SAJC 2021_Chapter 5 Sampling (Teacher version) V1

SAJC 2021 JC1 H1 Mathematics Sampling Page 1 of 36 Chapter 5 (Statistics) Sampling (Teacher’s copy) Objectives At the end of the chapter, you should be able to: (a) understand the concepts of population, simple random sample; (b) understand the use of and calculate the unbiased estimates of the population mean and variance, including cases where the data are given in summarised form x and 2 x , or  xa and   2 xa ; (c) understand that the sample mean X is a random variable with E( )X  and 2 Var( )X n  ; (d) use the fact that X has a normal distribution if X has a normal distribution; (e) apply the sampling distribution 2 N( , ) n  to solve statistical problems in real -world situations; (f) use the C entral Limit Theorem to treat sample mean as having a normal distribution when the population is not normally distributed and the sample size is sufficiently large. [Note: ‘Large’ samples will usually be of size at least 30, but students should know that using the approximation of normality can sometimes be useful with samples that are smaller than this.] Content 5.1 Introduction and Definitions 5.1.1 Population and Simple Random Sample 5.2 The Sample Mean as a Random Variable 5.2.1 Mean and Variance of the Distribution of Sample Mean 5.3 The Distribution of the Sample Mean and the Sample Sum 5.3.1 Distributions of Sample Mean and Sample Sum from a Normal population 5.3.2 Distributions of Sample Mean and Sample Sum from a Non-Normal population 5.4 Estimation 5.4.1 Some Definitions 5.4.2 Unbiased Estimates of Population Mean and Variance

SAJC 2021 JC1 H1 Mathematics Sampling Page 2 of 36 5.1 Introduction and Definitions In real life situations, we often need to conduct a statistical enquiry for different purposes. For example, a school may want to gather information about the studying habits of the students. A manufacturer of batteries may want to find out the length of lifespan of his product. For such purposes, we need information to draw valid conclusions about a group of individuals or objects. This group of individuals or objects is called a population. A population is defined as the entire collection of objects which a statistician is interested to study on. The size of a population can be finite or infi nite. In most situations, it is impossible or impractical to examine every individual or object of the whole population for various reasons, such as: (a) The population is large or infinite. E.g. the number of people who earned less than US$600 a month. (b) The collection of information may destroy the sample. E.g. when testing batteries, fireworks, electric fuses etc. (c) It may be impossible to gain access to every member of the population. E.g. measuring the lengths of ants of a particular species. Therefore it is more practical to conduct a care