Chapter 6 In order to know about some characteristic of a population with certainty, we must collect data from every element in the population of interest; such an effort is referred to as acensus.However, there are many potential difficulties associated with taking a census: A census may be expensive; if resources are limited, it may not be feasible to take a census. A census may be time consuming; if the data need be collected quickly, a census may not be suitable. A census may be misleading; if the population is changing quickly, by the time a census is completed the data may be obsolete. A census may be unnecessary; if perfect information about the characteristic(s) of the population of interest is not required, a census may be excessive. A census may be impractical; if observations are destructive, taking a census would destroy the population of interest. the objective of sampling is to gather data from a subset of the population that is as similar as possible to the entire population, so that what we learn from the sample data accurately reflects what we want to understand about the entire population.When we use the sample data we have collected to make estimates of or draw conclusions about one or more characteristics of a population (the value of one or more parameters), we are using the process ofstatistical inference. statistical inference The process of making estimates and drawing conclusions about one or more characteristics of a population (the value of one or more parameters) through the analysis of sample data drawn from the population. It is important to realize that sample results provide onlyestimatesof the values of the corresponding population characteristics Some error, or deviation of the sample from the population, is to be expected.With proper sampling methods, the sample results will provide "good" estimates of the population parameters Thesampled populationis the population from which the sample is drawn, and aframeis a list of the elements from which the sample will be selected random sampling can be used to select a sample from a finite population and we describe how a random sample can be taken from an infinite population that is generated by an ongoing process.
6.1 Selecting a Sample A measurable factor that defines a characteristic of a population, process, or system is called aparameter. working with a sample would be preferable to working with the entire population Sampling from a Finite Population Statisticians recommend selecting a probability sample when sampling from a finite population because a probability sample allows you to make valid statistical inferences about the population.The simplest type of probability sample is one in which each sample of sizenhas the same probability of being selected. It is called a simple random sample. Asimple random sampleof sizenfrom a finite population of sizeNis a sample selected such that each possible sample of sizenhas the same probability of being selected. Procedures used to select a simple random sample from a finite population are based on the use of random numbers.We can use Excel's RAND function to generate a random number between 0 and 1 by entering the formula =RAND()into any cell in a worksheet.The number generated is called a random number because the mathematical procedure used by the RANDfunction guarantees that every number between 0 and 1 has the same probability of being selected Sampling from an Infinite Population Sometimes we want to select a sample from a population, but the population is infinitely large or the elements of the population are being generated by an ongoing process for which there is no limit on the number of elements that can be generated.Thus, it is not possible to develop a list of all the elements in the population.This is considered the
infinite population case.With an infinite population, we cannot select a simple random sample because we cannot construct a frame consisting of all the elements.In the infinite population case, statisticians recommend selecting what is called a random sample. Arandom sampleof sizenfrom an infinite population is a sample selected such that the following conditions are satisfied. 1.Each element selected comes from the same population. 2.Each element is selected independently. Each case may require a different selection procedure. Let us consider two examples to see what we mean by the conditions: (1)Each element selected comes from the same population, and (2)each element is selected independently. A common quality-control application involves a production process for which there is no limit on the number of elements that can be produced. The conceptual population from which we are sampling is all the elements that could be produced (not just the ones that are produced) by the ongoing production process.Because we cannot develop a list of all the elements that could be produced, the population is considered infinite. Situations involving sampling from an infinite population are usually associated with a process that operates over time. 6.2 Point Estimation To estimate the value of a population parameter, we compute a corresponding characteristic of the sample, referred to as asample statistic.