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Discussion12%20Solutions - 9-10(a The critical values are...

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IE 300/GE 331 Discussion Problem Set 12 9-10 (a) The critical values are determined by 𝜇𝜇 0 ± 𝑧𝑧 𝛼𝛼 /2 𝜎𝜎 √𝑛𝑛 . It follows that 𝑧𝑧 𝛼𝛼 /2 2 9 = 1.5, 𝑧𝑧 𝛼𝛼 /2 = 2.25, 𝛼𝛼 = 2 Φ ( 2.25) = 0.0244 (b) 𝛽𝛽 = 𝑃𝑃 (98.5 ≤ 𝑋𝑋 101.5) = 𝑃𝑃 � 98.5 103 2 9 ≤ 𝑍𝑍 ≤ 101.5 103 2 9 = 𝑃𝑃 ( 6.75 ≤ 𝑍𝑍 ≤ − 2.25) = Φ ( 2.25) Φ ( 6.75) = 0.0122 (c) 𝛽𝛽 = 𝑃𝑃 � 98.5 105 2 9 ≤ 𝑍𝑍 ≤ 101.5 105 2 9 = Φ ( 5.25) Φ ( 9.75) = 7.6 𝐸𝐸 − 8 Since the true mean (105) is further away from the acceptance region, the probability of failing to reject the false null hypothesis is smaller (in discussion sections, draw a picture to show this). 9-12 (a) We reject the null hypothesis when 𝑥𝑥̅ is outside of the interval [ 𝜇𝜇 0 − 𝑧𝑧 𝛼𝛼 2 𝜎𝜎 √𝑛𝑛 , 𝜇𝜇 0 + 𝑧𝑧 𝛼𝛼 2 𝜎𝜎 √𝑛𝑛 ] .
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