ACJC 2026 Sampling Lecture Notes

1 16 SAMPLING SYLLABUS • Concepts of population and simple random sample • Concept of the sample mean X as a random variable with ( )E X = and ( ) 2 Var X n = • Distribution of sample mean 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. 30n ) • 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 ( )xa− and ( ) 2 xa−

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Concepts of Population and Sample ................................................. 3 2 Estimation ......................................................................................... 5 2.1 Unbiased Estimate of Population Mean .................................. 6 2.2 Sample Variance .................................................................... 6 2.3 Unbiased Estimate of Population Variance ............................ 7 2.4 Method of Coding ................................................................. 11 3 Sampling Distribution ................................................................... 12 3.1 Expectation and Variance of the Sample Mean X ............... 14 3.2 Distribution of the Sample Mean ( Samples taken from a Population that is Normally Distributed) .............................. 15 3.3 Distribution of the Sample Mean ( Samples taken from a Population that is not Normally Distributed) ........................ 17 4 Miscellaneous Examples ................................................................ 19 Annex A: Proof that 2s is an Unbiased Estimate of Population Variance 2 ......................................................................................... 22 Annex B: Practice Questions on Sampling ........................................... 23

16 Sampling 3 LECTURE 1 Lesson Outline • Understand the concepts of population and sample • Understand the concept of point estimation of population mean and population variance • Find the unbiased estimates of population mean and population variance 1 CONCEPTS OF POPULATION AND SAMPLE We have so far only considered random variables whose probability distribution is known. For example, random variables that follow a binomial distribution, a normal distribution etc. However, the probability distribution of the population is often not completely known. Very rarely in practice can we afford the luxury of examining the complete population. Almost invariably, sample observations are used to make inferences about the population probability distribution. This process is known as statistical inference. Sampling theory is the study of this relationship between a population and the samples drawn from that population. 1. A population is the complete set of items being investigated. 2. A sample is a subset of a population. 3. A sample can be chosen by selecting its members from a sampling frame (a suitable list of some sort, e.g. electoral register, class lists) which may or may not contain the entire population. 4. For the sample to be representative of the whole population and free of bias, random samples should be taken. Population Samples

ACJC 2025/26 H2 Mathematics (9758) 4 5. A random sample of size n is a sample which is taken in such a way that every sample of size n of the population has an equal chance (equal probability) of being chosen. This also means that for a random sample of size n, every member of the population has an equal chance of being chosen. Otherwise, the sample is a non- random sample. Examples of Random Samples (a) 50 balls are numbered from 1 to 50 and put into a box. 3 balls are drawn from the box to determine the winning numbers in a game. (b) 200 students are numbered from 1 to 200. Select 20 students by the following method: Select the first student by choosing an integer k randomly, where 1 10k . Subsequently, select the remaining 19 students by selecting the numbers 10k + , 20k + , …, 180k + , 190k + . Examples of Non-Random Sampling (a) A newly opened restaurant wanted to find out their customers’ dining experience in the restaurant. The manager of the restaurant suggested to complete the sampling survey in the fastest possible way. On day 1, the staff interviewed the first 20 customers to conduct the survey and collect feedback from the customers’ dining experience. 6. A sample statistic (or statistic) is any quantity obtained from a sample for the purpose of estimating a population parameter, while a population parameter is a quantity that measures some aspect (e.g., mean, variance, proportion) of the whole population. Examples of sample statistics are sample mean and sample variance, while examples of population parameters are population mean and population variance. 7. Often, we are interested in finding two population parameters, namely the population mean and population variance. Population parameters are fixed and usually unknown. Thus, the study of population often involves finding estimates of these parameters. The population mean is usually denoted by  and the population variance is denoted by 2 . 8. When a sample has been obtained, suitable sample statistics are calculated and used to estimate population parameters . For example, the sample mean x can be used as an estimate for the population mean .

16 Sampling 5 2 ESTIMATION There are many real-life situations where it is difficult, tedious and sometimes impractical to calculate population mean and population variance because of the large population size. However, we can make inferences about these population parameters by examining samples drawn from the population. This process of making inferences or generalisations about a population based on information obtained from samples selected from the population is known as statistical inference. There are two major areas of statistical inferences: (a) estimation and (b) tests of hypotheses. Suppose that a population has an unknown parameter (i.e. mean, variance, proportion). Then an estimate of this unknown parameter can be made from the information obtained from random sample(s) taken from the population. Suppose we want to find out the population mean of the heig ht of the male students in ACJC. One way of estimating this mean is to use the sample mean of a random sample made up of 5 male students from each class. In this chapter, we will study point estimation where a single value is used to infer information about the population. An estimate of a population parameter  is a single value t of a sample statistic T that is used to estimate . The sample statistic T is called an estimator of . Let X be a random variable and let 1 2 3, , , , nX X X X  be independent observations of X. Then, the sample mean 12 1 1 = + + +==   n n i i X X XXX nn is an estimator of the population mean . Consider a population with unknown parameter . Let T be some statistic derived from a random sample taken from a population to estimate the value of . Then T is an unbiased estimator of  if E( )T = .

ACJC 2025/26 H2 Mathematics (9758) 6 2.1 Unbiased Estimate of Population Mean Consider a random sample of size n taken from a population with unknown mean . For the n independent observations, 1 2 3, , , , nX X X X  , let 12 nX X XX n + + +=  . Proof ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 1 2 3 1 2 3 1 2 3 EE 1 E 1 E E E E 1 E( ) E( ) n n n X X X XX n X X X Xn X X X Xn nXn X  + + + +=  = + + + + = + + + + = = =    Therefore, X is an unbiased estimator of the population mean . ■ If 1 2 3, , , , nx x x x  are the values that 1 2 3, , , , nX X X X  assume, then an unbiased estimate of  is 12 ...+ + + 