信度理论——完全信度与部分信度
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(a)
Z60 = 0.528 Z100 =
,
M = 200. 10 Z60 = 0.682, 6
7
0.682 × 187.5 + (1 − 0.682) × 200 = 191.5. (b) Z60 = 0.467 Z100 = , M = 197.6.
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. (r, p), r > 0, 0 < p < 1, ¯ ≤ (1 + r)µ (1 − r)µ ≤ X p, µ . ¯ ≤ (1 + r)µ) ≥ p, P ((1 − r)µ ≤ X ¯ X (r, p)
7
µ
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(r, p), r > 0, 0 < p < 1, ¯ ≤ Zµ + rµ) = p P (Zµ − rµ ≤ Z X ¯ Z X (r, p)
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.
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n < nF . ¯ ≤ Zµ + rµ) P (Zµ − rµ ≤ Z X ¯ −µ −rµ X rµ =P ≤ ≤ ¯) ¯ ) Z var (X ¯) Z var (X var (X ¯ −µ X rµ =P = p, ≤ ¯ ¯ var (X ) Z var (X ) ¯ −µ X , yp = ¯) var (X 1+p Φ −1 2 √ rµ rµ n = yp = , ¯) Zσ Z var (X √ r nµ Z= = yp σ n . nF
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7.2.3 . N
X1, X2, · · · , Xm, · · · , λ
N
N .
7
S=
i=1
Xi,
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Xi yp = inf y : P
y
2 µ0, σ0 .
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S − E (S ) var (S ) σ0 µ0
r = 0.09 334.1 474.3 819.2 1336.3 r = 0.04 1691.3 2,401.0 4147.4 6765.1
7
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√ rµ n σ
≥ yp :
7
(1) √ r n σ ≤ = µ yp (2) :
7
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n ≥ nF
Z = 1, n , 1}, nF
,
7
Z = min{ nF
.
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: n , λ0
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n/λ0; ¯ X
¯) var (X
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σ2 µ2 ¯ var (X ) = ≤ . n λ0 ;
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(3) σ n ≥ λ0 ( )2 . µ σ 2 , λ0 µ . , σ n F = λ0 ( )2 . µ
7
100
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400-800 24 2400-2800 3
800-1200 32 2800-3200 1
1200-1600 21 3200-3600 1
1600-2000 10 3600+ 0o Back
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(r, p)
r = 0.1 270.6 384.2 663.6 1082.4 r = 0.05 1082.4 1536.6 2654.3 4329.6
λ0
r = 0.08 422.8 600.3 1036.8 1691.3 r = 0.03 3006.7 4268.4 7373.1 12026.8 r = 0.07 552.3 784.0 1354.2 2209.0 r = 0.02 6765.1 9604.0 16589.4 27060.3
≤y
≥p .
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: yp λ ≥ ( )2 1 + r , E (S ) . S=
N i=1 Xi
2
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(r, p)
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: (1) ,S :
7
P ((1 − r)E (S ) ≤ S ≤ (1 + r)E (S )) =P =P =P √
yp
yp 1.645 1.96 2.576 3.29
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7.2
p = 0.9 p = 0.95 p = 0.99 p = 0.999 p = 0.9 p = 0.95 p = 0.99 p = 0.999 r = 0.2 67.7 96.0 165.9 270.6 r = 0.06 751.7 1067.1 1843.3 3006.7
2 λ(σ0 y
+
µ2 0)
σ0 µ0
≥ p,
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≥ yp , 7.2.1 ,
λ ≥ ( rp )2 1 + n = 1,
.
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(2)
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7.3.
n < nF , ¯ X µ, , M . 0 < Z < 1, ,
, ¯ + (1 − Z )M u ˆ = ZX Z .
7.
7
7.1.
: (1) X2, · · · , Xn. ? (2) Xj . , j X1, X2, · · · , Xn ? n X1
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, X1, X2, · · · , Xn X1, X2, · · · , Xn µ. µ . µ.
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¯ ,X ¯ − µ| ≤ rµ) = P p ≤ P (|X √
(r, p) √ ¯ −µ X rµ n √ ≤ . σ σ/ n
7
rµ n ≥ yp . σ . ,
0
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=
yp r
2
.
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. , , ,
7
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¯ = X1 + · · · + Xn . X n µ, ¯ µ ˆ = (1 − Z )M + Z X, 0≤Z≤1 Z . Z=1 ¯, X 0<Z<1 ¯ ¯ X M, X , , 1, ¯ X . . , . ,M .
λrµ0 2 +µ2 ) λ(σ0 0
−rE (S )
var (S ) S − E (S ) var (S ) S − E (S ) var (S )
≤
S − E (S ) var (S ) rE (S )
≤
rE (S ) var (S )
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≤ ≤
var (S ) λrµ0
19305 340575 .
7
Z=
n = nF ,
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340575 × Z + 366833 × (1 − Z ) = 354662 ( 354662 19305
).
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= 18.37
.
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7.3.2 . (r, p) = (0.05, 90%). X 60 = 180.0 , 189.47 ; 20 X 80 = 185.0 , 190.88 ; 20 X 100 = 187.5 . , 100 ( : , n = 60 , , . ).
: n
7
n (r, p)
X
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nF X
, (r, p) .
n ≥ nF
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7.2.1 , . ,
2 . σ0 E (X1). n, µ0
X1, X2, · · · , Xn λ > 0, ¯ X (r, p) .
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7.3.1 , .
7.2.2 , 366833 ,
