Quantitative Methods Varsha Varde Quantitative Methods Sampling Techniques

Quantitative Methods Varsha Varde

Quantitative Methods Sampling Techniques

Sampling • The Process of Obtaining Information About a Whole by Examining Only a Part • Whole = Population Part = Sample • Everyday Life Concept • Example: Physician makes diagnosis on the basis of the findings of a small sample of blood Auditors use sampling to draw conclusions about large volumes of transactions Market researchers use sample of customers to determine market potential Sample Inspection is done to accept or reject a lot ? Varsha Varde 3

Why Sampling • Population too large to be studied in full • Sampling is Cheaper & Quicker as compared to Census • Necessary in destructive testing • Census not feasible-testing of medicines Varsha Varde 4

Purpose of Sampling • To Estimate Value of a Population Parameter (Mean, Variance, Proportions etc. ) on the basis of Value of the Corresponding Sample Statistic. • More Representative the Sample, More Accurate is the Estimate. • Bigger the Sample, Better the Estimate. • Bigger the Sample, Greater is Cost & Time Varsha Varde 5

Terminology Ä Population: Entire group of people, events, or objects of interest in context of research Ä Element: A single member of the population Ä Population Frame: List of all elements in the population from which a sample is drawn Ä Example: List of all students in a college, list of all ent. events in Mumbai in June 2010, list of all songs sung by Lata Mangeshkar Ä Population Parameters: Proportion, Mean & Variance. Varsha Varde 6

Terminology Ä Sample: A subset of population selected for data collection in the research study Ä Subject: A single member of the sample Ä Sampling: Process of selecting sufficient number of elements from the population Ä Sampling saves time & cost of research Ä Sample Statistics: Sample proportion, Sample mean (central tendency) & sample variance (dispersion). Varsha Varde 7

Concept of Sampling Error • Difference between the Actual Value of the Characteristic of Population and the Value Estimated from the Sample. • The Art & Science of Sampling is to Apply Appropriate Techniques to Minimize this Risk, i. e. Minimizing Sampling Error. Varsha Varde 8

Statistical Assurance About Minimum Sampling Error (Risk) is Provided by Two Parameters: 1. Precision: Quantum of Admissible Error 2. Reliability or Confidence Level: The Probability that the Sample Estimate Will Be In Fact Within the Stipulated Range of Precision Varsha Varde 9

Concept of Precision • Quantum of Admissible Error. Ideally Zero. • Cannot be Zero Unless Sample is 100%. • Precision Should be as small as possible Varsha Varde 10

Concept of Precision • Precision (i. e. Error or Risk in Statistics) Decreases as Sample Size Increases. • But, Cost & Time of Estimation Increases as Sample Size Increases. • This is an Issue of Resource Allocation. • Hence, You as Manager, Strike a Balance and Decide Optimal Level of Precision. • Note: Precision is Management Decision. Varsha Varde 11

Reliability or Confidence Level • It is the Probability that the Sample Estimate Will Be In Fact Within the Range of Precision Set by You. • This Prob Has to be Very High: 90%, 95%, 99%. • 100% Impossible Unless Sample is 100%. • In Any Sampling Scenario, You Must First Set Precision and Confidence Level. • They will determine Required Sample Size Varsha Varde 12

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Sample Size • How Big Should Be My Sample? • Sample Size Depends Upon the Sampling Technique Selected for the Purpose. • Therefore, First We Must Know About the Various Sampling Techniques. Varsha Varde 14

Sampling Techniques A Statistical /Probability Sample Should Be: • Selected Objectively so that Inferences Drawn from it are Reliable, • Free from Personal Biases, • Giving Equal or Known Chance of Selection to Every Unit of the Population. So, Sample Must Be Drawn Scientifically. Varsha Varde 15

Statistical Sampling Techniques • Many Techniques Available. • Selection of the Right One Depends Upon: - Nature of the Population, - Cost Budget, - Time Constraint, - Precision & Confidence Required • Hence, Selection Falls in Your Domain. Varsha Varde 16

Simple Random Sampling • Most Widely Used for Ease and Low Cost • Equal Probability of Selection to All Units in Population • Random Number Tables (RNT) Available • Internationally Tested for Randomness Varsha Varde 17

