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 careful study of a sample. A sample is a portion or subset of objects drawn from the population. For example, This process of obtaining samples of the population is called sampling and each object of the population is called a sampling unit. i.e. Sampling units are the individual members of the target population whose characteristics are to be measured. In the above example, a sampling unit is a student from SAJC. A sampling frame is a complete list of all the members of the population. i.e. the list of all the objects from which the sample is to be chosen. Examples are school registers, membership lists, Population: All students in SAJC Sample: a class of 20 students

SAJC 2021 JC1 H1 Mathematics Sampling Page 3 of 36 the electoral register etc. A problem, of course, is that the list may no t be up to date. In some cases, a list may not even exist. Usually, the objective of taking samples is to obtain estimates of population parameters, such as the population mean  or the population variance 2 . For example, a population parameter can be the height of a student in SAJC. Sample estimates of population parameters are ca led statistics. A sample statistic is said to be useful if it can be used to estimate an unknown parameter of the population. For example in the above example, to estimate the average height of students from SAJC, we can take a sample and compute the mean; this sample me an is a statistic that estimates the population mean (population parameter). The accuracy of our estimates will depend on the method of taking the sample and the sample size. This process of drawing valid conclusions about the population based on the results found from the sample is known as statistical inference (specific  general). What makes a good sample? To draw valid conclusions about the population, a sample must be free of selection bias and representative. A sample is said to be unbiased if every individual in the population has an equal chance of being selected. Eg: If we are interested in the distribution of the shoe sizes of male students in SAJC, then we should not stand at the finish line of a cross-country race and survey the first 50 boys who cross the line. The sample will be biased towards good runners. A sample is said to be maximally representative if the characteristics of the sample represent (as accurately as possible) the entire population. Eg: If we are interested in the characteristics of army officers, we should make sure each different type of officer is represented in order for the sample to be maximally representative. 5.1.1 Simple Random Samples Random Samples A sample is said to be a random sample (or a randomly chosen sample) if the method of selecting the sample is such that every member of the population has a n equal probability of being selected and each selection is independent of each other. When carrying out a random sample, you must ensure that all possible samples are equally likely to be chosen. The simplest type of random sample is a simple random sample of size n which is drawn from a population of size N in such a manner that every possible sample of size n is equally likely to be selected. By this method, all members of the population have an equal chance of being selected. Random sampling methods ensure that bias does not exist.

SAJC 2021 JC1 H1 Mathematics Sampling Page 4 of 36 Exercise 1 1. [H1 Math/N2011/A-level/Q7 modified] Two thousand students travel to college either by car, by bicycle or on foot. Any given student travels by the same method each day. A researcher carries out a survey to investigate the length of students’ journey times t o college, using a random sample of 100 students. Explain what is meant in this context by the term ‘a random sample’. Solution: A random sample is a sample drawn in such a way that each student who travels to college must have an equal chance of being selected as a member of the sample and the students must be selected independently i.e. one student being chosen does not have any effect on the chances of any other student then being chosen. In other words, the first student in the sample is ch osen at random from the population of 2000 students. Then for the second member of the sample, each of the remaining 1999 students has an equal chance of being selected, and so on. Note: [From Examiner’s report] Few candidates were able to give an appropriate explanation of the term ‘random sample’. The majority of answers consisted of either reversing the request and talking about a sample that was chosen randomly, or explaining how to find a random sample rather than explaining the meaning of the term. Some candidates did refer to there being an equal opportunity of selection, but hardly any mentioned the need for independence. 2. [H2 Math/N2019/A-level/Q6 (i) and (ii)] In a certain countr