Basic Concepts Random Sampling Concepts, Lecture notes of Business Statistics

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ETW

Business Data Modelling

Week 1

Review of some Basic Concepts

√ Random Sampling Concepts

Simple random sampling Systematic sampling Stratified random sampling Cluster sampling

Learning Objectives

• Understand the motivation for taking a sample.

• Understand the concept of a sampling design.

• Decide when and how to use the various sampling

techniques.

• Be aware of the different types of errors that can

occur in a survey.

Population : 200 students in a statistics class

Final Grade

Proportion of D & HD: p = 53/200 = 0.

p ˆ^  0. 1

p ˆ^  0. 3

2.1 Sampling Error

• Non-sampling Error
  • Errors not related to sampling
  • Failure to answer; wrong answers; selection bias;

interviewer error; recording error.

• Controlled by careful administration
• Sampling Error
• The inevitable error because a sample is taken (only

partial information of the population is available in a

sample)

• Controlled by choice of sample size and sampling

design.

2.2 Sampling Design

Sampling Terminology

• Population: A group of ALL members (or objects) about which a

study intends to make inferences or draw conclusions from.

• Frame: A list of all population members, called sampling units,

that helps in selecting population members to form a sample.

Example:

• Suppose the target population is all families living in Selangor. A

feasible frame would be the residential pages of the Selangor

telephone books.

• Frame would most likely differ from the population.
• Some families have no telephone. Other families have unlisted

numbers. Some families may have changed numbers or moved

out since the directory was printed.

2.2 Sampling Design

• Population and Frame
  • Sampling is done from the frame, not the

target population.

• In theory, the target population and the

frame are the same.

• In reality, a researcher’s goal is to minimize

the difference between the frame and the

target population.

Frame Generate 5 random numbers using RANDBETWEEN(1, 30)

11, 21, 9, 10, 25

Simple random sample:

A department store audit involves checking a random sample from a population of 30 outstanding credit- account balances. The 30 accounts are listed in the following table. Select five accounts at random.

Example:

Using random number generator in Excel: Consider the frame of 40 families with annual income shown in the Table below. Choose a simple random sample of size 10 from this frame.

Example:

2.2.2 Systematic Sampling - Procedure

How? • Identify a frame

• Choose k (the length of sampling interval) ; k = N/n
• Select a random number between 1 and k
• Select entry on the frame according to this random number.
• Select every k -th member thereafter.

. .

. .

. .

. .

Generate a random number between 1 and 20

Random start: 11

Systematic sample consists of units labeled: 11, 31, 51, … , 991

• Systematic sampling is easier to administer then simple random sampling.
• Suitability of systematic sampling depends on order of sampling units in the frame and purpose of study.

Systematic Sampling vs. Simple Random

Sampling

2.2.2 Stratified Random Sampling

How?

1. Divide population into mutually exclusive sets or strata using a distinguishing feature.
2. A simple random sample is selected from each stratum.

With respect to characteristics of interest

• Units within the same stratum are quite similar
• Units between the strata are dissimilar

Examples of distinguishing features/strata:

1. Gender – female, male
2. Age group – below 20, 20 – 30, 31 – 40, 41 – 50, above 50
3. Occupation – manager, clerical, blue-collar, others

The combined random samples form the stratified sample.

N 1

N 2

N 3

N (^4) n 4

n 3

n 2

n 1

Stratified sample

Population size: N Sample size: n

ni?

Proportional allocation:

n

n N

N (^) i i 

n N

n Ni That is, i  

2.2.2 Stratified Random Sampling – Determining

sample size for each stratum

2.2.3 Cluster Sampling

How?

• Divide the population into convenient groups - clusters
• Select a random sample of clusters.
• For each chosen cluster:

Select all units of each chosen cluster to form sample

Select a random sample from each chosen cluster to form sample

Proximity of sampling units is often a consideration in forming clusters

One-stage cluster sample

Two-stage cluster sample

2.2.3 Cluster Sampling

Population

1 2 cluster N

.... i

srs of n clusters