Random Number Table 73831 65465 98955 34797 87202 26365 82357 71994 27346 80200 80400 96962 42658 45336 73053 18361 51948 46016 19430 95116 38726 24437 46868 90743 67360 61384 99858 34025 90650 31247 45081 08987 62598 00033 04701 38088 30791 01058 19741 73503 16569 28686 61840 58777 70363 70553 04723 51435 04458 64807 20838 60008 35509 47978 05667 31668 28054 62915 67283 06875 21213 04621 64435 52836 51736 67232 92997 93798 87525 20419 Varsha Varde 18

Simple Random Sampling Steps 1. Assign Sequential Numbers to All Units 2. Open Any Page of RNT. Start Anywhere 3. From This Starting Point Proceed Vertically Downwards and Select As Many Numbers As Required Varsha Varde 19

Random Number Table 73831 65465 98955 34797 87202 26365 82357 71994 27346 80200 80400 96962 42658 45336 73053 18361 51948 46016 19430 95116 38726 24437 46868 90743 67360 61384 99858 34025 90650 31247 45081 08987 62598 00033 04701 38088 30791 01058 19741 73503 16569 28686 61840 58777 70363 70553 04723 51435 04458 64807 20838 60008 35509 47978 05667 31668 28054 62915 67283 06875 21213 04621 64435 52836 51736 67232 92997 93798 87525 20419 Varsha Varde 20

Example • Quality Controller wishes to select a random sample of 25 drums from the lot numbered from 312 to 9233. • Drums are already numbered • Largest Number: 9233. Hence 4 -digit Nos. • Randomly select the starting point: 7383 • Hence, First Sample is Drum No. 7383 • Next No. is 6546. feasible. Accept it. Varsha Varde 21 • Next No. is 9895. Infeasible. Discard it.

Random Sample of 25 Drums 73831 65465 98955 34797 87202 26365 82357 71994 27346 80200 80400 96962 42658 45336 73053 18361 51948 46016 19430 95116 38726 24437 46868 90743 67360 61384 99858 34025 90650 31247 45081 08987 62598 04701 38088 30791 01058 19741 16569 28686 61840 58777 70363 04723 51435 04458 64807 20838 35509 47978 05667 31668 28054 67283 06875 21213 04621 64435 51736 67232 92997 93798 87525 00033 73503 70553 60008 62915 52836 20419 Varsha Varde 22

Systematic Sampling • • • Use When Pop is Already Arranged in an Order. Example: Vouchers, Employee No. , Batch No. Variation of SRS. Faster. Speeds Up Sampling. Does Not Use Random Number Tables. Compute Skip Interval k = Ratio of Pop Size to Sample Size. • Randomly Select a Starting Number < k. • Then Systematically Selects Every kth Number. • Widely Used for Ease and Lower Cost. Varsha Varde 23

Example • Internal Auditor wishes to select a sample of 50 accounts receivable out of 520 such accounts in a sales office. • She opts for Systematic Sampling. • Skip Interval k = 520 / 50 = 10. 4 • Suppose Her Random no. below 10 is 7. • Sample: Acct Nos. 7, 17, 27, 37, ……, 497 Varsha Varde 24

Stratified Sampling • Example: 520 accounts receivable from 4 product divisions: Agro-Chemical (323), Leather (54), Textile(22), Plastic (121). • Sample of 50: 32, 5, 2 & 11 respectively • Population Discernibly Heterogeneous • Divide It into Several Parts (Called Strata) • Each Stratum Homogeneous Within Itself • Draw a Simple Random or Systematic Sample from Each Stratum. Varsha Varde 25

Cluster Sampling • Example: Hosiery Crates ( Each Crate Contains Full Assortment of Sizes), Bldgs in Apt Complex • Population Discernibly Heterogeneous • Divide It into Several Clusters • Each Cluster Heterogeneous Within Itself • Draw SRS or Systematic Sample of Clusters • Study Each Sampled Cluster Fully. • Use When Population is Inherently Divided into Heterogeneous Clusters. • Convenient. Saves Cost & Time. Varsha Varde 26

Multi-Stage Sampling • Samples are Drawn from Samples • Example: Select 4 Out of 25 Working Days, and Select Ten Sacks of Finished Product from Each Selected Day’s Output • This is 2 -Stage Sampling. • In Complex Situations, This Process Can Go On for 3, 4 or Even More Stages. Varsha Varde 27

Determining Sample Size Factors Influencing Sample Size: • Precision (Your Decision) • Confidence Level (Your Decision) • Sampling Technique (Your Decision) • Population Size (Known to You) • Pop Parameter to be Estimated (Kt. Y) • Dispersion in Population (Known to You) Varsha Varde 28

