Chapter 6: Sampling Distributions

Imagine a column of data which contains peoples' annual income, sorted from the smallest to the largest. That column is a distribution of data. In order to better visualize the data, suppose you take that column of income values and construct a histogram. With this histogram in mind, recall that a distribution of data has three important characteristics: (1) its shape, (2) its mean, and (3) its standard deviation. A sampling distribution is a particular type of dataset created by drawing different samples of the same size from a given population. Each time a new sample is drawn, a statistic (e.g., mean and proportion) is calculated and added to the dataset. A complete sampling distribution contains statistics from all possible samples of the same size taken from a single population. If the population is finite (i.e., a fixed number of values), then the number of unique samples of a given size is also finite. When a population is infinite, then the number of unique samples is infinite and so is the sampling distribution. As long as population sizes are very large and sample sizes are relatively small, we can treat finite and infinite populations equivalently. We will discuss what is meant by “large” and “small” later in this chapter. To illustrate the concept of a sampling distribution, suppose the population of interest is a large statistics class of 100 students. Let us say we take a random sample of 10 students from this population and calculate the average grade point average (GPA). What we just calculated is a statistic, and that statistic is random because it comes from a random sample. If we drew another random sample of 10 students, we would likely get a different average GPA value. If everyone in the class were identical, different samples would lead to the same average GPA. Of course, this is not the case. The population is made up of all kinds of students, from bookworms to slackers. And each has the same chance of being in the sample of 10

Ch 9: Understanding Sampling Distributions: Statistics as Random Variables

9.3: SD Module 0: An Overview of Sampling Distributions

Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 326 Chapter 9 Unless otherwise noted, all content on this page is © Cengage Learning. A sampling distribution is the probability distribution for a sample statistic. It plays the same role as probability distributions do for other random variables. We can specify the probability that a sample statistic will fall into a specific interval if we know all of the details about its sampling distribution. For instance, in Example 9.2, if we knew all of the details of the smooth curve in Figure 9.2, we could answer questions such as, “What is the probability that the mean hours of sleep (i.e., the sample mean) for a random sample of 190 students will be less than 6.9 hours?” or “What is the prob-ability that the mean hours of sleep for a random sample of 190 students will be between 7 and 8 hours?” Note that this is not the same question as “What is the prob-ability that one randomly selected student slept between 7 and 8 hours?” Sampling distributions give us information about sample statistics, not about individual values for the population. Note that sampling distributions give us information about sample statistics, given that we know all about the population already. For instance, to get the sampling distri-bution in Figure 9.2, we assumed that we knew that the mean and standard deviation of the population of sleep times were 7.1 hours and 2 hours, respectively. Knowing how to get information about possible sample statistics on the basis of known values of population parameters may not seem very helpful in practice. If we know about the population already, we don’t need to find sample statistics. Therefore, why do we want to know about sampling distributions? One answer is that we can turn the information around and use it for statistical in-ference.

CHAPTER 6 Continuous Random Variables and the Normal Distribution

Technology Assignments

268 garajstock/Shutterstock Sampling Distributions Chapters 5 and 6 discussed probability distributions of discrete and continuous random variables. This chapter extends the concept of probability distribution to that of a sample statistic. As we discussed in Chapter 3, a sample statistic is a numerical summary measure calculated for sample data. The mean, median, mode, and standard deviation calculated for sample data are called sample statistics. On the other hand, the same numerical summary measures calculated for pop- ulation data are called population parameters. A population parameter is always a constant (at a given point in time), whereas a sample statistic is always a random variable. Because every random variable must possess a probability distribution, each sample statistic possesses a prob- ability distribution. The probability distribution of a sample statistic is more commonly called its sampling distribution. This chapter discusses the sampling distributions of the sample mean and the sample proportion. The concepts covered in this chapter are the foundation of the inferential statistics discussed in succeeding chapters. You read about opinion polls in newspapers, magazines, and on the Web every day. These polls are based on sample surveys. Have you heard of sampling and nonsampling errors? It is good to be aware of such errors while reading these opinion poll results. Sound sampling methods are essential for opinion poll results to be valid and to minimize effects of such errors. 7.1 Sampling Distribution, Sampling Error, and Nonsampling Errors 7.2 Mean and Standard Deviation of x 7.3 Shape of the Sampling Distribution of x 7.4 Applications of the Sampling Distribution of x 7.5 Population and Sample Proportions; and the Mean, Standard Deviation, and Shape of the Sampling Distribution of p ̭ 7.6 Applications of the Sampling Distribution of p ̭ CHAPTER 7 268

