Sample means and random variables

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1 Empirical Loop Sample means and random variables Descriptive Statistics Collect Data Research Design Inferential Statistics Hypothesis Sample, Population, Estimate of Population Types of Populations Real-All potential observations are accessible at the time of sampling Sample Population Estimate of Population from a Sample Hypothetical-All potential observations NOT accessible at the time of sampling 3 Good Samples Sample, Population, Estimate of Population Representative Random As large as you can afford Sample Population Estimate of Population from a Sample 5 6 1

2 What is? What is? Random Variables - Discrete - Continuous Normal Distribution Rules Independent Random Variables Conditional Frequentist-The percentage of time some event will happen. Bayesian-Your belief that something is true. 7 8 What is? Random Variables - Discrete - Continuous Normal Distribution Rules Independent Random Variables Conditional Random Variable A variable whose numeric value is a randomly chosen member of a population. X=x X=1, X=2, X=3 X=, X=5, X= Each outcome has a probability of occurring Discrete Random Variable A variable whose possible values are separated by gaps Die Face

3 Mass Function (PMF) Mass Function (PMF) The probability of each outcome must be greater than or equal to 0. The probability of all outcomes must add up to Die Face 13 1 Discrete Uniform Distribution Binomial Distribution Die Face 15 Number of Heads in Two Coin Tosses 16 What is? Random Variables - Discrete - Continuous Normal Distribution Continuous Random Variable A variable whose possible values are always separated by another possible value. Rules Independent Random Variables Conditional

4 p(179<=x<=181) p(x<=178) Discrete Continuous p(a x b) = b a f (x)dx p(x=x)= Density Function (pdf) Continuous Uniform Distribution The pdf must never go below 0. The area under the pdf must equal Normal Distribution 23 2

5 Normal Distribution f (x) = (x µ ) 1 2 σ 2π e 2σ 2 p(a x b) = b a f (x)dx 25 Rules: or What is? Random Variables - Discrete - Continuous Normal Distribution Rules Independent Random Variables Conditional What s the the probability of rolling a 1 OR a 2 on a fair die? What s the the probability of rolling an odd number OR a number less than on a fair die? Rules: or Rules: not P(X=a or X=b)=P(X=a)+P(X=b)-P(X=a & b) What s the the probability of NOT rolling a 1 on a fair die? Special Case: Mutually Exclusive P(X=a or X=b)=P(X=a)+P(X=b)

6 Rules: not Compute the following probabilities. P(X a)=1-p(x=a) Assume that the probability that a fatal car accident involves a driver who has been drinking alcohol is.0, who has been smoking pot is.15, and who has been both drinking alcohol and smoking pot is % 83% What is the probability that a driver involved in a fatal car accident was NOT drinking alcohol? What is the probability that a driver involved in a fatal car accident was drinking alcohol OR smoking pot? What is? Random Variables - Discrete - Continuous Normal Distribution Rules Independent Random Variables Conditional P(X=h)=1/2 P(X=t)=1/2 First Flip Second Flip P(X=h, Y=t)=? 33 3 P(X=h)=1/2 First Flip Second Flip P(X=ace)=/52 P(X=h, Y=t)=1/ 1st Draw 2nd Draw P(X=t)=1/2 P(X=h)P(Y=t)=1/ P(X=ace, Y=ace)=?

7 Independent Events Sampling without replacement P(X=ace)=/52 (/52) 2 =~.006 1st Draw 2nd Draw P(X=h, Y=t)=P(X=h)P(Y=t)=1/ Dependent Events P(X=ace, Y=ace)=*3/(52*51)=~ P(X=ace, Y=ace)=*3/(52*51) 38 Independent Random Variables Exercise 13.2: Assume that people are equally likely to be born in each of the 12 months and that the birthdays of married couples are independent. Find the following probabilities: Two random variables are independent if the value taken by one variable tells you nothing about the value taken by the other variable. A) The probability that the wife is born in February and the husband is born in January. B) The probability that both wife & husband were born in Spring (i.e., April or May). P(X=x,Y=y)=P(X=x)P(Y=y) 39 0 What is? Random Variables - Discrete - Continuous Normal Distribution Rules Independent Random Variables Conditional 1 2 7

8 Conditional Conditional The probability of a random variable taking a particular value, given the value of some other random variable. given [that] P(X=x Y=y) P(rain clouds) P(0 turning in blank test) P(hyper coffee) P(height=5 8 gender=female) 3 The probability of a random variable taking a particular value, given the value of some other random variable. P(rain & clouds) P(rain clouds) = P(clouds) P(X = x Y = y) = P(X = x,y = y) P(Y = y) Conditional Ex. 13.3: Among 100 couples who had undergone marital counseling 60 couples described their relationships as improved and, among this group, 5 couples had children. The remaining couples described their relationships as unimproved, and among this group 5 couples had children. What is the probability of an improved relationship given that the couple has children? Self-Defense: Lifestyle The CASA (Center for Addiction and Substance Abuse at Columbia) study establishes a clear progression that begins with gateway drugs and leads to cocaine use: nearly 90% of people who have ever tried cocaine used all three gateway substances [alcohol, marijuana, & cigarettes] first. Source: record html 5 6 Conditional & Independent Random Variables First Flip Second Flip First Flip Second Flip P(flip2=t flip1=h)=p(flip2=t)=.5 P(Y=t X=h)=? P(X = x Y = y) = P(X = x,y = y) P(X = x)p(y = y) = P(Y = y) P(Y = y) 7 8 8

