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L23posted - Using OC Curve to Determine Sample Size Example...

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Using OC Curve to Determine Sample Size Example 4.5-1, p 255 . Standard pro- duction setup. Average time μ = 30 minutes to complete task. Assume X N (30 , 1) . Change suggested. New X N ( μ, 1) where hoped μ < 30 . Sample size n workers to test H 0 : μ = 30 H 1 : μ < 30 1
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Definition . The rejection (or critical region) CR is the set of data values for which H 0 is rejected. Natural choice for this problem is the region: If c too small, then tend not to reject and have large Type II error probability. If c too large, then tend to reject and have large Type I error probability. 2
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Suppose try . CR = { ¯ x 29 . 5 } , n = 4 . OC ( μ ) = P (accept H 0 | μ ) = P ( ¯ X > 29 . 5 | μ ) = P ( ¯ X - μ 1 / n > 29 . 5 - μ 1 / 2 | μ ) = P ( Z > 29 . 5 - μ 1 / 2 ) = 1 - Φ( 29 . 5 - μ 1 / 2 ) Selected values: μ 28.5 29 29.5 30 OC ( μ ) 0.0228 0.1587 0.500 0.8413 Read off table: Type I error probability is 1-0.8413 = 0.1587. Type II error probability at μ = 29 . 5 is 0.500. Suppose take n = 16 . See graph. 3
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28 28.5 29 29.5 30 30.5 31 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Operating Characteristic Curve Section 4.5 3 OC( μ ) μ n=4 n=16 4
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Solving for n . Requirements . CR = { ¯ x c } Type I error prob- ability is α. Type II error probability at μ = 29 . 5 is β where α, β specified. Can c, n be deter- mined to satisfy these two conditions? α = P (Type I error) = P (Reject H 0 | μ = 30) = 1 - P (Accept H 0 | μ = 30)
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  • Fall '10
  • drera
  • Type I and type II errors, α, type ii error, error probability, Operating Characteristic Curve, Characteristic Curve Section

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