5 Sampling Distributions and Confidence Intervals

Fortunately, we already know something about those distributions because of the material we have covered on the binomial and normal distributions. The next section reveals how the binomial and normal distributions are related to sampling distributions for proportions and means. A population statistic (or sometimes parameter ) is a quantity based on data from the entire population, whereas a sample statistic (or estimate ) is based on a sample of data taken from the population. SUMMARY A sampling distribution of a sample statistic includes the probability of getting each possible value that the sample statistic can take. The theoretical average of any sample statistic, under repeated sampling, is known as its expected value . A sample statistic is an unbiased estimator if its expected value is always the same as the corresponding population statistic's value. The extent to which sample statistics vary around the population statistic's value is called sampling error . Sampling distributions for proportions Proportions and percentages (which are proportions multiplied by 100) are widely used in psychology, other social sciences, and in the popular media. Sampling Distributions and Confidence Intervals âä Quite often, proportions are estimated by samples drawn from a population. It is common practice, for instance, for pollsters to forecast election results by estimating the proportion of the electorate who will vote for each candidate on the basis of a random sample from that electorate. In order to know how precise their estimates are, they need to know the sampling distribution of a proportion. Many psychological experiments require us to know the sampling distribu-tion of a proportion for somewhat different reasons. Experiments and clinical trials usually involve making independent observations under identical condi-tions, rather like someone ¯ipping a coin over and over again to see whether it is an unbiased coin.

Chapter 8. Sampling and the Sampling Distribution of a Statistic

8.4 The Sampling Distribution of a Statistic

Hence sampling distributions aid us in discerning the patterns and magnitudes of sampling errors; they describe how a statistic varies, due to chance, under random sampling. For instance, let T be an estimator for θ with t the realization of T. We take a random sample of size n and record t (= t 1 ). Then we take a second random sample of the same size and again find t (= t 2 ). Typically t 1 = t 2 . As we take more and more random samples of the same size n, we build up a relative frequency distribution of various t’s or estimates of θ . As the number of random samples gets larger and larger, this relative frequency distribution gets closer and closer to 308 Chapter 8 Sampling and the Sampling Distribution of a Statistic the true theoretical sampling distribution of T, which reflects the inherent sample- to-sample variability of the estimates of θ , and which is generally not of the same form as the population probability density function (probability mass function) f (x; θ ). However, as we shall soon see, a sampling distribution is always related in some specific way to the parent population distribution. To formalize the preceding discussion, let us extract all possible random samples of a given size from a specific population distribution and compute the corresponding realizations of a statistic T. The resulting relative frequency dis- tribution yields an approximation to the theoretical probability distribution that would have been obtained had the number of samples been infinite rather than just large. Hence we may define the sampling distribution of a statistic T as a the- oretical probability distribution that associates with each possible realization t of T its probability of occurrence (or probability density value). As such, it provides us with a theoretical model for the relative frequency distribution of the possible realizations of T that we would obtain under repeated sampling from the same population.

