MGMT 242 Random Sampling and Sampling Distributions Chapter 6 “He stuck in his thumb, Pulled out a...
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MGMT 242 Random Sampling and Sampling Random Sampling and Sampling Distributions Distributions Chapter 6 Chapter 6 “ “ He stuck in his thumb, He stuck in his thumb, Pulled out a plum Pulled out a plum and said and said ‘ ‘ what a good boy am I!’” what a good boy am I!’” old nursery rhyme old nursery rhyme
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
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
Slide 2
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
Slide 3
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.
Slide 4
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.
Slide 5
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
Slide 6
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.
Slide 7
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)
Slide 8
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.
Slide 9
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.
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. .