Inference on Single Mean

We will assume that the distribution remains normal

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Unformatted text preview: with a standard deviation of 300 hours.) deviation If we randomly sample 1 bulb, will we realize that the average life has decrease? What if we sample 3 bulbs? 9 bulbs? sample G. Baker, Department of Statistics G. University of South Carolina; Slide 9 University Why Averages Instead of Single Why Readings? Readings? µ = 1500 800 1300 σ = 300 µ = 2000 1800 2300 2800 Single Readings Y < 1400 would signal shift G. Baker, Department of Statistics G. University of South Carolina; Slide 10 University 10 Why Averages Instead of Single Why Readings? Readings? µ = 1500 800 1300 σ = 173 µ = 2000 1800 2300 2800 Averages of n = 3 Y < 1654 would signal shift G. Baker, Department of Statistics G. University of South Carolina; Slide 11 University 11 Why Averages Instead of Single Why Readings? Readings? µ = 1500 µ = 1500 µ = 1500 800 1300 µ = 2000 µ = 2000 µ = 2000 1800 2300 σ = 100 2800 Averages of n = 9 Y < 1800 would signal shift G. Baker, Department of Statistics G. University of South Carolina; Slide 12 University 12 What if the original distribution What is not normal? Consider the roll of a fair die: of Rolling A Fair Die Probability 0.20 0.15 0.10 0.05 0.00 1 2 3 4 5 6 # of Dots G. Baker, Department of Statistics G. University of South Carolina; Slide 13 University 13 Suppose the single measurements Suppose are not normally Distributed. Let Y = time to fail of a light bulb in constant failure rate mode constant Y is exponentially distributed is with λ = 0.0005 = 1/2000 with 0.0005 0 1000 2000 3000 4000 5000 6000 G. Baker, Department of Statistics G. 8000 University of South Carolina; Slide 14 University 14 7000 Single measurements Averages of 2 measurements Averages of 4 measurements Averages of 25 measurements Source: Lawrence L. Lapin, Statistics in Modern Business Decisions, 6th ed., 1993, Dryden Press, Ft. Worth, Texas. G. Baker, Department of Statistics G. University of South Carolina; Slide 15 University 15 n=1 As n increases, what happens to the variance? n=2 n=4 A.Variance increases. B.Variance decreases. C.Variance remains the same. n=25 G. Baker, Department of Statistics G. University of South Carolina; Slide 16 University 16 n=1 n=2 n=4 n = 25 G. Baker, Department of Statistics G. University of South Carolina; Slide 17 University 17 Central Limit Theorem If n iis sufficiently large, the sample If s means of random samples from a population with mean µ and standard and deviation σ are approximately normally are distributed with mean µ and standard and deviation σ / n . deviation G. Baker, Department of Statistics G. University of South Carolina; Slide 18 University 18 Random Behavior of Means Summary If Y is distributed n(µ, σ), then yn n( ), is distributed n(µ, σ / n ). n( If Y is distributed non-n(µ, σ), then y x≥30 If ), is distributed approximately n(µ, σ / n ). n( ). G. Baker, Department of Statistics G. University of South Carolina; Slide 19 University 19 If We Can Consider y to be Normal … If Recall: If Y is distributed normally with mean µ and standard deviation σ, then and Y −µ Z= σ So if y is distributed normally with mean µ and standard deviation σ / n , then then Y −µ Z= σ/ n G. Baker, Department of Statistics G. U...
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This note was uploaded on 01/12/2014 for the course STA 509 taught by Professor Wang during the Fall '13 term at South Carolina.

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