average. Dispersion Measures: Z-score Age data: 25, 30, 20, 40 and 37 Mean: 30.4 Variance: 68.3 Standard deviation: 8.264 Suppose your age is 39. How much older are you compared to the average? Your age Mean = 39 30.4 = 8.6 Standard Score (or Z-score ) We use this Is being 8.6 yrs older than average makes you really old in the sample? 2 Dispersion Measures: Z-score Z-score: Measures the deviation of each value from the mean. Z-score The data value - Mean Standard Deviation X i X S 8.6 8.264 The data value - Mean Standard Deviation 1. 04 Your Z-score If your value is equal to mean, then the your Z-score = 0 Usefulness of Standard Normal Distribution Suppose the variable, IQ, which is known to be normally distributed Approximately 2/3 (or 68%) has the Z-scores within 1. 95% of a sample is within 2 ( 1.96, to be specific). 2.5% Normal distribution Histogram is (perfectly) symmetric 2.5% 68% -1.96 +1.96 3 Important Fact about Z-score IF distribution of the variable you are looking at (e.g. IQ, test score) is normally distributed , THEN the Z-score of a particular observation tells you how unusual (measured by deviation from the mean) the observation is. Z-score of your IQ = +2.0 (you re within top 2.5% => you have unusually high IQ). Z-score of your income is -2.0 => you re unusually poor (assuming that income is normally distributed). But to be able to say this, IQ (or the variable of your interest needs to be normally distributed Less Important Random Variable and Prob. Distribution Random variable (X): a variable whose value is determined by the outcome of an experiment in which outcome is subject to chance (= values determined by stochastic or probabilistic events). Examples of possible random variable: Coin toss: {0,1} Bernoulli distribution Dice toss: {1,2,3,4,5,6} Uniform distribution Customer arrivals to a bank teller: {could be anything} Poisson distribution Customer service time: {could be anything} Exponential distribution The probabilities associated with these random events are often represented by a particular type of probability distribution. 4 Less Important Type of Distributions Types of distribution Discrete random variable and distributions: Used to describe discrete events like arrival rate. Mathematically summation concept Bernoulli, Binomial, Poisson, Geometric, Hypergeometric, etc. Continuous random variable and distributions: Used to describe continuous events like service time. Mathematically integral concept Normal, Uniform, Chi-Sq, Exponential, etc. Discrete Probability Distribution n p x i 1 i 1 The probability distribution is like a histogram, but, for each x-axis point, the associated y-value represents the percentage (or probability), as opposed to the frequency. 5 Continuous Probability Distribution Prob. 1 2 Income ( 000) Ex. Income distribution Each area is obtained by integrating (in stead of aggregating) the range. Suppose that income of the US population is distributed like the left figure. The probability of drawing someone who earns more than $40,000 is given by the area 2 . The probability of drawing someone who earns less than $40 000 is given by the area 1 . p x i 1 Note: The prob. associated with any particular value of x (income) is zero (mathematically). Less Important p.d.f. and c.d.f. Exponential p.d.f. Ex. Service time Probability density function (p.d.f.) 2/3 1/3 1/6 0 1 2 3 4 5 6 Uniform p.d.f. Ex. dice toss Cumulative distribution function (c.d.f.) Uniform c.d.f. 1 5/6 4/6 3/6 2/6 1/6 0 What is the probability of obtaining x<4? A. 2/3 What is the probability of obtaining x>4? A. 1-2/3=1/3 Exponential c.d.f. 1 2 3 4 5 6 6 Less Important Shape of p.d.f. ( moments ) Parameters of distribution functions <= determines the shape Mean (a measure of central tendency) Variance and standard deviation (measures of dispersion) Skewness (a measure of the location of the peak point) +ve skewness -ve skewness Kurtosis (a measure of the spread) lower kurtosis higher kurtosis Summary: r.v. and Distributions Outcomes of a random variable are used to form a (probability) distribution. Distributions Types: Discrete vs. (depending Continuous on the phenomenon you d like to describe) p.d.f. or c.d.f Shapes: Parameters (mean, variance, etc.) Adds (or integrated) to 1 7 Less Important Normal Distribution 1 2 Gauss (1809) P x e x 2 /2 2 2 parameter family (mean, s.d.) N( , 2) Normal distribution with the mean, and the variance, 2) N(50,10) Perfectly symmetric N(50,20) N(50,30) Z-score and Normal Distribution Z-score The data value - Mean Standard Deviation X i X S Standardization makes N(0,1) if the original distribution is normally distributed. Standard Normal Distribution N(0,1) N(50,5) Standardization 50 0 8 SPSS Practice Creating Standard Normal Distribution Data