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MGMT 242 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

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

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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.)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).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.If we take a random sample, it is equally likely that any member of the population will be included in the sample.

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MGMT 242 Random Sampling Sample represents population only if each member of population equally likely to be included in sample.Sample represents population only if each member of population equally likely to be included in sample. Types of random sampling (see also Chapter 16):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.

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MGMT 242 Sample Statistics Sample Mean: xbar = (1/N) x i, 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.Sample Mean: xbar = (1/N) x i, 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)] (x i - xbar) 2 } (1/2Sample Standard Deviation: s = {[1/ (N-1)] (x i - xbar) 2 } (1/2 s is an unbiased estimate of the population standard deviation, . s is an unbiased estimate of the population standard deviation, . Note that for large samples (large N), N-1 N

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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 withIn general (which means almost always), no matter what distribution the population follows, the distribution of the sample means follows a normal distribution with mean sample means (for the population of sample means) equal to , the mean for the parent population, andmean sample means (for the population of sample means) equal to , 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.standard deviation of the means sample means = / N. This means that the larger the sample size, the more accurately we estimate the mean.

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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)

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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 dev- iation of the mean.

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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 1st data point out of 100 has a fraction approximately 1/101 lower; the 100th 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.

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MGMT 242 Normal Probability Plots (P-plots) Examples Exam 2 scores were negatively skewed (range: 49-100, Q1=90, median=92, Q3= 94 rank ordered valuez-scoreExam 2 1 0.02 -2.1049 2 0.04 -1.8072 3 0.05-1.6178 4 0.07-1.4779 5 0.09-1.3581 6 0.11-1.2485 7 0.13-1.1586 8 0.14-1.0787 9 0.16-0.9989 etc. .

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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 inter- changed from the previous (conventional) order: Nscore is y-axis, actual score is x-axis rank ordered valuez-scoreExam 2 1 0.02 -2.1049 2 0.04 -1.8072 3 0.05-1.6178 4 0.07-1.4779 5 0.09-1.3581 etc. .

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MGMT 242 Qualitative Appearance of P-plots