Determining Sample Size Effect of Factors Influencing Sample Size • Lower Precision – Bigger Sample • Higher Confidence – Bigger Sample • Wider Dispersion in Pop – Bigger Sample • Ironically, Population Size Affects Sample Size Only Marginally Varsha Varde 29

Probability Sampling v Example: A sample of 100 TVs to be drawn from 10, 000 TVs produced in June 2010 v Each TV has 100 ÷ 10, 000 = 0. 01 i. e. 1% chance of being chosen v Sampling Design tells researcher precisely how to pick up 100 TVs Varsha Varde 30

1: Simple Random Sampling n. Two lucky numbers to be drawn out of 100 tokens. Put all 100 tokens in a basket. Stir well. Close eyes and pick up two tokens n. For larger population, assign serial numbers to each element. Use a standard table of random numbers. Select the required number of elements one after other n. But, enlisting large p pulations is tedious. Varsha Varde 31

A Case Study HR Director of a software firm with 1926 engineers wants to find out desirability of changing the current 10 – 6 working hours to flexitime along with its benefits & drawbacks perceived by the engineers before the next board meeting She would pick up a few engineers randomly & ask them appropriate questions. Varsha Varde 32

2: Systematic Sampling n. A sample of 50 cars to be selected from 10, 000 cars produced in 2009 n 10, 000 ÷ 50 = 200. Select every 200 th car n. More precisely, select a random number between 1 and 200, say 30. Select 30 th car n. Starting from 30 th car, select every 200 th car: 30, 230, 430, 630, 830, 1030, 1230, 1430… Varsha Varde 33

A Case Study Maruti Suzuki Ltd. wants to check response of prospective buyers to the new features introduced in its small car segment From the dealers alphabetical list, the Company selects every 50 th dealer & sends a senior marketing manager to talk to them. Varsha Varde 34

3: Stratified Random Sampling n. If population contains identifiable subgroups of elements, researcher must provide proper representation to each subgroup n. Ex. : Population: All students of a college n. Identifiable Subgroups: males / females; arts/ science / commerce; brilliant / average / poor n. Lata M. songs: By language, solo / duet Varsha Varde 35

. 3: Stratified Random Sampling n. Process: Divide the population into mutually exclusive identifiable subgroups (strata) n. Draw a simple random sample (or systematic sample) from each stratum n. Size of sample from each stratum directly proportional to size of the stratum n. Homogeneity within each stratum n. Heterogeneity between strata. Varsha Varde 36

Study of Absenteeism (2% sample) Category (Stratum) 5 Strata Managers Total Number 7750 250 Sample Size 155 5 Junior Managers 500 10 Assistants 2000 40 Skilled Workers 4000 80 Unskilled Labour 1000 20 Varsha Varde 37

. 3: Stratified Random Sampling n. Stratified random sampling involves dividing population into strata n. Hence, it needs higher time and cost n. But, it provides desired precision with smaller sample than sampling from nonstratified population Varsha Varde 38

4: Cluster Sampling n. Used when population consists of several groups of elements in such a manner that: n. Groups are similar to each other and n. Each group (CLUSTERS) is heterogeneous n. So, population has inter-group homogeneity and intra-group heterogeneity n. Exactly opposite of stratified population n. Process: Select a. Varsha few clusters randomly. 39 Varde

. 4: Cluster Sampling Examples n Complex of many identical buildings. We can select 5 out of 50 buildings n A Mgmt Inst: 2000 students per year. 50 per batch. 40 batches run concurrently. Each has some active, some ordinary & some passive students, and 75% boys, 25% girls. Choose 4 batches and talk to all 200 students without disturbing other 36 batches. Varsha Varde 40

. 4: Cluster Sampling Examples n. A truckload of mangoes in 4 dozen boxes. Each box has upper layer of top quality fruits. Quality & size drops layer by layer. n. Thus, homogeneity between boxes & heterogeneity within each box. n. Draw a random or systematic sample of a few boxes, open them and study them. n. No need to open other boxes from the truck. Varsha Varde 41

. 4: Cluster Sampling n. Convenient n. Sample size smaller n. Less time and cost n. But, restrictive in application: You don’t frequently get such populations. Varsha Varde 42

A Case Study } Under a community health program for tribals, it was necessary to discover their current state of nutrition, health & beliefs } Since adivasi padas are located at long distances from each other in tribal areas, a few adivasi padas were selected at random and all residents from infants to old ones were checked. Varsha Varde 43