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Dr. C. Ertuna 1
Statistical Sampling & Analysis of Sample Data
(Lesson - 04/A)
Understanding the Whole from Pieces
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Sampling
Sampling is :
• Collecting sample data from a population and
• Estimating population parameters
Sampling is an important tool in business decisions since it is an effective and efficient way obtaining information about the population.
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Sampling (Cont.)
How good is the estimate obtained from the sample?
• The means of multiple samples of a fixed size (n) from some population will form a distribution called the sampling distribution of the mean
• The standard deviation of the sampling distribution of the mean is called the standard error of the mean
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Sampling (Cont.)
• Estimates from larger sample sizes provide more accurate results
• If the sample size is large enough the sampling distribution of the mean is approximately normal, regardless of the shape of the population distribution - Central Limit Theorem
n/x • Standard Error of the mean =
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Sampling Distribution of the Mean
THE CENTRAL LIMIT THEREOMFor samples of n observations taken from a population with mean and standard deviation , regardless of the population’s distribution, provided the sample size is sufficiently large, the distribution of the sample mean , will be normal with a mean equal to the population mean
. Further, the standard deviation will equal the population standard deviation divided by the square-root of the sample size .
The larger the sample size, the better the approximation to the normal distribution.
( ) x
x
x n
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Sampling Statistics
Sampling statistics are statistics that are based on values that are created by repeated sampling from a population,
such as:
•Mean of the sampling means
•Standard Error of the sampling mean
•Sampling distribution of the means
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Sampling: Key Issues
Key Sampling issues are:
• Sample Design (Planning)
• Sampling Methods (Schemes)
• Sampling Error
• Sample Size Determination.
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Sampling: Design
Sample Design (Sample Planning) describes:
• Objective of Sampling
• Target Population
• Population Frame
• Method of Sampling
• Statistical tools for Data Analysis
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Sampling: Methods
Subjective Methods– Judgment Sampling– Convenience
Sampling
Probabilistic Methods• Simple Random Sampling• Systematic Sampling• Stratified Sampling• Cluster Sampling
Sampling Methods (Sampling Schemes)
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Sampling: Methods (Cont.)
Simple Random Sampling Method
• refers to a method of selecting items from a population such that every possible sample of a specified size has an equal chance of being selected
• with or without replacement
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Sampling: Methods (Cont.)
Stratified Sampling Method:
• Population is divided into natural subsets (Strata)
• Items are randomly selected from stratum
• Proportional to the size of stratum.
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PopulationPopulation
Cash holdings of All Financial Institutions in the Country
Large Institutions
Medium Size Institutions
Small Institutions
Stratified PopulationStratified Population
Stratum 1
Stratum 2
Stratum 3
Select n1
Select n2
Select n3
Stratified Sample of
Cash Holdings of Financial
Institutions
Stratified Sampling Example
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Cluster sampling refers to a method by which the population is divided into groups, or clusters, that are each intended to be mini-populations. A random sample of m clusters is selected.
Cluster Sampling
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Cluster Sampling Example
42 22 105 20 36 52 76
Algeria Scotland California Alaska New York Florida Mexico
Mid-Level Managers by Location for a Company
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Sampling Error
SAMPLING ERROR-SINGLE MEANThe difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed
from a population.
Where: -x Error Sampling
mean Population
mean Samplex
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Sampling: Error (Cont.)
Sampling Error is inherent in any sampling process due to the fact that samples are only a subset of the total population.
• Sampling Errors depends on the relative size of sample
• Sampling Errors can be minimized but not eliminated.
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Sampling: Error (Cont.)
If Sampling size is more than 5% of the population
• “With Replacement” assumption of Central Limit Theorem and hence, Standard Error calculations are violated
• Correction by the following factor is needed.
1N
nN
18Dr. C. Ertuna
Sampling: Size
Sample Size Determination.
2222/ E/szn
where,n = sample sizez = z-score = a factor representing probability
in terms of standard deviationα = 100% - confidence levelE = interval on either side of the mean
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Estimation
Estimation (Inference) is assessing the the value of a population parameter using sample data
Two types of estimation:
•Point Estimates
•Interval Estimates
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Estimation
FOR ESTIMATION USE ALLWAYS STANDARD NORMAL
DISTRIBUTION
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Estimation (Cont.)
• Most common point estimates are the descriptive statistical measures.
• If the expected value of an estimator equals to the population parameter then it is called unbiased.
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Estimation (Cont.)
Unbiased EstimatorsPoint Estimate Population Parameter (sample mean) (population mean) s (sample Std. Dev.) (population Std. Dev) p (sample proportion) (population proportion)
x
That means that we can use sample estimates as if they were population parameters without committing an error.
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Estimation (Cont.)
Interval Estimate provides a range within which population parameter falls with certain likelihood.
Confidence Level is the probability (likelihood) that the interval contains the population parameter. Most commonly used confidence levels are 90%, 95%, and 99%.
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Confidence Interval
Confidence Interval (CI) is an interval estimate specified from the perspective of the point estimate.