set: electric.sav Select everyone whose weight is less than 180 lbs (variable body weight - wt58 ) Make a histogram of serum cholesterol (chol58). Create Z-scores for each value of chol58. Make a histogram of the Z-scores (Zchol58). Usefulness of Standard Normal Distribution For any given Z-score, you can tell that the observation is in top or bottom %. You can calculate the probability of any random variable taking below or above any given value. Useful values (called critical values later) to remember. Prob. of a randomly selected person s IQ falling into either of these areas is about 0.025. Prob. (two tails) 0.10 (10%) 0.05 (5%) 0.01 (1%) Z-score 1.64 1.96 2.57 2.5% (0.025) 2.5% (0.025) You can calculate the probability corresponding to any Z-score. http://www.statsoft.com/textbook/sttable.html#z http://www.oswego.edu/~srp/stats/normal_cdf.htm 9 Practice: Standard Normal Table Your Z-score for age was 1.22. What is the probability of your neighbor being older than you? P(Z>1.22) = 0.5 - 0.3888 = 0.1112 1.0 - 0.8888 = 0.1112 How about the probability of your neighbor being younger than you? P(Z<1.22) = 0.5 + 0.3888 = 0.8888 or simply, 1 - 0.1112 = 0.8888 What is the probability of both of your neighbors are older than you? 0.1112 * 0.1112 = 0.0124 Independence of two random events Usefulness of Standard Normal Distribution So far, we assumed that the variable of your interest (e.g., age, income, IQ) is normally distributed In reality, this is often not the case. But we know that something (means of a variable) is always normally distributed. Law of Large Numbers Central Limit Theorem 10 Law of Large Number and Central Limit Theorem Suppose distribution of population IQ : N(100,1) Take 500 random samples of size 50 Sample size, n 100 No. of Samples 1 2 3 4 ... 50 1 100 120 110 98 90 2 99 106 118 102 100 3 101 96 108 120 100 . . . . . . . . 500 110 90 99 105 103 Sum 5000 5600 5100 . . . . 4900 X 100 112 102 98 What is the mean of the sample means, X? A. 100 Law of Large Numbers What is the distribution (histogram) of X would look like? A. Normal D Central Limit Theorem Central Limit Theorem Uniform Distribution 2/3 Dice example Uniform p.d.f. 1/3 Random sampling 1 1 4 6 4 6 3 5 . . . . 4 6 3 5 2 4 2 6 3 4 6 4 ...50 Sum X 2 39 3.9 1 40 4.0 1 53 5.3 . . . . . . . . 6 46 4.6 1/6 0 1 2 3 4 5 6 3.5 1 2 3 . . . . 500 3.5 : True mean = Expected Value 11 Example: Salary Distribution of Major League Baseball Players Skewed to the left: Positive Sampling Distribution Suppose that we take a random sample of size n out of these 747 baseball players. Calculate the mean salary of these n players. Repeat the above steps 100 times. Sampling distribution of mean salaries. 1 n=2 2 n=5 3 n = 10 4 n = 40 As a rule of thumb, the distribution looks approx. normal when n 30. But it depends on the shape of original distribution. 12 Central Limit Theorem Let X1, ,Xn be a random sample from a population with mean and variance 2. Let X X 1 .... X n n be the sample mean. Then if n is sufficiently large (n 30), the SAMPLE MEANS forms a normal distribution with the mean and the s.d. / n regardless of the shape of the original distribution. X N , follows True (pop) mean = expected value 2 n When the sample size goes up, the variance goes down. Note on your project data Sample size (the number of observations >30) is just fine. No need to be ambitious data collection and cleaning takes time! 13 Less Important Expected Value Expected value (E(X)= ): The expected value is the long-run average or the average obtained when you have many observations (<= law of large number). Suppose a random variable has N possible outcomes: E(X) = X1*P(X1)* + X2 *P(X2)+ + XN*P(XN) Example: Dice Toss E(X) = 1*(1/6) + 2*(1/6) + 3*(1/6)+ 4*(1/6) + 5*(1/6) + 6*(1/6) = 3.5 Less Important Expected Value Pay $2/time to play: E Payoff 0 0. 123967 . . . 2 0. 789538 0. 0697 Law of large number asserts that casinos always make profit... Always ve. 14 Where Are We Heading? Hypothesis Testing We can do a hypothesis testing only because we know that the law of large numbers and the central limit theorem are true. Two Important Facts When n is sufficiently large, SAMPLE MEANS are always normally distributed. Then, the Z-scores of the sample means would form a standard normal distribution, N(0,1). Hypothesis Testing A typical question you can answer in Hypothesis Testing: The mean internet time for US college students is 22 hours/week. The standard deviation is known as 4 hours. At GMU, a random sample of 64 students was drawn. The sample mean was slightly higher and 23 hours. Is the mean of internet time for GMU students statistically significantly different from that of typical US college students? 15

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