# ch18 - STAT 2023 - Holbrook Sampling Distribution Models...

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Unformatted text preview: STAT 2023 - Holbrook Sampling Distribution Models Chapter 18 18 - 1 STAT 2023 - Holbrook Inferential Statistics 18 - 2 Statistical Methods STAT 2023 - Holbrook Statistical Methods Descriptive Statistics 18 - 3 Inferential Statistics Inferential Statistics STAT 2023 - Holbrook 1. Involves: 2. Estimation Hypothesis Hypothesis Testing Testing Purpose 18 - 4 Make Decisions Make about Population Characteristics Characteristics Population? Inference Process STAT 2023 - Holbrook 18 - 5 Inference Process STAT 2023 - Holbrook Population 18 - 6 Inference Process STAT 2023 - Holbrook Population Sample 18 - 7 Inference Process STAT 2023 - Holbrook Population Sample Sample statistic (Y) (Y) 18 - 8 Sample Inference Process STAT 2023 - Holbrook Estimates Estimates & tests tests Sample Sample statistic (Y) (Y) 18 - 9 Population Sample Estimators STAT 2023 - Holbrook 1. Random Variables Used to Estimate a Random Population Parameter Population Sample Mean, Sample Proportion, Sample Median 2. Example: Sample Mean Y Is an Estimator of Example: Population Mean µ If Y = 3 then 3 Is the Estimate of µ If Estimate 3. What allows us to say that Y is a good 3. What estimate of µ? Sampling Distributions 18 - 10 10 STAT 2023 - Holbrook Sampling Distributions 18 - 11 11 Sampling Distribution STAT 2023 - Holbrook 1. Theoretical Probability Distribution 2. Random Variable is Sample Statistic Random Sample Sample Mean, Sample Proportion etc. 3. Results from Drawing All Possible Results All Samples of a Fixed Size Fixed 4. List of All Possible [Y, P(Y) ] Pairs 18 - 12 12 Sampling Distribution of Mean STAT 2023 - Holbrook Note: 1. Your book develops sampling distributions for sample proportions. (pages 458­465) (pages 336­343, first edition) 2. We will develop the sampling distribution of the sample mean. 18 - 13 13 STAT 2023 - Holbrook Developing Sampling Distributions Suppose There’s a Suppose Population ... Population Size, N = 4 Random Variable, Y, Random Is # Errors in Work Is Values of Y: 1, 2, 3, 4 Values Uniform Distribution © 1984-1994 T/Maker Co. 18 - 14 14 Population Characteristics STAT 2023 - Holbrook Summary Measures Population Distribution N µ= ∑Yi i =1 N = 2 .5 N σ= ∑(Yi − µ ) i =1 18 - 15 15 .3 .2 .1 .0 N 1 2 = 1.12 2 3 4 STAT 2023 - Holbrook All Possible Samples of Size n = 2 16 Samples 1st 2nd Observation Obs 1 2 3 4 1 1,1 1,2 1,3 1,4 2 2,1 2,2 2,3 2,4 3 3,1 3,2 3,3 3,4 4 4,1 4,2 4,3 4,4 Sample with replacement 18 - 16 16 STAT 2023 - Holbrook All Possible Samples of Size n = 2 16 Samples 16 Sample Means 1st 2nd Observation Obs 1 2 3 4 1st 2nd Observation Obs 1 2 3 4 1 1,1 1,2 1,3 1,4 1 1.0 1.5 2.0 2.5 2 2,1 2,2 2,3 2,4 2 1.5 2.0 2.5 3.0 3 3,1 3,2 3,3 3,4 3 2.0 2.5 3.0 3.5 4 4,1 4,2 4,3 4,4 4 2.5 3.0 3.5 4.0 Sample with replacement 18 - 17 17 STAT 2023 - Holbrook Sampling Distribution of All Sample Means 16 Sample Means Sampling Sampling Distribution Distribution 1st 2nd Observation Obs 1 2 3 4 1 1.0 1.5 2.0 2.5 2 1.5 2.0 2.5 3.0 3 2.0 2.5 3.0 3.5 4 2.5 3.0 3.5 4.0 18 - 18 18 P( X) P( y) .3 .2 .1 .0 X 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y Summary Measures of All Sample Means STAT 2023 - Holbrook N ∑Yi i =1 1.0 + 1.5 + + 4.0 µy = = = 2.5 N 16 ∑ (Y N σy = i =1 i − µy ) 2 N (1.0 − 2.5) + (1.5 − 2.5) 2 = 18 - 19 19 16 2 + + ( 4.0 − 2.5) 2 = 0.79 Comparison STAT 2023 - Holbrook Sampling Distribution Population PP(y) ( X) P(Y) P(X) .3 .2 .1 .0 1 2 3 4Y .3 .2 .1 .0 X 1 1.5 2 2.5 3 3.5 µ = 2.5 σ = 1.12 18 - 20 20 µy = 2.5 σ y = 0.79 4 y Standard Error of Mean σy STAT 2023 - Holbrook 1. Standard Deviation of All Possible Standard Sample Means Sample Measures Scatter in All Sample