Anthony J Greene1 Where We Left Off What is the probability of randomly selecting a sample of three individuals, all of whom have an I.Q. of 135 or more? Find the z-score of 135, compute the tail region and raise it to the 3 rd power. XX So while the odds chance selection of a single person this far above the mean is not all that unlikely, the odds of a sample this far above the mean are astronomical z = 2.19P = =

Anthony J Greene2 Sampling Distributions I What is a Sampling Distribution? A If all possible samples were drawn from a population B A distribution described with Central Tendency µ M And dispersion σ M, the standard error II The Central Limit Theorem

Anthony J Greene3 Sampling Distributions What you’ve done so far is to determine the position of a given single score, x, compared to all other possible x scores x

Anthony J Greene4 Sampling Distributions The task now is to find the position of a group score, M, relative to all other possible sample means that could be drawn M

Anthony J Greene5 Sampling Distributions The reason for this is to find the probability of a random sample having the properties you observe. M

Anthony J Greene6 1.Any time you draw a sample from a population, the mean of the sample, M, it estimates the population mean μ, with an average error of: 2.We are interested in understanding the probability of drawing certain samples and we do this with our knowledge of the normal distribution applied to the distribution of samples, or Sampling Distribution 3.We will consider a normal distribution that consists of all possible samples of size n from a given population Sampling Distributions

Anthony J Greene7 Sampling Error Sampling error is the error resulting from using a sample to estimate a population characteristic.

Anthony J Greene8 Sampling Distribution of the Mean For a variable x and a given sample size, the distribution of the variable M (i.e., of all possible sample means) is called the sampling distribution of the mean. The sampling distribution is purely theoretical derived by the laws of probability. A given score x is part of a distribution for that variable which can be used to assess probability A given mean M is part of a sampling distribution for that variable which can be used to determine the probability of a given sample being drawn

Anthony J Greene9 The Basic Concept Extreme events are unlikely -- single events For samples, the likelihood of randomly selecting an extreme sample is more unlikely The larger the sample size, the more unlikely it is to draw an extreme sample

Anthony J Greene10 The original distribution of x: 2, 4, 6, 8 Now consider all possible samples of size n = 2 What is the distribution of sample means M

Anthony J Greene11 The Sampling Distribution For n=2 Notice that it’s a normal distribution with μ = 5

Anthony J Greene12 Heights of the five starting players

Anthony J Greene13 Possible samples and sample means for samples of size two M

Anthony J Greene14 Dotplot for the sampling distribution of the mean for samples of size two (n = 2) M

Anthony J Greene15 Possible samples and sample means for samples of size four M

Anthony J Greene16 Dotplot for the sampling distribution of the mean for samples of size four (n = 4) M

Anthony J Greene17 Sample size and sampling error illustrations for the heights of the basketball players

Anthony J Greene18 Dotplots for the sampling distributions of the mean for samples of sizes one, two, three, four, and five M M M M M

Anthony J Greene19 The possible sample means cluster closer around the population mean as the sample size increases. Thus the larger the sample size, the smaller the sampling error tends to be in estimating a population mean, , by a sample mean, M. For sampling distributions, the dispersion is called Standard Error. It works much like standard deviation. Sample Size and Standard Error

Anthony J Greene20 Standard Error of M For samples of size n, the standard error of the variable x equals the standard deviation of x divided by the square root of the sample size: In other words, for each sample size, the standard error of all possible sample means equals the population standard deviation divided by the square root of the sample size.

Anthony J Greene21 The Effect of Sample Size on Standard Error The distribution of sample means for random samples of size (a) n = 1, (b) n = 4, and (c) n = 100 obtained from a normal population with µ = 80 and σ = 20. Notice that the size of the standard error decreases as the sample size increases.

Anthony J Greene22 Mean of the Variable M For samples of size n, the mean of the variable M equals the mean of the variable under consideration:  M . In other words, for each sample size, the mean of all possible sample means equals the population mean.

Anthony J Greene23 The standard error of M for sample sizes one, two, three, four, and five Standard error = dispersion of M σ M

Anthony J Greene24 The sample means for 1000 samples of four IQs. The normal curve for x is superimposed

Anthony J Greene25 Suppose a variable x of a population is normally distributed with mean  and standard deviation . Then, for samples of size n, the sampling distribution of M is also normally distributed and has mean  M =  and standard error of Sampling Distribution of the Mean for a Normally Distributed Variable

Anthony J Greene26 (a) Normal distribution for IQs (b) Sampling distribution of the mean for n = 4 (c) Sampling distribution of the mean for n = 16

Samples Versus Individual Scores

Anthony J Greene28 Frequency distribution for U.S. household size

Anthony J Greene29 Relative-frequency histogram for household size

Anthony J Greene30 Sample means n = 3, for 1000 samples of household sizes.

Anthony J Greene31 The Central Limit Theorem For a relatively large sample size, the variable M is approximately normally distributed, regardless of the distribution of the underlying variable x. The approximation becomes better and better with increasing sample size.

Anthony J Greene32 Sampling distributions for normal, J-shaped, uniform variable M MMM MMM MMM

Anthony J Greene33 APA Style: Tables The mean self-consciousness scores for participants who were working in front of a video camera and those who were not (controls).

Anthony J Greene34 APA Style: Bar Graphs The mean (±SE) score for treatment groups A and B.

Anthony J Greene35 APA Style: Line Graphs The mean (±SE) number of mistakes made for groups A and B on each trial.

Anthony J Greene36 Summary We already knew how to determine the position of an individual score in a normal distribution Now we know how to determine the position of a sample of scores within the sampling distribution By the Central Limit Theorem, all sampling distributions are normal with

Anthony J Greene37 Sample Problem 1 Given a distribution with μ = 32 and σ = 12 what is the probability of drawing a sample of size 36 where M > 48 Does it seem likely that M is just a chance difference?

Anthony J Greene38 Sample Problem 2 In a distribution with µ = 45 and σ = 45 what is the probability of drawing a sample of 25 with M >50?

Anthony J Greene39 Sample problem 3 In a distribution with µ = 90 and σ = 18, for a sample of n = 36, what sample mean M would constitute the boundary of the most extreme 5% of scores? z crit = ± 1.96 z M

Anthony J Greene40 Sample Problem 4 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? What information are we missing? n = 9

Anthony J Greene41 Sample Problem 5 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 16

Anthony J Greene42 Sample Problem 6 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 25

Anthony J Greene43 Sample Problem 7 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 36

Anthony J Greene44 Sample Problem 8 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 81

Anthony J Greene45 Sample Problem 9 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 169

Anthony J Greene46 Sample Problem 10 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 625

Anthony J Greene47 Sample Problem 10 In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 1

Anthony J Greene48

Anthony J Greene49 Sample Problem 11 In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 1 z = ±2.58

Anthony J Greene50 Sample Problem 12 In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 4 z = ±2.58

Anthony J Greene51 Sample Problem 13 In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 16 z = ±2.58

Anthony J Greene52 Sample Problem 14 In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 64 z = ±2.58

Anthony J Greene53 Sample Problem 15 In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 258 z = ±2.58

Anthony J Greene54

Anthony J Greene55 n = 1 z M

Anthony J Greene56 n = 4 z M

Anthony J Greene57 n = 16 z M

Anthony J Greene58 n = 64 z M

Anthony J Greene59 n = 258 z M

Anthony J Greene60 n = ∞