Statistical Sampling

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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.)

• Standard Error of the mean

• 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

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Sampling Distribution of the Mean
THE CENTRAL LIMIT THEREOM For samples of n
observations taken from a population with mean ?
and standard deviation ?, regardless of the
populations 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.
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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
Sampling Methods (Sampling Schemes)

• Subjective Methods
• Judgment Sampling
• Convenience Sampling

• Probabilistic Methods
• Simple Random Sampling
• Systematic Sampling
• Stratified Sampling
• Cluster Sampling

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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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Stratified Sampling Example
Population Cash holdings of All Financial
Institutions in the Country
Stratified Sample of Cash Holdings of Financial
Institutions
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Cluster Sampling

• 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.

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Cluster Sampling Example
Mid-Level Managers by Location for a Company
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Sampling Error

• SAMPLING ERROR-SINGLE MEAN
• The difference between a value (a statistic)
  computed from a sample and the corresponding
  value (a parameter) computed from a population.
• Where

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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.

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Sampling Size

• Sample Size Determination.

where, n sample size z z-score a factor
representing probability in terms of standard
deviation a (100 - confidence level) E
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 z or t 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.)
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
Lower Confidence Limit
Upper Confidence Limit
Point Estimate
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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
  (p) is the sample statistics

p x/n where, x number of observations in the
sample with desired characteristics
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Confidence Interval- From General to Specific
Format -
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CI of the Mean (Cont.)
where, E Margin of Error
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Confidence Interval- From Statistical Expression
to Excel Formula -

• Where
• z a/2 Normsinv(1 a/tails)
• and when n lt 30 AND s is not known , then z ?
  t so
• t a/2 n-1 Tinv(2a/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) a
(100 - confidence level)
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Confidence Interval

• PHStat2

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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

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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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Students 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
populations standard deviation. If k parameters
must be estimated before the populations
standard deviation can be calculated from a
sample of size n, the degrees of freedom are
equal to n - 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

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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

12.0704 ounces
12.1096 ounces
n400
n100
12.051 ounces
12.129 ounces
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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

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From Margin of Error to Sampling Size
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Sampling Size

• Sample Size Determination.

where, n sample size z z-score a factor
representing probability in terms of standard
deviation a 100 - confidence level E
interval on either side of the mean
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Sampling Size

• PHStat2

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Pilot Samples
A pilot sample is a sample taken from the
population of interest to provide and estimate
for the population standard deviation. Normally
its size is smaller than the anticipated sample
size.
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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.

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Next Lesson

• (Lesson - 04/B)
• Hypothesis Testing