12 Sampling Distributions Part 1 (1)

Sampling distribution should become tighter around

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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? 26
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Comparing our “sampling distribution” to theoretical S a m p le s ize : n = 1 0 Av e ra g e o f x ̅ 27 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 ̅ ̅ 28 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 2 5000 35000 4 5 000 5 5000 65000 750 0 0 8 5000 95 000 1050 00 1 150 00 1 250 0 0 1350 00 1 450 0 0 1 55000 1 65000 1 75000 1 8 5 0 0 0 195 0 00 2 0 5 000 2 150 0 0 2 25000 2 350 0 0 2 450 00 2 550 0 0 2 65000 2 7 5 0 00 2 850 0 0 29 5 0 00 3 0500 0 315 000 3250 0 0 335 0 00 345 00 0 Mo re Annual salary, $ Relative frequency, % Population n=10 n=51
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Comparing our “sampling distribution” to theoretical In our example of 25 samples of n=51 Mean of x̅’s = $111,234 μ = $111,208 Pretty close! Standard deviation What is the theoretical standard error of sample means? 2 9 86 . 594 , 6 $ 15 . 652 , 6 $ 51 15 . 506 , 47 ) 25 ( = = = = means sample X x s n σ σ
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Salary example: main points Sampling error: none of the sample means was exactly the population mean Larger standard error (standard deviation) of sample means for smaller samples (n=10), than for larger ones (n=51). Larger sample size: distribution of sample means becomes tighter around true mean. We know the theoretical mean and standard error for sampling distribution, regardless of how many samples we take: With just 25 samples, and n=51, we came close! This is all an illustration. In practice we have only one sample But with larger sample size, get a bonus benefit  sampling distribution normal. 3 0 X x μ μ = n X x σ σ =
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Central Limit Theorem In our salary example, even though population distribution not “normal,” distribution of sample means looks close for n = 51 Central Limit Theorem As long as sample size is large enough : Sampling distribution (distribution of sample means) follows Normal distribution: x̅~ N( μ X, σ/√n) How large is large enough? True regardless of sample size if population distribution is normal If not, n ≥ 30 usually works May need larger sample size if distribution is highly skewed Sampling without replacement ok if sample size < 5% of population If population distribution is normal; sampling distribution normal regardless of sample size 3 1
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Central Limit Theorem Population distribution n=2 n=5 n ≥ 30 n=2 n=5 n ≥ 30 Population distribution 32
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Effect of Sample Size on Sampling Distribution 33 Central Limit Theorem 0 1 2 0 2.5 5 7.5 10 X Density Population n=5 n=25 n=100
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Example: Family Sedan Purchase Price Mini Dooper, a car manufacturer specialized on manufacturing small and cute cars, considers entering into the family sedan business. In order to competitively price their product, they need to estimate the parameters of family sedan prices in the market. You have a sample of customers who
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