9 Conditional Conditional P(X = x Y = y) = P(X = x,y = y) = P(Y = y) P(Y = y X = x) P(X = x) P(Y = y) Bayes s Rule Clarence goes to student health to get tested for a disease called stinkfoot. Stinkfoot is a horrific, but thankfully rare, disease that is only contracted by 1 out of every 1,000,000 people. A doctor runs an accurate test for stinkfoot on Clarence, but the test isn t perfect % of the time the test correctly detects stinkfoot. But.02% of the time, the test says that a randomly selected person has stinkfoot. Sadly, Clarence s test results come back positive. Should Clarence undergo treatment? P(X = x Y = y) = P(Y = y X = x) P(X = x) P(Y = y) 9 50 Conditional P(X = x Y = y) = P(X = x,y = y) = P(Y = y) P(Y = y X = x) P(X = x) P(Y = y) Bayes s Rule Source: Utts, J. (1999) Seeing Through Statistics Summary Or P(X=a or X=b)=P(X=a)+P(X=b)-P(X=a & b) Or: Mutually Exclusive P(X=a or X=b)=P(X=a)+P(X=b) Independence P(X=x,Y=y)=P(X=x)P(Y=y) Not P(X=a)=1-P(X=a) 53 Summary Conditional P(X = x Y = y) = Special Case: Independence P(X=x Y=y)=P(X=x) 5 P(X = x,y = y) P(Y = y) 9

10 What s the mean? Expected Value of a Random Variable Variance of a Random Variable Sum/Mean of Independent Samples from a Random Variable e.g., Binomial Distribution - Sampling Distribution of the Mean - Standard Error of the Mean - Central Limit Theorem Die Face Expected Value What is the expected value of the outcome of this gambling game? I give you $2 If I rolled this die an infinite number of times, what would be the mean value? µ = E(X) = P(X = x i )x i You give me $ What is the expected value of the outcome of this gambling game? Expected Value of a Random Variable 1 $8 2 -$1 3 -$1 -$1 5 -$1 6 -$1 Variance of a Random Variable Sum/Mean of Independent Samples from a Random Variable - e.g., Binomial Distribution - Sampling Distribution of the Mean - Standard Error of the Mean - Central Limit Theorem

11 Variance What is the variance of the outcome of this gambling game? I give you $2 Expected value of squared deviation from the mean. σ 2 = Var(X) = P(X = x i )(x i E(X)) 2 You give me $1 σ = Std(X) = Var(X) What is the variance of the outcome of this gambling game? Expected Value of a Random Variable 1 $8 2 -$1 3 -$1 -$1 5 -$1 6 -$1 Variance of a Random Variable Sum/Mean of Independent Samples from a Random Variable - e.g., Binomial Distribution - Sampling Distribution of the Mean - Standard Error of the Mean - Central Limit Theorem 63 6 Two Coin Flips Three Coin Flips

12 Binomial Distribution Binomial Distribution P(k n, p) = n p k (1 p) n k k n choose k n n! = k k!(n k)! Number of Successes in Three Coin Tosses Binomial Distribution Use the binomial distribution formula to compute the following probabilities: Assume that there s a 75% chance that someone will say yes if you ask him/her to marry you. If you ask 5 people, what is the probability that 1 person will say Yes? If you ask 5 people, what is the probability that 1 person OR LESS will say Yes? If you ask 5 people, what is the probability that MORE than 1 person will say Yes? Expected Value of a Random Variable Variance of a Random Variable Sum/Mean of Independent Samples from a Random Variable e.g., Binomial Distribution - Sampling Distribution of the Mean Standard Error of the Mean - Central Limit Theorem # of Heads

13 # of Heads Mean Heads Sampling Distribution of the Mean: Chapter 9 The sampling distribution of the mean is the probability distribution of sample means for all possible random samples of a given size from some population Mean Heads (Two Flips) # of Heads Mean Heads # of Heads

14 + + + µ = E(X) =.5 σpopulation 2 = Var(X) =.25 Distribution σ =.25 =.5 1 Flip µ X = E(X) =.5 Sampling σ 2 x = Var(Y) = Var(X) =.0625 Distribution σ x = Var(X) =.25 of the Mean Flips Mean Heads 82 # of Heads 83 Mean Heads Sampling Distribution of the Mean: Chapter 9 The sampling distribution of the mean is the probability distribution of sample means for all possible random samples of a given size from some population. Mean of Independent Samples X = n i=1 n X i µ X = E(X ) = E(X) = µ X µ x =Mean of the sampling distribution of the mean σ x =Standard error of the mean σ X = Var(X) n = σ X n 8 85 µ = E(X) =.5 σ 2 = Var(X) =.25 σ =.25 =.5 1 Flip µ X = E(X) =.5 σ 2 x = Var(Y) = Var(X) =.0625 σ x = Var(X) =.25 Flips Central Limit Theorem For large N (25-100), the sum of n independent samples of random variable X is approximately normally distributed Flip Flips Mean Heads 86 Mean Heads

15 88 89 Continuous Uniform Distribution 1 sample mean of 20 samples 90 From: R. R. Wilcox (2003) Applying Contemporary Statistical Techniques 91 1 sample mean of 20 samples From: R. R. Wilcox (2003) Applying Contemporary Statistical Techniques 92 From: R. R. Wilcox (2003) Applying Contemporary Statistical Techniques 93 15

16 1 sample mean of 25 samples From: R. R. Wilcox (2003) Applying Contemporary Statistical Techniques 9 From: R. R. Wilcox (2003) Applying Contemporary Statistical Techniques 95 mean of 50 samples From: R. R. Wilcox (2003) Applying Contemporary Statistical Techniques 96 Expected Value of a Random Variable Variance of a Random Variable Sum/Mean of Independent Samples from a Random Variable - e.g., Binomial Distribution - Sampling Distribution of the Mean - Standard Error of the Mean - Central Limit Theorem 97 16