Means 2. Less Than Population Standard Less Deviation Deviation 3. Formula 18 - 21 21 σy = σ n STAT 2023 - Holbrook Properties of Sampling Distribution of Mean (optional) 18 - 22 22 STAT 2023 - Holbrook Properties of Sampling Distribution of Mean 1. Unbiasedness Mean of Sampling Distribution Equals Population Mean Mean Mean 2. Efficiency Sample Mean Comes Closer to Population Mean Sample Than Any Other Unbiased Estimator Than 3. Consistency 18 - 23 23 As Sample Size Increases, Variation of Sample As Mean from Population Mean Decreases Mean STAT 2023 - Holbrook Sampling from Normal Populations 18 - 24 24 Sampling from Normal Populations STAT 2023 - Holbrook Central Tendency Population Distribution σ = 10 µy = µ Dispersion σy = 18 - 25 25 σ n µ = 50 XY Sampling Distribution n=4 σ y = 5 n =16 σ y = 2.5 µ X- = 50 µ y = 50 XY STAT 2023 - Holbrook Sampling from Non­Normal Populations 18 - 26 26 Sampling from Non­Normal Populations STAT 2023 - Holbrook Central Tendency Population Distribution σ = 10 µy = µ Dispersion σy = 18 - 27 27 σ n µ = 50 XY Sampling Distribution n=4 σ y = 5 n =30 σ y = 1.8 µ X- = 50 µ y = 50 XY STAT 2023 - Holbrook Central Limit Theorem 18 - 28 28 Central Limit Theorem STAT 2023 - Holbrook 18 - 29 29 Central Limit Theorem STAT 2023 - Holbrook As As sample size gets large enough (n ≥ 30) ... 30) X Y 18 - 30 30 Central Limit Theorem STAT 2023 - Holbrook As As sample size gets large enough (n ≥ 30) ... 30) sampling sampling distribution becomes almost normal. normal. X Y 18 - 31 31 Central Limit Theorem STAT 2023 - Holbrook As As sample size gets large enough (n ≥ 30) ... 30) 18 - 32 32 σ σy = n sampling sampling distribution becomes almost normal. normal. µy = µ X Y Central Limit Theorem STAT 2023 - Holbrook 1. The Central Limit Theorem also The allows to answer the following: allows • What is the probability that the sample What mean g.p.a. of our class is between 2.5 and 3.5? and P(2.5 < Y < 3.5) 18 - 33 33 STAT 2023 - Holbrook Standardizing Sampling Distribution of Mean Z= Sampling Distribution Y − µy σy Y −µ = σ Standardized n Normal Distribution σ X σy σ =1 µµ X y 18 - 34 34 Xy µ =0 Z Thinking Challenge STAT 2023 - Holbrook You’re an operations You’re analyst for AT&T. Longanalyst distance telephone calls distance are normally distribution with µ = 8 min. & σ = 2 min. If you select random samples of 25 calls, what percentage of the sample sample means would be between 7.8 & 8.2 minutes? 7.8 minutes? © 1984-1994 T/Maker Co. 18 - 35 35 STAT 2023 - Holbrook Sampling Distribution Solution* Y − µ 7.8 − 8 Z= = = − .50 σ n 2 25 Y − µ 8.2 − 8 Z= = = .50 Standardized Sampling σ n 2 25 Distribution Normal Distribution σ σ Xy = .4 7.8 18 - 36 36 σ =1 8 8.2 Xy -.50 0 .50 Z For P(Z < ­0.50) (from Z­Table) STAT 2023 - Holbrook Y − µ 7.8 − 8 Z= = = − .50 σ n 2 25 Y − µ 8.2 − 8 Z= = = .50 Standardized Sampling σ n 2 25 Distribution Normal Distribution σ σ Xy = .4 σ =1 .3085 7.8 18 - 37 37 8 8.2 Xy -.50 0 .50 Z For P(Z < 0.50) (from Z­Table) STAT 2023 - Holbrook Y − µ 7.8 − 8 Z= = = − .50 σ n 2 25 Y − µ 8.2 − 8 Z= = = .50 Standardized Sampling σ n 2 25 Distribution Normal Distribution σ σ Xy = .4 σ =1 .6915 7.8 18 - 38 38 8 8.2 Xy -.50 0 .50 Z STAT 2023 - Holbrook For Area Between P(–0.50 < Z < 0.50) Y − µ 7.8 − 8 Z= = = − .50 σ n 2 25 Y − µ 8.2 − 8 Z= = = .50 Standardized Sampling σ n 2 25 Distribution Normal Distribution σ σ Xy = .4 σ =1 (.6915 - .3085) = .3830 7.8 18 - 39 39 8 8.2 Xy -.50 0 .50 Z End of Chapter Any blank slides that follow are blank intentionally. ...
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## This note was uploaded on 12/07/2011 for the course STA 2023 taught by Professor Staff during the Fall '11 term at Santa Fe College.

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