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Statistics · 200 Level
This is the complete lecture note for STA 221 Statistical Inference II, prepared for Statistics students at 200 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: Sampling and Sampling Distributions Suggested Outline of Main Parts and Sub-Parts • 1.1 Introduction to Sampling • 1.1.1 Definition of sampling • 1.1.2 Importance of sampling in statistics • 1.1.3 Types of populations and samples • 1.2 Methods of Sampling • 1.2.1 Simple random sampling • 1.2.2 Stratified sampling • 1.2.3 Systematic sampling • 1.2.4 Cluster sampling • 1.3 Sampling Distributions • 1.3.1 Concept of a sampling distribution • 1.3.2 Sampling distribution of the mean • 1.3.3 Sampling distribution of the proportion • 1.4 Properties of Sampling Distributions • 1.4.1 Mean and variance of the sampling distribution • 1.4.2 Standard error • 1.4.3 Relationship to population parameters • 1.5 Applications of Sampling Distributions • 1.5.1 Role in estimation • 1.5.2 Role in hypothesis testing • 1.5.3 Practical examples in socio-economic studies Introduction Statistics often deals with large populations, yet it is rarely possible to collect data from every individual. Instead, we rely on sampling, which allows us to study a
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS smaller group and make inferences about the whole population. Understanding how samples behave, and how their distributions relate to population parameters, is fundamental to statistical inference. This Series will therefore provide you with the foundation needed to appreciate the role of sampling in estimation and hypothesis testing, which are central themes of this course. The aim of this Series is to introduce you to the principles of sampling and to explain how sampling distributions are constructed and applied in statistical inference. By the end of the Series, you should be able to describe different sampling methods, explain the concept of a sampling distribution, and evaluate its properties and applications. We shall commence this Series by examining the basic ideas of sampling, including definitions, importance, and the distinction between populations and samples. Next, we will explore various sampling methods such as simple random, stratified, systematic, and cluster sampling. Following that, we will study the concept of sampling distributions, focusing on the mean and proportion. Subsequently, we will analyse the properties of sampling distributions, including mean, variance, and standard error. Finally, we will conclude by considering practical applications of sampling distributions in estimation, hypothesis testing, and socio-economic studies. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define the concept of sampling and explain its importance in statistical inference, • SLO 1.2 describe and differentiate between major methods of sampling including simple random, stratified, systematic, and cluster sampling, • SLO 1.3 explain the concept of a sampling distribution and illustrate how it applies to the mean and proportion, • SLO 1.4 analyse the properties of sampling distributions by evaluating their mean, variance, and standard error, and • SLO 1.5 apply the principles of sampling distributions to practical contexts such as estimation, hypothesis testing, and socio-economic studies. 1.1 Introduction to Sampling
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Sampling is a fundamental concept in statistics because it is often impractical or impossible to collect data from an entire population. A population is the complete set of individuals, items, or data points that we are interested in studying, while a sample is a smaller subset of that population selected for analysis. By studying samples, we can make inferences about populations without the need for exhaustive data collection. Fill in the blank: A __________ is the complete set of individuals or items under study, while a sample is a smaller subset of it. 1.1.1 Definition of Sampling Sampling is the process of selecting a subset of individuals or items from a population to represent the whole. It allows researchers to gather information efficiently and make generalisations about the population. Diagram showing population and sample relationship 1.1.2 Importance of Sampling in Statistics Sampling is important because it saves time, reduces cost, and makes data collection feasible. For example, instead of surveying every household in a country, a government agency can select a representative sample to estimate average income levels. Sampling also enables researchers to apply statistical methods to test hypotheses and estimate parameters.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: Sampling is important because it saves __________, reduces cost, and makes data collection feasible. 1.1.3 Types of Populations and Samples Populations can be finite, such as the number of students in a university, or infinite, such as the number of possible outcomes when rolling a die repeatedly. Samples can be random, where every member of the population has an equal chance of selection, or non-random, where selection is based on convenience or judgement. Examples of finite and infinite populations
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: A population of students in a university is an example of a __________ population. Pilot Questions 1.1 1. A population refers to: A. A subset of individuals selected for study B. The complete set of individuals or items under study C. A group chosen by convenience D. None of the above 2. Sampling is defined as: A. Collecting data from the entire population B. Selecting a subset of individuals or items to represent the population C. Estimating parameters without data D. None of the above 3. Sampling is important because it: A. Saves time and reduces cost B. Always gives exact population values C. Eliminates the need for statistics D. None of the above 4. A population of households in a country is: A. Infinite B. Finite C. Random D. None of the above 5. A sample where every member of the population has an equal chance of selection is called: A. Non-random sample B. Random sample C. Convenience sample D. None of the above 1.2 Methods of Sampling
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS When conducting statistical studies, the way in which a sample is selected from a population is crucial. Different sampling methods provide different levels of accuracy and reliability. A good sampling method ensures that the sample is representative of the population, thereby allowing valid inferences to be made. Fill in the blank: A good sampling method ensures that the __________ is representative of the population. 1.2.1 Simple random sampling Simple random sampling is a method where every member of the population has an equal chance of being selected. This can be achieved using random number tables, lottery methods, or computer-generated randomisation. It is considered the most straightforward and unbiased method of sampling. Illustration of simple random sampling process 1.2.2 Stratified sampling
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Stratified sampling involves dividing the population into distinct groups, called strata, based on shared characteristics such as age, gender, or income level. Samples are then drawn from each stratum, often proportionally, to ensure representation of all groups. Fill in the blank: Stratified sampling divides the population into __________ based on shared characteristics. 1.2.3 Systematic sampling Systematic sampling selects members of the population at regular intervals. For example, if you have a list of 1,000 households and you want a sample of 100, you might select every 10th household after choosing a random starting point. This method is simple to apply but may introduce bias if there is a hidden pattern in the population list. Illustration of systematic sampling with every 10th unit selected 1.2.4 Cluster sampling Cluster sampling involves dividing the population into groups, or clusters, and then randomly selecting entire clusters for study. This is often used when populations are spread across wide geographical areas. For example, a
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