1. Statistical Sampling & Analysis of Sample Data
(Lesson - 04/A)
Understanding the Whole from Pieces
Dr. C. Ertuna

2. 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.
Dr. C. Ertuna

3. 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
Dr. C. Ertuna

4. 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
Dr. C. Ertuna

5. 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 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.
Dr. C. Ertuna

6. 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
Dr. C. Ertuna

7. Sampling: Key Issues Key Sampling issues are: Sample Design (Planning)
Sampling Methods (Schemes)
Sampling Error
Sample Size Determination.
Dr. C. Ertuna

8. Sampling: Design Sample Design (Sample Planning) describes:
Objective of Sampling
Target Population
Population Frame
Method of Sampling
Statistical tools for Data Analysis
Dr. C. Ertuna

9. Sampling Methods (Sampling Schemes)
Subjective Methods
Judgment Sampling
Convenience Sampling
Probabilistic Methods
Simple Random Sampling
Systematic Sampling
Stratified Sampling
Cluster Sampling
Dr. C. Ertuna

10. 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
Dr. C. Ertuna

11. 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.
Dr. C. Ertuna

12. Stratified Sampling Example
Population
Cash holdings of All Financial Institutions in the Country
Large Institutions
Medium Size Institutions
Small Institutions
Stratified Population
Stratum 1
Stratum 2
Stratum 3
Select n1
Select n2
Select n3
Stratified Sample of Cash Holdings of Financial Institutions
Dr. C. Ertuna

13. 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.
Dr. C. Ertuna

14. Cluster Sampling Example
Mid-Level Managers by Location for a Company 42 22
105 20 36 52 76
Algeria
Scotland
California
Alaska
New York
Florida
Mexico
Dr. C. Ertuna

15. 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:
Dr. C. Ertuna

16. 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.
Dr. C. Ertuna

17. 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.
Dr. C. Ertuna

18. Sampling: Size Sample Size Determination. n = sample size
where,
n = sample size
z = z-score = a factor representing probability in terms of standard deviation
α = 100% - confidence level
E = interval on either side of the mean
Dr. C. Ertuna

19. Estimation
Estimation (Inference) is assessing the the value of a population parameter using sample data
Two types of estimation:
Point Estimates
Interval Estimates
Dr. C. Ertuna

20. FOR ESTIMATION USE ALLWAYS STANDARD NORMAL DISTRIBUTION
Dr. C. Ertuna

21. 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.
Dr. C. Ertuna

22. Estimation (Cont.)
That means that we can use sample estimates as if they were population parameters without committing an error.
Dr. C. Ertuna

23. 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%.
Dr. C. Ertuna

24. 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
Dr. C. Ertuna

25. Lower Confidence Limit Upper Confidence Limit
Confidence Intervals
Lower Confidence Limit
Upper Confidence Limit
Point Estimate
Dr. C. Ertuna

26. 95% Confidence Intervals
0.95
z.025= -1.96
z.025= 1.96
Dr. C. Ertuna

27. CI for Proportions p = x/n
For categorical variables having only two possible outcomes proportions are important.
An unbiased estimation of population proportion (π) is the sample statistics
p = x/n
where,
x = number of observations in the sample with desired characteristics
Dr. C. Ertuna

28. Confidence Interval - From General to Specific Format -
Point Estimate  (Critical Value)(Standard Error)
(Based on CL) {
CI unite value =
CI proportion =
Dr. C. Ertuna

29. 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)
Dr. C. Ertuna

30. Confidence Interval of the Mean
Logic of the CImean computation
CLT
Unbiased Estimator
Dr. C. Ertuna

31. 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
Dr. C. Ertuna

32. CI of the Mean (Cont.) E = Margin of Error E unite value =
where,
E = Margin of Error
E unite value =
E proportion =
Dr. C. Ertuna

33. 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
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.
Dr. C. Ertuna

35. 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.
Dr. C. Ertuna

36. 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 - k.
Dr. C. Ertuna

37. Example of a CI Interval Estimate for 
A sample of 100 cans, from a population with  = 0.20, produced a sample mean equal to A 95% confidence interval would be:
ounces
ounces
Dr. C. Ertuna

38. 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 A 95% confidence interval would be:
ounces
ounces
n=400
Dr. C. Ertuna
n=100
ounces
ounces

39. 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:
0.54
0.70
Dr. C. Ertuna

40. From Margin of Error to Sampling Size
E unite value =
E proportion =
Dr. C. Ertuna

41. Sampling: Size Sample Size Determination. where, n = sample size
z = z-score = a factor representing probability in terms of standard deviation
α = 100% - confidence level
E = interval on either side of the mean
Dr. C. Ertuna

42. 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.
Dr. C. Ertuna

43. 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.
Dr. C. Ertuna

44. 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?
Dr. C. Ertuna

45. 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)
Dr. C. Ertuna

46. NORMDIST versus CI (Cont.)
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)
Dr. C. Ertuna

47. Next Lesson
(Lesson - 04/B)
Hypothesis Testing
Dr. C. Ertuna