12 Sampling Distributions Part 1 (1)

# Px12 e 1 03679 memoryless so same ? 2 arrivalshr 2 1

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P(X>1/2) = e-1 = 0.3679 Memoryless, so same λ =2 arrivals/hr = ) 2 1 | 1 ( X X P

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9 Where we’re going Populations and samples Sampling distributions Sampling error Mean and standard deviation Central Limit Theorem! Estimation: point v. interval Confidence Intervals Means Proportions Sample size Goal: Inference - using sample data to learn something about population!
Population and sample: Parameter vs statistic Parameter: A measure computed from the entire population. Statistic: A measure computed from the sample. Population: The set of all objects or individuals of interest or the measurements obtained from all members Sample: A subset of the population (observed data) Use statistic to make inferences about parameter! μ (population mean) p (population proportion) σ (population standard deviation) 10 deviation) standard (sample ) proportion (sample ˆ mean) (sample s p x

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Parameters and statistics x : # of successes in population x : # of successes in the sample Mean Variance Proportion Population, size N Sample, size n Note: In the textbook, π is used for p and p is used for 11 N x N i i = = 1 μ ( 29 N x N i i = - = 1 2 2 μ σ n x x n i i = = 1 = - - = n i i x x n s 1 2 2 ) ( 1 1 N x p = n x p = ˆ ˆ p
Properties of Statistics Unbiased A sample statistic is unbiased if the expected value (mean) of the sample statistic is equal to the population parameter Consistent A sample statistic is consistent if the variance of the sample statistic goes to 0 as the sample size becomes large 12

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Sampling Distribution Sampling Distribution: Distribution of possible values for the sample statistic over all possible samples Random sampling is a very important assumption Samples are independent of one another Example: Salaries of professional employees 13
Mean of all possible x̅’s (sampling distribution) is population mean Standard deviation (called standard error) of all possible x̅’s (sampling distribution) is: Sampling Distribution 14 X x μ μ = n X x σ σ =

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Sampling distribution illustration: Salaries Population All “professional classification” salaried employees in a large organization N = 2,597 μ = \$111,208 σ =\$45,506.15 What if we didn’t have the population data and wanted to estimate the mean salary of professional employees? We would have used sample, rather than population data Consider two cases 1. Random samples of n=10 2. Random samples of n=51 15
16 Professional classification salaries (\$) 2006 Population data, N=2597 0 0.02 0.04 0.06 0.08 0.1 0.12 25000 35000 45 000 55 00 0 65 00 0 75 00 0 8 500 0 95 00 0 105 000 115 000 125 000 13 5000 14500 0 155000 16 50 00 1750 00 18 5000 1 95000 2050 00 2 15000 225000 235000 245000 255000 265000 275 0 00 285000 295 000 30 500 0 315 00 0 3250 00 33 50 00 34 500 0 Mo r e Annual salary, \$ Relative frequency, %

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Draw a random sample from the population Draw a random sample of n=10 cases from population Sample: X̅ = \$85,381 Population: N=2,597 μ =\$111,208 17
Sampling error ̅ 18 Professional classification salaries (\$) 2006 Population data, N=2597 0 0.02 0.04 0.06 0.08 0.1 0.12 2500 0 35000 4500 0 5 500 0 6 5000 75000 85000 95000 10 5000 11 5000 12 50 00 13 50 00 1450 00 1 55000 165000 175000 1 85000 195000 205 000 215000 225 000 23500 0 245 000 25500 0 26 500 0 27 500 0 28500 0 29 5 0 00 3 05000 3150 0 0 3250 0 0 3 35000 34 5000 M ore Annual salary, \$ Relative frequency, %

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Sampling error Sampling Error: The difference between a statistic computed from a
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