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Assignment 8.docx - 4.15 A textile ber manufacturer is...

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4.15 A textile fiber manufacturer is investigating a new drapery yarn, which has a standard deviation of 0.3 kg. The company wishes to test the hypothesis H0:14 against using a random sample of five specimens. (a) What is the P-value if the sample average is H 0 : μ = 14 kg H 1 : μ < 14 kg σ = 0.3 kg σ / ( n ) = 0.3/ ( 5 ) = 0.134164 kg ´ x = 13.7 kg z = 13.7 14 0.134164 =− 2.236 z = -2.236 is 0.0127 (using the table) P = 0.0127 (b) Find for the case where the true mean elongation force is 13.5 kg and we assume that 0.05. Using the table 0.05 = -1.64485 ´ x 14 0.134164 = ¿ -1.64485 Solve for ´ x
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. ´ x = 13.7793 kg μ = 13.5 kg z = 13.7793 13.5 0.134164 = 2.082 1-P (Z < 2.082) = 1- 0.9813 = 0.0187 (c) What is the power of the test from part (b)? The power of a statistical test is the probability of correctly rejecting a false null hypothesis H 0 . From part (b), the probability of failing to reject a false null hypothesis H 0 : μ = 14 kg is β = 0.0187 . the power of is 1- β = 1 – 0.0187 = 0.9813 4.19 The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2. We wish to test H0:100 versus with a sample of n 9 specimens. (a) If the rejection region is defined as or find the type I error probability H 0 : μ = 100 H 1 : μ≠ 100 σ = 2
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σ / ( n ) = 2/ ( 9 ) = 0.6667 z 1 = 98.5 100 0.6667 =− 2.25 and z 2 = 101.5 100 0.6667 = 2.25 α = P ( Z
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