CHAPTER 5 Introduction to Probability Distributions

5.5 Expected values (mathematical expectation)

S A M P L I N G D I S T R I B U T I O N S F O R M E A N S A N D P R O P O R T I O N S 6 6.1 Statistical inference and the sampling distribution of the mean 6.2 Sampling distribution of proportions 6.3 Some desirable properties of estimators Chapter Objectives Having carefully studied this chapter and completed the exercises you should be able to do the following  Know the symbols for population parameters and the corresponding sample statistics  Calculate and plot the sampling distribution of the mean for small populations  State and apply the Central Limit Theorem  Calculate probabilities for x for large samples  Calculate the two limits that contain a given percentage of sample means.  Calculate a means control chart and comment on the stability of the process mean  Calculate and plot the sampling distribution of proportions for large n  Calculate the two limits that contain a given percentage of sample proportions  Plot a p -chart and comment on the stability of the process proportion of successes 6.1 Statistical inference and the sampling distribution of the mean Inference about population parameters Recall that the definition of statistics given in Collins Concise Dictionary was given in the intro- duction: ... a science concerned with the collection, classification and interpretation of quantitative data and with the application of probability theory to the analysis and estimation of population parameters. [ 266 ] C H A P T E R 6 Now that you have studied the ‘collection, classification and interpretation of quantitative data’ and ‘probability theory’ in Chapters 1 to 5, you have a solid foundation for the study of the really interesting part of statistics—statistical inference which is based on the ‘analysis and estimation of population parameters’. Parameters such as the population mean will be estimated by a sample mean, calculated from a random sample.

Chapter 7: Random Variables and Probability Distributions

Cumulative Review Exercises

427 8 T he inferential methods presented in Chapters 9–15 use sample data to learn about population characteristics. For example, let m denote the true mean fat content of quarter-pound hamburgers marketed by a national fast-food chain. To learn something about m , we might select a sample of n 5 50 hamburgers and determine the fat content for each one. The sample data might produce a mean of x 5 28.4 grams. How close is this sample mean to the population mean, m ? If we select another sample of 50 quarter-pound burgers and then determine the sample mean fat content, would this second value be near 28.4, or might it be quite different? These questions can be addressed by studying what is called the sampling distribution of x . Just as the probability distribution of a numerical variable describes its long-run behavior, the sampling distribution of x provides information about the long-run behavior of x when samples are selected from a population. In this chapter, we also consider the sampling distribution of a sample proportion (the fraction of individuals or objects in a sample that have some characteristic of interest). The sampling distribution of a sample proportion, p / , provides information about the long-run behavior of the sample proportion. This knowledge allows us to make inferences about a population proportion. LEARNING OBJECTIVES Students will understand: ● ● That the value of a sample statistic varies from sample to sample. ● ● That a sampling distribution describes sample-to-sample variability in the values of a statistic. ● ● How the standard deviations of the sampling distributions of x and p / are related to sample size. Students will be able to: ● ● Describe general properties of the sampling distribution of x . ● ● Describe general properties of the sampling distribution of p / . ● ● Determine when the sampling distribution of x or p / is approximately normal.

9 The Role of Sampling in Inferential Statistics

Sampling with and without Replacement

However, by assuming that all pop-ulations are at least 100 times as large as the samples being drawn from them, we will be able to use sampling without replacement and obtain quite accurate results. SAMPLING DISTRIBUTIONS Now that we are somewhat more familiar with what a random sample is and how one is selected, we will turn to the question of how it is used in inferential statistics. As mentioned previously, the basic idea in inferential statistics is to use a statistic calcu-lated on a sample in order to estimate a parameter of a population. This procedure is made difficult by the unavoidable fact that in most cases, the value we get for any statistic will vary somewhat from sample to sample, even when all the samples are randomly selected from the same “parent” population. TABLE 9.3. Probabilities of selecting the second card (Population size = 5,200; Sample size = 2) With replacement Without replacement Probability of Ace of hearts 100 5 200 0192307 , . = 99 5 199 0190421 , . = Probability of all other cards 100 5 200 0192307 , . = 100 5 199 0192344 , . = TABLE 9.2. Probabilities of selecting the second card (Population size = 52; Sample size = 2) With replacement Without replacement Probability of Ace of hearts 1 52 0192307 = . 0 52 0 = . Probability of all other cards 1 52 0192307 = . 1 51 0196078 = . DESCRIBING THE SAMPLING DISTRIBUTION OF MEANS EMPIRICALLY 251 DESCRIBING THE SAMPLING DISTRIBUTION OF MEANS EMPIRICALLY Imagine the process of deciding on a particular sample size N to use, randomly choosing and listing all possible samples of size N from the parent population, and for each sample, recording the value of the statistic we are interested in. (Of course, each sample of size N is replaced in the population before the next sample of size N is selected.) If we then take all the values we have obtained for this statistic, we can construct a frequency distribution of them just as we can construct a frequency distribution of any collection of numbers.

