Random Sampling and Sampling Distributions Chapter 6 He

Random Sampling and Sampling Distributions Chapter 6 “He stuck in his thumb, Pulled out a plum and said ‘what a good boy am I!’” old nursery rhyme MGMT 242

Topics and Goals for Chapter 6 • Random Sampling • Sample Statistics and Relation to Population Parameters • Sampling Distribution for Sample Mean-”The Central Limit Theorem” • Checking Normality-The Normal Probability Plot – – samples from normal distributions positively skewed distributions negatively skewed distributions with outliers MGMT 242

Populations and Samples • A population is a large collection (theoretically, for the mathematician, infinite) of the individuals or items of interest (e. g. consuming public, machine line production items, etc. ) • To measure characteristics of the population we have to take a sample (smaller number). • If we take a random sample, it is equally likely that any member of the population will be included in the sample. MGMT 242

Random Sampling • Sample represents population only if each member of population equally likely to be included in sample. • Types of random sampling (see also Chapter 16): – Simple Random Sampling (SRS)-sample whole population – Stratified Random Sampling divide population into groups and sample from each group; for example, in polls, divided country into four geographical regions and sample from each – Cluster Sampling Divide population into groups and take a sample of a few groups from the total--e. g. , looking at hospital performance, sample patients in few hospitals randomly chosen from all hospitals in the state. MGMT 242

Sample Statistics • Sample Mean: xbar = (1/N) xi , where “xbar” is x with a bar over it; the sum is taken over all values of the random variable X measured in the sample of N units. xbar is an estimator of the population mean, . • Sample Standard Deviation: s = {[1/ (N-1)] (xi- xbar)2 }(1/2 s is an “unbiased estimate” of the population standard deviation, . Note that for large samples (large N), N-1 N MGMT 242

Sampling Distribution for Sample Means: The Central Limit Theorem--1 • In general (which means almost always), no matter what distribution the population follows, the distribution of the sample means follows a normal distribution with • mean µsample means (for the population of sample means) equal to µ, the mean for the parent population, and • standard deviation of the means sample means= / N. This means that the larger the sample size, the more accurately we estimate the mean. MGMT 242

Sampling Distribution for Sample Means: The Central Limit Theorem-2 • The histogram on the left is for a sample from a uniform distribution (0 to 100). The sample mean is 50. 2 and the sample standard deviation is 29. 3 ( 100/ 12) MGMT 242

Sampling Distribution for Sample Means: The Central Limit Theorem-2 • The histogram on the left is for the means of 150 samples, each size 9 (N = 9). The average of these 150 means is 49. 4 and the standard deviation of these 150 sample means is 9. 8 which is about (100/[ 12 9]), the population standard deviation of the mean. MGMT 242

Normal Probability Plots (“P-plots”) • The procedure to get this plot, which tests whether data follow a normal distribution procedure, is the following: – 1) order the N data; – 2) assign a rank from 1 --the lowest--to N--the highest value; – 3) find the centile score of the mth data point from the relation centile score = m/(N+1)--e. g the 1 st data point out of 100 has a fraction approximately 1/101 lower; the 100 th data point has a fraction 100/101 lower; – 4) find the z-value (standard normal variate) corresponding to the centile score (this would be the z-score or N-score). – 5) plot the observed points versus the z-score; • If the points fall approximately on a straight line, the distribution is a normal distribution. MGMT 242

Normal Probability Plots (“P-plots”) Examples Exam 2 scores were negatively skewed (range: 49 -100, Q 1=90, median=92, Q 3= 94 rank ordered value z-score Exam 2 1 0. 02 2 0. 04 3 0. 05 4 0. 07 5 0. 09 6 0. 11 7 0. 13 8 0. 14 9 0. 16 etc. …. 49 72 78 79 81 85 86 87 89 -2. 10 -1. 80 -1. 61 -1. 47 -1. 35 -1. 24 -1. 15 -1. 07 -0. 99 MGMT 242

Normal Probability Plots (“P-plots”) Examples (cont. ) This Pplot for Exam 2 scores is from the “Statplus” addin; note that the axes are interchanged from the previous (conventional) order: Nscore is y-axis, actual score is xaxis rank ordered value z-score Exam 2 1 0. 02 2 0. 04 3 0. 05 4 0. 07 5 0. 09 etc. …. 49 72 78 79 81 -2. 10 -1. 80 -1. 61 -1. 47 -1. 35 MGMT 242

Qualitative Appearance of P-plots MGMT 242