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 reasonable alternative would be to select and study a subset or a sample of the units in the population. As in the previous example, a possible sample can be formed by selecting all the Year 6 students from Raffles Institution to obtain information about the YouTube channels they subscribed to. In order to derive meaningful and reliable information about the population, it is necessary to select a sample from the population carefully. More importantly, it should be a representative sample of the population. For example, if we are studying heights of teenagers in Singapore, our sample should “resemble” the population and contain members with a good spread of heights. The method of selecting the sample is called the sampling method or sampling plan.

Raffles Institution H2 Mathematics 2025 Year 6 __________________ Chapter S4: Sampling Page 3 of 24 1.2 Random Samples A sample is a random sample if it is chosen in such a way that: 1. every element in the population has an equal chance of being selected, and 2. the selections are independent of each other A sample that is not random is known as a non-random sample. Example 1 In a school there are 30 classes, each with 30 students, and 20 classes, each with 20 students. To find out students’ perception of the food sold in the canteen, the school takes a random sample of 100 students. Explain what is meant in this context by the term ‘a random sample’. State, with a reason, whether each of the following methods would give a random sample. Method 1: 2 students are randomly chosen from each of the 50 classes. Method 2: Assign every student in the school a number, from 1 to 1300, by arranging their names in alphabetical order. Use a computer to generate 100 random numbers. The 100 students with the corresponding numbers would be the ones chosen to be in the sample. Method 3: First 100 students who patronize the noodle stall are selected as the sample. Solution A random sample is a sample chosen in such a way that every student in the school has an equal chance of being selected, and the selection of students are independent of each other. Method 1: P(a particular student in a class of 20 being selected for the sample) 1 19 2220 19 20    or 1 19 1 1 20 2 1 10 C C C  P(a particular student in a class of 30 being selected for the sample) 1 29 2230 29 30    or 1 29 1 1 30 2 1 15 C C C  Since the probability of choosing any one student is not equal, method 1 does not give a random sample. Method 2: Since the children from all classes are grouped together and 100 children corresponding to the 100 numbers randomly selected by the computer forms a sample, each child has an equal chance of 100 1300 of being selected and the selections are approximately independent* of each other. Hence method 2 gives a random sample.

Raffles Institution H2 Mathematics 2025 Year 6 __________________ Chapter S4: Sampling Page 4 of 24 *Note: In method 2, since 1300 is reasonably large, the selection of 100 students will be approximately independent of each other. For example, P(a student is included in the sample) 100 0.0769231300  P(student A is included in the sample given that student B is included) 99 0.0762121299  Method 3: It gives a non-random sample as students who do not eat noodles at all will not patronize the noodle stall and hence have zero probability of being included in the sample. Having seen a few random and non-random samples, we will now discuss what to do with the information obtained from these samples. One of the main aims in the study of statistics is to be able to make inferences about the population based on sample information. 1.3 Population Parameters - Population Mean and Population Variance A population parameter is a number that is associated with a population characteristic. In many cases, we are concerned with two population parameters, namely the population mean and the population variance. These are constants of a population and they are often unknown. The study of a population often involves finding estimates of these parameters. Notation: The population mean is denoted by  and the population variance is denoted by 2 . 1.4 Random Sample of size n and Sample Statistics To define a sample statistic, let’s first look at the definition of a random sample of size n. A random sample of size n (from some given distribution) is a set of n random variables X1, X2, X3, . . . , Xn , satisfying the following 2 conditions: (a) Each Xi (i = 1, 2, 3, … , n) has exactly the same distribution. (b) The set X1, X2, X3, . . . , Xn are mutually independent. Suppose X1, X2, X3, . . . , Xn is a random sample of size n taken from some distribution, then a sample statistic is a function of these random variables. Thus once the random sample is known, the numerical value of the statistic is known, a