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Unformatted text preview: rease the spread (σ ). Again, there might be multiple solutions, we are looking for the
REASONABLE smaller σ (0 is not a reasonable choice). ”Try and Error” will be in force
for this task.
σ
0.100
0.090
0.087
0.086
↓ P(8 ≤ X ≤ 8.3)
0.819
0.854
0.864
0.868
↓ So as long as σ ≤ 0.086, AUC([8, 8.3]) > 0.865.
63. Do Exercise 20 on p.254.
SOLUTION: X ∼ N(200, 102 )
a. P(X ≤ 160) = P(Z ≤ 160−200
10 b. Yes, it is unusually small. = P(Z ≤ −4) ≈ 0. c. Yes, the only chance to observe an event which almost never happen by assumption (X ∼
N(200, 102 )) is that the assumption is inappropriate, i.e. the process is no longer functioning correctly.
d. P(X ≥ 203) = P(Z ≥ 203−200
10 = P(Z ≥ 0.3) = 1 − P(Z ≤ 0.3) = 1 − 0.6179 = 0.3821. e. No, since an event with probability 0.3821 is not regarded as an unusual event. f. No, observation (203 or more) on an event with probability of 0.3821 for happening (according to X ∼ N(200, 102 )) is reasonable, which cannot provide convincing evidence against
the assumption.
g. P(X ≤ 195) = P(Z ≤ 195−200
10 = P(Z ≤ −0.5) = 0.3085. h. No, since an event with probability 0.3085 is not regarded as an unusual event.
i. No, an event (195 or less) with probability of 0.3085 for happening (according to X ∼
N(200, 102 )) is reasonable, which cannot provide convincing evidence against the assumption....
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This document was uploaded on 02/15/2014 for the course MATH 231 at Lehigh University .
 Fall '11
 DANIELCONUS
 Statistics, Standard Deviation

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