12 Sampling Distributions Part 1 (1)

Sample and a parameter computed from a population

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sample and a parameter computed from a population Sampling error = statistic – parameter Note: In the textbook, π is used for p p is used for For the mean For the proportion 19 μ - = x error Sampling p p - = ˆ error Sampling ˆ p
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Sample no. x ̅ 1 85,381 2 112,811 3 85,978 4 110,937 5 99,735 6 108,442 7 114,943 8 134,668 9 127,726 10 111,002 11 86,158 12 99,614 13 103,733 14 102,513 15 97,659 16 115,060 17 81,178 18 105,959 19 111,047 20 133,037 21 110,792 22 103,816 23 112,513 24 101,395 25 114,193 Average 106,812 What if we draw more samples of n=10? 20
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Comparing our “sampling distribution” to theoretical 21 Professional classification salaries ($) 2006 Population data, N=2597 0 0.02 0.04 0.06 0.08 0.1 0.12 25000 35000 45000 55000 65000 75000 85000 95000 105000 115000 125000 135000 145000 155000 165000 175000 185000 195000 205000 215000 225000 235000 245000 255000 265000 275000 285000 295000 305000 315000 325000 335000 345000 More Annual salary, $ Relative frequency, %
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Comparing our “sampling distribution” to theoretical ̅ ̅ 22 Distributions of Population and Sample Means, Professional classification salaries ($) 2006 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 25000 35000 45000 55000 65000 75000 85000 95000 105000 115000 125000 135000 145000 155000 165000 175000 185000 195000 205000 215000 225000 235000 245000 255000 265000 275000 285000 295000 305000 315000 325000 335000 345000 More Annual salary, $ Relative frequency, % Population n=10
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Comparing our “sampling distribution” to theoretical In our example of several* samples of n=10 x̅ = $106,812 μ = $111,208 Standard deviation What is the theoretical standard error of sample means? 2 3 49 . 630 , 13 $ 76 . 022 , 15 $ 10 15 . 506 , 47 ) 25 ( = = = = means sample X x s n σ σ
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S a m p le n o . n= 5 1 x ̅ n = 1 0 x ̅ 1 1 2 5 ,6 7 1 1 1 2 ,8 1 1 2 1 0 5 ,9 9 7 8 5 ,3 8 1 3 1 0 7 ,0 7 2 8 5 ,9 7 8 4 1 1 7 ,9 0 3 1 1 0 ,9 3 7 5 1 1 3 ,5 8 1 9 9 ,7 3 5 6 1 0 5 ,8 3 8 1 0 8 ,4 4 2 7 1 1 2 ,6 5 6 1 1 4 ,9 4 3 8 1 1 9 ,0 1 4 1 1 1 ,0 0 2 9 1 1 4 ,8 6 1 8 6 ,1 5 8 1 0 1 0 0 ,6 9 1 1 3 4 ,6 6 8 1 1 1 0 1 ,1 0 5 9 9 ,6 1 4 1 2 1 1 5 ,9 7 2 1 0 3 ,7 3 3 1 3 1 1 0 ,0 7 1 1 0 2 ,5 1 3 1 4 1 0 7 ,2 0 9 1 2 7 ,7 2 6 1 5 1 0 7 ,9 9 6 9 7 ,6 5 9 1 6 1 1 4 ,2 8 2 1 1 5 ,0 6 0 1 7 1 0 6 ,3 8 2 8 1 ,1 7 8 1 8 1 1 2 ,6 0 3 1 0 5 ,9 5 9 1 9 1 1 3 ,0 7 5 1 1 1 ,0 4 7 2 0 1 2 2 ,4 2 8 1 3 3 ,0 3 7 2 1 1 1 1 ,2 0 2 1 1 0 ,7 9 2 2 2 1 0 5 ,5 3 7 1 0 3 ,8 1 6 2 3 1 0 6 ,2 1 7 1 1 2 ,5 1 3 2 4 1 0 3 ,0 7 9 1 0 1 ,3 9 5 2 5 1 2 0 ,4 0 3 1 1 4 ,1 9 3 Av e ra g e 1 1 1 ,2 3 4 1 0 6 ,8 1 2 Standard error = σ x ̅ = σ / √n What do you think will happen? Sampling distribution should become ______ around the true mean of X. What if we increase sample size? As n increases, standard error of sampling distribution: Mean of sampling distribution? 24
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Sample no. n=51 x ̅ n=10 x ̅ 1 125,671 112,811 2 105,997 85,381 3 107,072 85,978 4 117,903 110,937 5 113,581 99,735 6 105,838 108,442 7 112,656 114,943 8 119,014 111,002 9 114,861 86,158 10 100,691 134,668 11 101,105 99,614 12 115,972 103,733 13 110,071 102,513 14 107,209 127,726 15 107,996 97,659 16 114,282 115,060 17 106,382 81,178 18 112,603 105,959 19 113,075 111,047 20 122,428 133,037 21 111,202 110,792 22 105,537 103,816 23 106,217 112,513 24 103,079 101,395 25 120,403 114,193 Average 111,234 106,812 Standard error = σ x ̅ = σ / √n What do you think will happen? Sampling distribution should become tighter around the true mean of X. What if we increase sample size? As n increases, standard error of sampling distribution: Mean of sampling distribution? 25
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Sample no. n=51 x ̅ n=10 x ̅ 1 125,671 112,811 2 105,997 85,381 3 107,072 85,978 4 117,903 110,937 5 113,581 99,735 6 105,838 108,442 7 112,656 114,943 8 119,014 111,002 9 114,861 86,158 10 100,691 134,668 11 101,105 99,614 12 115,972 103,733 13 110,071 102,513 14 107,209 127,726 15 107,996 97,659 16 114,282 115,060 17 106,382 81,178 18 112,603 105,959 19 113,075 111,047 20 122,428 133,037 21 111,202 110,792 22 105,537 103,816 23 106,217 112,513 24 103,079 101,395 25 120,403 114,193 Average 111,234 106,812 Standard error = σ x ̅ = σ / √n What do you think will happen?
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