RI S4 Sampling_Lecture Notes

Raffles Institution H2 Mathematics (9758) 2025 Year 6 ___________________ Chapter S4: Sampling Page 1 of 24 Chapter S4 :Sampling SYLLABUS INCLUDES  Concepts of population and simple random sample  Concept of the sample mean X as a random variable with  E X  and   2 VarX n  .  Distribution of sample means from a normal population  Use of the Central Limit Theorem to treat sample mean as having normal distribution when the sample size is sufficiently large (e.g. 30n )  Use of unbiased estimates of the population mean and variance from a sample, including cases where the data are given in summarised form x and 2 x , or  x a and   2 x a PRE-REQUISITES  Normal distribution CONTENT 1 Terminology 1.1 Population and Sample 1.2 Random Samples 1.3 Population Parameters – Population Mean and Population Variance 1.4 Random Sample of size n and Sample Statistics 2 Estimates of Population Parameters 2.1.1 Unbiased Estimator of Population Mean 2.1.2 Unbiased Estimator of Population Variance 3 The Sample Mean X as a Random Variable 4 The Distribution of the Sample Mean 4.1 Sampling from a Normal Population 4.2 Sampling from a Non-Normal Population Appendix 1: Unbiased Estimators of Population Mean and Variance Appendix 2: Use of GC to simulate Sampling from a Normal Population

Raffles Institution H2 Mathematics 2025 Year 6 __________________ Chapter S4: Sampling Page 2 of 24 INTRODUCTION Many real-world problems require the collection of data and studying/analyzing them. The objective of such statistical investigations is inference – that is, making decisions, estimations, generalizations or predictions about an unknown aspect of a population based on information obtained from a sample. 1 TERMINOLOGY 1.1 Population and Sample A population is the entire collection of data (persons or items or individuals) that we want to study. In studying a population, we may focus on one or more characteristics (of the persons or items or individuals) in the population. For example, we may be interested in the YouTube channels teenagers in Singapore subscribed to, so the population refers to all teenagers in Singapore while the characteristic refers to the various youtube channels. If the population we wish to study is small in size, then it may be feasible to measure every unit in the population (this process is called a census). Very often, it is impossible or impractical to study an entire population by examining every unit in the population. This could be due to reasons such as: (a) size of population is too big or infinite, (b) population is dynamic and intractable, (c) time and funding constraints. Besides, if we are not too concerned with being exact, then it may not be worthwhile investing time and effort to the study of every data in the population. A more reaso