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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? So while the odds chance selection of a single person this far • Find the z-score of 135, compute the tail above the mean is not all that unlikely, the odds of a sample this region it to the 3rd power. far above theand meanraise are astronomical z = 2.19 P = 0.0143 0.01433 = 0.0000029 X X Anthony J Greene 1 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 µM And dispersion σM ,the standard error II The Central Limit Theorem Anthony J Greene 2 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 Greene 3 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 Greene 4 Sampling Distributions • The reason for this is to find the probability of a random sample having the properties you observe. M Anthony J Greene 5 Sampling Distributions 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: n 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 Anthony J Greene 6 Sampling Error Sampling error is the error resulting from using a sample to estimate a population characteristic. Anthony J Greene 7 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 Anthony J Greene being drawn 8 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 Greene 9 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 Greene 10 The Sampling Distribution For n=2 Notice that it’s a normal distribution with μ = 5 Anthony J Greene 11 Heights of the five starting players Anthony J Greene 12 Possible samples and sample means for samples of size two M Anthony J Greene 13 Dotplot for the sampling distribution of the mean for samples of size two (n = 2) M Anthony J Greene 14 Possible samples and sample means for samples of size four M Anthony J Greene 15 Dotplot for the sampling distribution of the mean for samples of size four (n = 4) M Anthony J Greene 16 Sample size and sampling error illustrations for the heights of the basketball players Anthony J Greene 17 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 Greene 18 Sample Size and Standard Error 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, m, by a sample mean, M. For sampling distributions, the dispersion is called Standard Error. It works much like standard deviation. Anthony J Greene 19 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: M n 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 Greene 20 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 Greene 21 Mean of the Variable M For samples of size n, the mean of the variable M equals the mean of the variable under consideration: mM m . In other words, for each sample size, the mean of all possible sample means equals the population mean. Anthony J Greene 22 The standard error of M for sample sizes one, two, three, four, and five Standard error = dispersion of M σM Anthony J Greene 23 The sample means for 1000 samples of four IQs. The normal curve for x is superimposed Anthony J Greene 24 Sampling Distribution of the Mean for a Normally Distributed Variable Suppose a variable x of a population is normally distributed with mean m and standard deviation . Then, for samples of size n, the sampling distribution of M is also normally distributed and has mean mM = m and standard error of M n Anthony J Greene 25 (a) Normal distribution for IQs (b) Sampling distribution of the mean for n = 4 (c) Sampling distribution of the mean for n = 16 Anthony J Greene 26 Samples Versus Individual Scores Distribution Distribution of Individual of Samples Scores Distribution Consists of Observed Central Tendency Theoretical Central Tendency Observed Dispersion Theoretical Dispersion M x MM M mM m sM s M Frequency distribution for U.S. household size Anthony J Greene 28 Relative-frequency histogram for household size Anthony J Greene 29 Sample means n = 3, for 1000 samples of household sizes. Anthony J Greene 30 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 Greene 31 Sampling distributions for normal, J-shaped, uniform variable M M M M M M M M Anthony J Greene M M 32 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 Greene 33 APA Style: Bar Graphs The mean (±SE) score for treatment groups A and B. Anthony J Greene 34 APA Style: Line Graphs The mean (±SE) number of mistakes made for groups A and B on each trial. Anthony J Greene 35 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 mean m and dispersion of M Anthony J Greene n 36 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 12 M 2 36 M m 48 32 z 8 M 2 Anthony J Greene Does it seem likely that M is just a chance difference? 37 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? 45 M 9 25 50 45 z 0.55 9 Anthony J Greene 38 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? zcrit = ± 1.96 M m 18 1.96 M 3 M 36 -1.96 M 1.96 * 3z 90 OR 1.96 *+1.96 3 90 M M 84.12 95.88 M 98.55 OR Anthony MJ Greene 84.12 39 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 n = 9information are we missing? M z 18 18 6 n 9 3 M m M 93 90 0.50 Anthony J Greene 6 P 0.308540 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 18 18 M 4.5 n 16 4 M m 93 90 z 0.67 M 4.5 Anthony J Greene P 0.2514 41 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 18 18 M 3.6 n 25 5 M m 93 90 z 0.83 M 3.6 Anthony J Greene P 0.2033 42 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 18 18 M 3.0 n 36 6 M m 93 90 z 1.00 M 3.0 Anthony J Greene P 0.1587 43 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 M z 18 18 2.0 n 81 9 M m M 93 90 1.50 Anthony J Greene 2.0 P 0.0668 44 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 M z 18 18 1.38 n 169 13 M m M 93 90 2.17 Anthony J Greene 1.38 P 0.0146 45 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 M z 18 18 0.72 n 625 25 M m M 93 90 4.17 Anthony J Greene 0.72 P 0.00001 46 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 M z 18 18 18.0 n 1 1 M m M 93 90 0.17 Anthony J Greene 18.0 P 0.5675 47 0.60 0.50 P-value 0.40 0.30 0.20 0.10 0.00 0 100 200 300 400 500 600 700 Sample Size Anthony J Greene 48 z = ±2.58 Sample Problem 11 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? 20 20 M 20.0 n=1 n 1 1 M m z M z * M m M M 2.58 * 20 200 M 2.58 * 20 200 M 251.6, 148.4 Anthony J Greene 49 z = ±2.58 Sample Problem 12 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? 20 20 M 10.0 n=4 z M m M n 4 2 M z * M m M 2.58 *10 200 M 2.58 *10 200 M 225.8, 174.2 Anthony J Greene 50 z = ±2.58 Sample Problem 13 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? 20 20 M 5.0 n = 16 z M m M n 16 4 M z * M m M 2.58 * 5 200 M 2.58 * 5 200 M 212.9, 187.1 Anthony J Greene 51 z = ±2.58 Sample Problem 14 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? 20 20 M 2.5 n = 64 n z M m M 64 8 M z * M m M 2.58 * 2.5 200 M 206.4, 193.6 M 2.58 * 2.5 200 Anthony J Greene 52 z = ±2.58 Sample Problem 15 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? 20 20 M 1.25 n = 258 n 256 16 z M m M M z * M m M 2.58 *1.25 200 M 203.2, 196.8 M 2.58 *1.25 200 Anthony J Greene 53 270 250 Score 230 210 190 170 150 0 50 100 150 200 250 300 Sample Size Anthony J Greene 54 n=1 150 z M -2.58 148 .4 175 200 Anthony J Greene 225 250 +2.58 251.6 55 n=4 150 175 z M -2.58 174.2 200 Anthony J Greene 225 +2.58 225.8 250 56 n = 16 150 175 200 z -2.58 +2.58 187.1 212.9 Anthony J Greene M 225 250 57 n = 64 150 175 200 z -2.58 +2.58 193.6 206.4 Anthony J Greene M 225 250 58 n = 258 150 175 200 z -2.58 +2.58 196.8 203.2 Anthony J Greene M 225 250 59 n=∞ 150 175 200 Anthony J Greene 225 250 60

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