In other words CI is
• an interval on either side (+/-) of the point estimate
• based on a fraction (t or z-score) of the Std. Dev. of the point estimate
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Confidence Intervals
Point EstimateLower Confidence
LimitUpper Confidence
Limit
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95% Confidence Intervals
0.95
z.025= -1.96 z.025= 1.96
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CI for Proportions
For categorical variables having only two possible outcomes proportions are important.
An unbiased estimation of population proportion (π) is the sample statistics
p = x/nwhere,x = number of observations in the sample with
desired characteristics
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Confidence Interval- From General to Specific Format -
Point Estimate (Critical Value)(Standard Error)
(Based on CL)
CI unite value = n/szx 2/
CI proportion = n/p1(pzp 2/
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Confidence Interval- From Statistical Expression to Excel Formula -
Where
z α/2 = Normsinv(1 – α/tails)
and when n < 30 z t , then
t α/2 n-1 = Tinv(2α/tails, n-1)
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CI of the Mean (Cont.)
where,z = z-score = a critical factor representing
probability in terms of Standard Deviation (for sampling Standard Error) (valid for normal distribution) (critical value)
t = t-score = a factor representing probability in terms of standard deviation (or Std. Error) (valid for t distribution) (critical value)
α = 100% - confidence level
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CI of the Mean (Cont.)
where,
E = Margin of Error
E unite value = n/sz 2/
E proportion = n/)p1(pz 2/
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Z-score
A z-score is a critical factor, indicating how many standard deviation (standard error for sampling) away from the mean a value should be to observe a particular (cumulative) probability.
There is a relationship between z-score and probability over p(x) = (1-Normsdist(z))*tails and
There is a relationship between z-score and the value of the random variable over
X
Z
Dr. C. Ertuna 34
Z-score (Cont.)
Since the z-score is a measure of distance from the mean in terms of Standard Deviation (Standard Error for sampling), it provides us with information that a cumulative probability could not. For example, the larger z-score the unusual is the observation.
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Student’s t-Distribution
The t-distribution is a family of distributions that is bell-shaped and symmetric like the Standard Normal Distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the normal distribution.
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Degrees of freedom
Degrees of freedom (df) refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - kn - k.
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Example of a CI Interval Estimate for
A sample of 100 cans, from a population with = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be:
039.009.12100
20.096.109.12
n
zx
12.051 ounces
12.129 ounces
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Example of Impact of Sample Size on Confidence Intervals
If instead of sample of 100 cans, suppose a sample of 400 cans, from a population with = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be:
0196.009.12400
20.096.109.12
n
zx
12.051 ounces
12.129 ounces
12.0704 ounces
12.1096 ouncesn=400
n=100
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Example of CI for Proportion
62 out of a sample of 100 individuals who were surveyed by Quick-Lube returned within one month to have their oil changed. To find a 90% confidence interval for the true proportion of customers who actually returned:
62.0100
62
n
xp
100
)62.01)(62.0(645.162.0
0.50.544
0.70.700
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From Margin of Error to Sampling Size
E unite value = n/sz 2/
E proportion = n/)p1(pz 2/
2222/ E/szn
222/ E/p1pzn
41Dr. C. Ertuna
Sampling: Size
Sample Size Determination.
2222/ E/szn
where,n = sample sizez = z-score = a factor representing probability in terms of
standard deviationα = 100% - confidence levelE = interval on either side of the mean
222/ E/p1pzn
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Pilot Samples
A pilot sample is a sample taken from the population of interest of a size smaller than the anticipated sample size that is used to provide and estimate for the population standard deviation.
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Example of Determining Required Sample Size
The manager of the Georgia Timber Mill wishes to construct a 90% confidence interval with a margin of error of 0.50 inches in estimating the mean diameter of logs. A pilot sample of 100 logs yield a sample standard deviation of 4.8 inches.
25038.24950.0
)8.4(645.12
22
n
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RANGE versus CI
Example:
The customer’s demand is normally distributed with a mean of 750 units/month and a standard deviation of 100 units/month. What is the probability that the demand will be within 700 units/month and 800 units/month?
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RANGE versus CI (Cont.)
1) A RANGE is GIVEN, probability asked (population and given)
• The customer’s demand is normally distributed with a mean of 750 units/month and a standard deviation of 100 units/month. What is the probability that the demand will be within 700 units/month and 800 units/month?
Answer: p(x≤800) - p(x≤700) ; p(700≤x≤800) = NORMDIST(800,750,100,true) -
NORMDIST(700,750,100,true)
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NORMDIST versus CI (Cont.)
2) PROBABILITY IS GIVEN, Upper and Lower limits are asked (sample mean, s, n)
• What would be the Confidence Interval for an expected sales level of 750 units/month if you whish to have a 90% confidence level based on 30 observations?
U/LL(x) = x NORMSINV(1-(/tails))*(s/SQRT(n))U/LL(x) = 750 NORMSINV(0.95)*100/SQRT(30)
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Next Lesson
(Lesson - 04/B) Hypothesis Testing
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