Chapter 6 - Sampling and Sampling Distributions

Note that the dispersion for the sampling distribution of X is smaller under Case 2 (SRS) than it is with Case 1 (RS). 6.5 Sampling Distribution for a Statistic from an Infinite Population Consider a population from which k random samples, each of size n , are taken. In general, if given k samples, k di ↵ erent values for the sample mean will result. If k is very large, theoretically infinite, the values of the means from each of the samples, denoted X i for each sample i , will be random variables with a resulting distribution referred to as the sampling distribution of the sample mean. The sampling distribution of a statistic, t ( X ), is the resulting probability distribution for t ( X ) calculated by taking an infinite number of random samples of size n . The resulting sampling distribution will typically not coincide with the distribution of the parent population. 6.5.1 Sampling Distribution for the Sample Mean The sample mean is the average of a sample and the best measure of center when a distribution is bell shaped. The sampling distribution of the sample mean has some rather 368 Probability and Statistics with R , Second Edition ¯ X S 2 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 Random Sample Simple Random Sample 0 4 8 12 0 4 8 12 FIGURE 6.1: Sampling distributions of X and S 2 under random sampling (RS) and simple random sampling (SRS) for Example 6.9 amazing properties. If one is sampling from a normal population, the sampling distribution of the sample mean is also a normal distribution. Even more incredible, as long as the variance of the population is finite, the sampling distribution of the sample mean is also normal when sampling from any other distribution as long as the sample size is su ffi ciently large.

Chapter 3 Introduction to Hypothesis Testing: Permutation Tests

3.5 Cause and Effect

The most likely outcome is 0.5, followed by 0.4 and 0.6, and so on – values farther from 0.5 are less likely. The distribution is bell-shaped and centered approxi- mately at 0.50; the sample mean is 0.504. The standard deviation is 0.167; we call the standard deviation of a statistic a standard error. Mathematical Statistics with Resampling and R, Third Edition. Laura M. Chihara and Tim C. Hesterberg. © 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc. 78 4 Sampling Distributions 0 4 8 12 16 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Count Figure 4.1 Distribution of ̂ p after 50 sets of 10 tosses. 0 28 56 84 112 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Count Figure 4.2 Distribution of ̂ p after 500 sets of 10 tosses. Definition 4.1 Let X be a random sample and let T = h(X ) denote some statistic. The sampling distribution of T is its probability distribution. ∥ (Here X may be a vector (X 1 , … , X n ); it may also represent samples from mul- tiple groups or populations.) The permutation distributions in Chapter 3 are sampling distributions, as is the above example. The key point is that a sampling distribution is the distribution of a statistic that summarizes a data set and represents how the statistic varies across many random data sets. A histogram of one set of observations drawn from a population does not represent a sampling distribution. A histogram of permutation means, each from one sample, does represent a sampling distribution. Definition 4.2 A statistic which estimates a quantity of interest is an estima- tor. If X 1 , X 2 , … , X n are random variables from a distribution with parameter 𝜃 and g (X 1 , X 2 , … , X n ) an expression used to estimate 𝜃, then we call this function an estimator. ∥ For example, if X is the number of heads in 50 throws of a coin, then ̂ p = X∕50 is an estimator for the true proportion of head p, whereas X is a statistic that is not an estimator.

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