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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.)

• 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

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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

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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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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

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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