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. . . . ... ... ... ...

2014 5555

, s2, ( 2 ). (pointestimation) (interval estimation) .

x

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, . - , .... . , .

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(point estimation), , . , , , (intervalestimation), , , , .

, , . , 25 25 ( 1

).

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X (..) FX (x; ) F(x; ) . {x1, . . . , xn} X n. g(x1, . . . , xn) .

(estimator) = g(x1, . . . , xn).

{x1, . . . , xn} n, {x1, . . . , xn} .. {X1, . . . , Xn}, F(x; ). .. {x1, . . . , xn} ( .. {X1, . . . , Xn}) ( ..). ( {x1, . . . , xn}) , ........ = E ( ) 2 = Var( ).

.. X 2.

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2

2

; :

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(unbiased) () ,

, .

.. X .

s2 2

2

x

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, . (consistent) ( )

. (effective) ,

(adequate) .

, s2 2, .

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.. X , F(x; ) f (x; ), X X . {x1, . . . , xn}. .

, , , , , .

, F(x; ) 2. , 2 .

[a, b], a b 2

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( 1, 2) 2, :

1. s2 2 ,

2. s2

( 1, 2) 2.

(method ofmoments). : 2 .

. . . .

1 25 . 1. s2

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5.67 2 0.375.

.. X ( ) N(, 2), X .

, , , .

X = xi

f (xi ; ) ( X P(X = xi)). {x1, . . . , xn} , n (likelihood function)

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, {x1, . . . , xn} . L(x1, . . . , xn; 1) > L(x1, . . . , xn; 2) 1 2 , 1 2 {x1, . . . , xn}. , L(x1, . . . , xn; ) ( ) log L(x1, . . . , xn; ). (maximum likelihoodestimator)

2 .. ,

( n)

.

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. .. n. , . . ..... . . . . . . . [ [ [ [1111, , , , 2] 2] 2] 2] .... , .

.. X , , = E( ) = . , , n . n . 2

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( )

(standarderror)

.. =

, .

, X .

. . . . 2 .. X .

:

1. .. X N(, 2) (n > 30)

.. N(, 2/n).

2. , .. X

.

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, .. X .. X1, . . . , Xn .

N(, 2/n), .. z

[z/2, z1/2], z 1 1 1 1 ,

, z/2 z1/2, . /2 1/2 z/2 z/2

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(z) . z < z/2 z > z1/2 . % . z [z/2, z1/2] 1.

0 z/2 = z1/2.

[z/2, z1/2] .

, 1 .. z z1/2 z1/2 = 1(1/2) .

= 0.05 [1.96, 1.96] .. z 0.95, z0.975 =

1(0.975) = 1.96. [ , , . z1/2 z/2 z/2 z/2].

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[z/2, z1/2] 1 .

1 (confidence level) (confidence interval) 1.

:

1 1 1 1 () () () () ....

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:

. . . . 95% 1. 2=0.38.

, .. , ( ).

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Q1 Q3.

.

.

X N(, 0.38) N(, 0.38/25), n = 25 .

:

1. 1 = 0.95 ( = 0.05).

2. z0.975= 1(0.975) = 1.96.

3. ( = 5.67)

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95% [5.43, 5.91].

= 5.67 95% .

2 .. X s2. (n > 30) s2 2

s2 . .

n X , .. t

student t n1

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. n t z . , 2 s2 ttttn1,1/2n1,1/2n1,1/2n1,1/2 z1/2 student. [ t 1/2 n1. student n1 =24 1 = 0.95.] 1

( ). .. z1/2. tn1,1/2 student. s2 = 0.375

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student, 1 /2 = 0.975 n 1 = 24,

t24, 0.975 = 2.064

n student .

.. X . [ X , .. .] ( ) .

= X .. X = X .

. Wilcoxon . ( ).

x~

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. :

2 .. X [ , ..].

n [ n , , ].

1 [ , ]. 95%.

.

.. X n .

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.. .. . , n n n n .. .. n. , .. X , .. . ..

n .. w

tn1,1/2 n . n n tn1,1/2 z1/2. n

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. . . . 95% .. student ( ) [5.42, 5.92]. .. w = 5.92 5.42 = 0.50. ( w = 0.25) 25

2

2 s2 2

X2

n 1

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X2 . X2 .

X2 5, 24 50

1 X2

X2 : 2 n1,/2 P(

2

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X2 n1=24 1 = 0.95.

2 (1 )% 2

X2 24

1 = 0.95.

. . . . (.1) 2 . ( ) s2=0.375. n1=24 = 0.05 X2 222224, 0.02524, 0.02524, 0.02524, 0.025= 12.4 2222 24, 0.97524, 0.97524, 0.97524, 0.975 = 39.4 ( ). 95% 2

95% ..

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s2 = 0.375 . 2

s2, X2 student. ( ) X2 . .

p

.. , ( 1) ( 0). p. [ p Bernoulli]. . p n

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p

m n . n

, p :

p

.. p

n:

0.25. w ..

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. . . . n = 100 m = 12 .

p

95%

z1/2 = z0.975 = 1.96)

95% [0.056, 0.184] , .

p 95% w = 0.05 ( 5 ). n ( z1/2 = z0.975 = 1.96 w = 0.05)

. 95% .. 1/3 15