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Sampling distributions BPS chapter 11 © 2006 W. H. Freeman and Company Objectives (BPS chapter 11) Sampling distributions Parameter versus statistic The law of large numbers What is a sampling distribution? The sampling distribution of The central limit theorem x Reminder: Parameter versus statistic Population: the entire group of individuals in which we are interested but can’t usually assess directly. A parameter is a number describing a characteristic of the population. Parameters are usually unknown. Sample: the part of the population we actually examine and for which we do have data. A statistic is a number describing a characteristic of a sample. We often use a statistic to estimate an unknown population parameter. Population Sample The law of large numbers (page 273) Law of large numbers: As the number of randomly-drawn observations (n) in a sample increases, the mean of the sample (x) gets closer and closer to the population mean m (quantitative variable). ˆ the sample proportion (p ) gets closer and closer to the population proportion p (categorical variable). Problem 11.29 (page 296) Roll 2 fair six-sided dice and consider the total number of dots on the up-faces. Question: If we considered all possible rolls, what would be the average number of dots on the up-faces? Population? Parameter? m – the average number of dots on the up-faces – population mean We will sample using the Law of Large Numbers Applet to see the Law of Large Numbers in action! Sample? All possible rolls Rolls done via the Applet (each additional roll increases our sample size by 1) Statistic? x The average number of dots on the up-faces of all of the rolls in the sample – sample mean What is a sampling distribution? (page 276) The sampling distribution of a statistic is the distribution of all possible values taken by the statistic when all possible samples of a fixed size n are taken from the population. It is a theoretical idea—we do not actually build it. But we will simulate it! The Big Ideas: Averages are less variable than individual observations. Averages are more normal than individual observations. Note: When sampling randomly from a given population, the law of large numbers describes what happens when the sample size n is gradually increased. The sampling distribution describes what happens when we take all possible random samples of a fixed size n. Sampling distribution of x (the sample mean) We take many random samples of a given fixed size n from a population with mean m and standard deviation s. Some sample means will be above the population mean m and some will be below, making up the sampling distribution. Sampling distribution of “x bar” Histogram of some sample averages Let’s Simulate Building a Sampling Distribution We will use the program Sampling Sim Be sure you understand the difference between the population window, the sample window, and the sampling distribution window! For any population with mean m and standard deviation s: x Sampling distribution of s/√n m x In English: 1 2 1. The mean of the sample means is the population mean. 2. The standard deviation of the sample means is the population standard deviation divided by the square root of the sample size. What does 2. say about the variation of the sample mean versus the variation of the original population variable?? Averages are less variable than individual measurements! Mean of a sampling distribution of x: There is no tendency for a sample mean to fall systematically above or below m, even if the distribution of the raw data is skewed. Thus, the mean of the sampling distribution of x is an unbiased estimate of the population mean m —it will be “correct on average” in many samples. Standard deviation of a sampling distribution of x: The standard deviation of the sampling distribution measures how much the sample statistic x varies from sample to sample. It is smaller than the standard deviation of the population by a factor of √n. Averages are less variable than individual observations. For normally distributed populations When a variable in a population is normally distributed, then the sampling distribution of x for all possible samples of size n is also normally distributed. Sample means If the population is N(m,s), then the sample means distribution is N(m,s/√n). Population The central limit theorem (page 281) Central Limit Theorem: When randomly sampling from any population with mean m and standard deviation s, when n is large enough, the sampling distribution of x is approximately normal: N(m,s/√n). Sampling distribution of x for n = 2 observations Population with strongly skewed distribution (Figure 11.5 page 283) Sampling distribution of x for n = 10 observations Sampling distribution of x for n = 25 observations Averages are more normal than individual measurements! IQ scores: population vs. sample In a large population of adults, IQ scores have mean 112 with standard deviation 20. Suppose 200 adults are randomly selected for a market research campaign. The distribution of the sample mean IQ is A) exactly normal, mean 112, standard deviation 20. B) approximately normal, mean 112, standard deviation 20. C) approximately normal, mean 112 , standard deviation 1.414. D) approximately normal, mean 112, standard deviation 0.1. C) approximately normal, mean 112, standard deviation 1.414. Population distribution: N (m = 112; s = 20) Sampling distribution for n = 200 is N (m = 112; s /√n = 1.414) What if IQ scores are normally distributed for the population?? Application Hypokalemia is diagnosed when blood potassium levels are low, below 3.5mEq/dl. Let’s assume that we know a patient whose measured potassium levels vary daily according to a normal distribution N(m = 3.8, s = 0.2). If only one measurement is made, what's the probability that this patient will be misdiagnosed hypokalemic? If instead measurements are taken on four separate days, what is the probability of such a misdiagnosis? Practical note Large samples are not always attainable. Sometimes the cost, difficulty, or preciousness of what is studied limits drastically any possible sample size. Blood samples/biopsies: no more than a handful of repetitions acceptable. Often we even make do with just one. Opinion polls have a limited sample size due to time and cost of operation. During election times, though, sample sizes are increased for better accuracy. Not all variables are normally distributed. Income is typically strongly skewed for example. Is x still a good estimator of m then? Income distribution Let’s consider the very large database of individual incomes from the Bureau of Labor Statistics as our population. It is strongly right-skewed. We take 1000 SRSs of 100 incomes, calculate the sample mean for each, and make a histogram of these 1000 means. We also take 1000 SRSs of 25 incomes, calculate the sample mean for each, and make a histogram of these 1000 means. Which histogram corresponds to the samples of size 100? 25? How large a sample size? It depends on the population distribution. More observations are required if the population distribution is far from normal. A sample size of 25 is generally enough to obtain a normal sampling distribution from a strong skewness or even mild outliers. A sample size of 40 will typically be good enough to overcome extreme skewness and outliers. In many cases, n = 25 isn’t a huge sample. Thus, even for strange population distributions we can assume a normal sampling distribution of the mean, and work with it to solve problems. Further properties The Central Limit Theorem is valid as long as we are sampling many small random events, even if the events have different distributions (as long as no one random event has an overwhelming influence). Why is this cool? It explains why so many variables are normally distributed. Example: Height seems to be determined by a large number of genetic and environmental factors, like nutrition. So height is very much like our sample mean x. The “individuals” are genes and environmental factors. Your height is a mean. Now we have a better idea of why the density curve for height has this shape.