Lecture 10. Random Sampling and Sampling Distributions David R. Merrell Intermediate Empirical Methods for Public Policy and Management

Agenda Normal Approximation to Binomial Poisson Process Random sampling Sampling statistics and sampling distributions Expected values and standard errors of sample sums and sample means

Binomial Random Variable Binomial random variable X is the number of “successes” in n trials, where Probability of success remains the same from trial to trial Trials are independent

Binomial Probability Distribution Discrete distribution with: P(X=x) = ( n!/(x!(n-x)!)) p x q n-x n is number of trials x is number of successes in n trials (x = 0, 1, 2,..., n) p is the probability of success on a single trial q is the probability of failure on a single trial

Properties of the Binomial RV Mean:  = np Variance:   = npq Standard Deviation: 

Binomial(n = 10, p =.4) xP(X=x)

Approximation to Binomial Distribution Use normal distribution when: n is large np > 10 n(1 - p) > 10 Parameters of the approximating normal distribution are the mean and standard deviation from the binomial distribution

Approximation of Binomial Distribution n = 80, p =.4

How Good is the Approximation? Normal with mean = and standard deviation = x P( X <= x) x P( X <= x) Binomial with n = 80 and p = x P( X <= x) P(X < 29)

Application 1 The Chicago Equal Employment Commission believes that the Chicago Transit Authority (CTA) discriminates against Republicans. The records show that 37.5% of the individuals listed as passing the CTA exam were Republicans; the remainder were Democrats (no one registers as an independent in Illinois). CTA hired 30 people last year, 25 of them were Democrats. What is the probability that this situation could exist if CTA did not discriminate?

Application 1 (cont.) Success: a Republican is hired The probability of success, p = The number of trials, n = 30 The number of successes, x = 5 P(x  5) = ???

Application 1 (cont.) Mean:  = np = 30*.375 = Variance:   = npq = 30*.375*.625 = 7.03 Standard Deviation:  = 2.65 Normal with mean = and standard deviation = 2.65 x P( X <= x)

Poisson Process time homogeneity independence no clumping rate xxx 0 time Assumptions

Poisson Process Earthquakes strike randomly over time with a rate of = 4 per year. Model time of earthquake strike as a Poisson process Count: How many earthquakes will strike in the next six months? Duration: How long will it take before the next earthquake hits?

Count: Poisson Distribution What is the probability that 3 earthquakes will strike during the next six months?

Poisson Distribution Count in time period t

Minitab Probability Calculation Click: Calc > Probability Distributions > Poisson Enter: For mean 2, input constant 3 Output: Probability Density Function Poisson with mu = x P( X = x)

Duration: Exponential Distribution Time between occurrences in a Poisson process Continuous probability distribution Mean =1/ t

Exponential Probability Problem What is the probability that 9 months will pass with no earthquake?  t = 1/12, t = 1/3 1/ t = 3

Minitab Probability Calculation Click: Calc > Probability Distributions > Exponential Enter: For mean 3, input constant 9 Output: Cumulative Distribution Function Exponential with mean = x P( X <= x)

Exponential Probability Density Function MTB > set c1 DATA > 0:12000 DATA > end Let c1 = c1/1000 Click: Calc > Probability distributions > Exponential > Probability density > Input column Enter: Input column c1 > Optional storage c2 Click: OK > Graph > Plot Enter: Y c2 > X c1 Click: Display > Connect > OK

Exponential Probability Density Function

Sampling Population - entire set of objects that we are interested in studying Sample - a chosen subset of a population

Some Samples Are... random -- each item in the population has an equal chance of being selected to be part of the sample representative -- has the same characteristics as the population under study, a microcosm of the population

Population Parameters and Sample Statistics Population Parameter Numerical descriptor of a population Values usually uncertain e.g., population mean (  ), population standard deviation (  ) Sample Statistics Numerical descriptor of a sample Calculated from observations in the sample e.g., sample mean, sample standard deviation S

What is a sampling distribution? Sample statistics are random variables Sample statistics have probability distributions “Sampling distribution” is the probability distribution of a sample statistic

MTB > Retrieve 'C:\MTBWIN\DATA\RESTRNT.MTW'. Retrieving worksheet from file: C:\MTBWIN\DATA\RESTRNT.MTW Worksheet was saved on 5/31/1994 MTB > info Information on the Worksheet Column Name Count Missing C1 ID C2 OUTLOOK C3 SALES C4 NEWCAP C5 VALUE C6 COSTGOOD C7 WAGES C8 ADS C9 TYPEFOOD C10 SEATS C11 OWNER C12 FT.EMPL C13 PT.EMPL C14 SIZE

MTB > desc 'sales' Descriptive Statistics Variable N N* Mean Median TrMean StDev SEMean SALES Variable Min Max Q1 Q3 SALES MTB > boxp 'sales' * NOTE * N missing = 25

MTB > hist 'sales' * NOTE * N missing = 25

MTB > let c15 = loge('sales') J *** Values out of bounds during operation at J Missing returned 1 times MTB > let c15 = loge('sales' + 1) MTB > name c15 'logsales' MTB > desc 'logsales' Descriptive Statistics Variable N N* Mean Median TrMean StDev SEMean logsales Variable Min Max Q1 Q3 logsales MTB > boxp 'logsales' * NOTE * N missing = 25

Four Samples of Size 50 From Restaurant “Logsales” Data--Histograms

MTB > Desc c16-c19 Descriptive Statistics Variable N N* Mean Median TrMean StDev SEMean C C C C Variable Min Max Q1 Q3 C C C C Random Samples from Restaurant “Logsales” Data--Summary

Next Time... Central Limit Theorem--”Sample averages are approximately normally distributed”