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Extended solution of review 3rd prob (2)

# Extended solution of review 3rd prob (2) - a or a Type II...

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Extended solution of prob 8.25 8.25 (p.305) Focus: percentage of SiO2 in aluminous concrete. Desired value: 5.5. Test for true average using 16 independent samples, which are found to be normally distributed with x-bar = 5.25 and σ = 0.3. a) Does true average differ conclusively from 5.5? Set up and carry out appropriate test at α = 0.01. b) If true µ = 5.6, what is probability of detecting departure from H o ? c) What value of n would be required to satisfy given α and β = 0.01? Part a: H 0 : 5 . 5 = μ v.. H a : 5 . 5 μ ; for a level .01 test, (not specified in the problem description), reject H 0 if either 58 . 2 z or 58 . 2 - z . Since 58 . 2 33 . 3 075 . 5 . 5 25 . 5 - - = - = z , reject H 0 . Part b: Note that the probability of detecting departure from H o is the opposite of a Type II error; recall the 2x2 matrix, if true µ = 5.6, then H a
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Unformatted text preview: a or a Type II error. We can call the Type II error (5.6). Using info on p298 in Devore: P(detect departure from H o ) = 1 - (5.6) ( 29 ( 29 ( 29 -+- + -+ - -+-- -+ -=-075 . 1 . 58 . 2 075 . 1 . 58 . 2 1 / ' / ' 1 6 . 5 1 2 / 2 / n z n z o o ( 29 ( 29 105 . 91 . 3 25 . 1 1 =- + -= Note that there is therefore a very high probability of a Type II error occurring if H a is true. Part c: again using info on p298 -- ( 29 ( 29 97 . 216 1 . 33 . 2 58 . 2 3 . ' 2 2 2 / = -+ = -+ = o z z n , so use n = 217....
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