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  -2024
 5 
   
(    -V)
:    :100
:          ,           
           
-
1.      :
(i)      
(ii)   
(iii)   1    II  
2. (a)      (MLE)     ?     
 (OLS)       ?      
(b)                  
  
-
3. x² ( - )     ?    
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4.    -          
-
5. (a) 500     , 280            
             ?
(b) Z       
6.     ,       :

 
  
    
 
1
10
15
90
2
12
14
108
5%           
-
7.                
                
(   )            :
  1
  2
  3
  4
20
19
21
15
23
15
19
17
18
17
20
16
17
20
17
18
Easy2Siksha.com
16
16
               
8.    ?          
 ANOVA       
  -2024
 5 
   
(    -V)
:    :100
:          ,           
           
-
1.      :
(i)      
(ii)   
(iii)   1    II  
:         :    
                
   ,     ,          
;  ,          
              " " ,
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      ,          , 
        : -  -
,  ,   I   II   ,      
  
(i) -  - :       
                   
                  
    ,            ,
       ,           ,
     -   
-        (   )    
   ,    ""        
       ,      (   
     )       
,                
 ;              ,  
 ,             
-   -  ,        ()
              ,   
 
 :
- :   (  )    
-   :  (  )   
        : -       ,  
        ;  -  -      ,  
        
(ii)  : 
,                ,  
   ,         , "!    
  "  ,        
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               
                  

  ,             
   50 ,     (α) 5% ,       :
-   (    )  , 60      

-   (     )  , 40    60   
    
     "  "   ,       
  ,          
,           -     
       
(iii)   I    II  :       
         ,    
                :
 I    II 
  I :
 
          ,       
       ,           
  ,    I            
 ,       
       :          , 
          ,    I    
    (α)   ,   0.05  5%      I  
      ,     5%     

  II :  
           ,     
      ,            
      ,     
Easy2Siksha.com
   ,         ,     
       ,    II       β 
   β            
  ,    I           
   :
  I  (α):    ,      ",    !" 
 
  II  (β):  ,    !”     
   
             α    (  
 )       ,       
    ( II ) α           
,        
    
   - -/- ,  ,   I/II  - 
       
1.   :    (-)       (-) ?
2.     :         
     
3.  :      ( I   II )  
     
               
  ,      ,       
 ,  ,      

 ,          -  ,   
   
-  -        
            
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  I    II        ,    
   
          ,   
,         ,      ,
             
2. (a)      (MLE)     ?     
 (OLS)       ?      
(b)                  
  
:                
   ( )    ,        
     -         ""  
     (       )  ,   :
"    ,        ?" ( 
    )         ,   -
    
          : ()   
()      ()      ()   
    ,      
󷊆󷊇  ():      ()     
1.      () ?
          
  :          
  , MLE  : "         ,    
        ?"
    :
1.      10     7   
2.   ()      
3. 10   7          
4. MLE  : p          
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5.   , p = 0.7 MLE
, MLE                
       
2.      () ?
           
               
  
   , OLS           
      
 :                
  ,  OLS             
     
3.      

     (OLS)

      
      

    
    
  
      
     

 
-   
   
 
FLEXIBILITY
     
     
 
      
     
 ?”
4.      
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               , OLS 
MLE               
         
,             
󷈷󷈸󷈹󷈺󷈻󷈼  ():            
1.      
 :
o              
,   
o           
o :   \bar{x}   μ    
    :
o              
 
o :     xˉ=52\bar{x} = 52 ,  52, μ  
 :       ,          
   ()         
2.     
        ,    :
() 
             
:  ,       
   ?  ,     
() 
     , -    ,   
   
:       ,     
    
   ?             
() 
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  ,          
:     ,      -  ,  
  
   ?       
()  
               
 
:           
   ?       
()  (   )
           
   ?         
 
󼩺󼩻       
             :
 :      
 :     ,       
 :    ,    
  :             
 :    ,    
          - , ,  

󹶓󹶔󹶕󹶖󹶗󹶘
       :
     ()        
       
     (OLS)          , 
    ,  MLE    
     :     ;     
     
  : , , ,     
Easy2Siksha.com
 , MLE  OLS        -     
,                
       ,         
   
-
3. x² ( - )     ?    
: -  (χ²)   
                 
       ,         
           :     ,
   ,          
      - ,   χ² (-)  

 , -               
    -   ,       
 ,                
                 
        ,        
  
   
         -  20      
                   
       ,       100 
 ,   50    50         , 
47    53             
,     ?            - 
 
  , χ²              
     :
1.      Z₁, Z₂, ..., Zₙ          
( 0,  1)         " "
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2.                ,  
    :
  , χ², n   (df)   -     
            
 ,  , χ²               
 -   
,  -  , -      ,    
    :
1.      
-      ( -)      - 
(χ² ≥ 0)             :      ,
       
    (df)  ,   
-     ,         
  
2.   
   (df),  ν    , -     
            
  ,  χ² 5       ,  df = 5.
  ,           , 
    
3.    
χ²           :
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      :
                    

4.   
-        :
     -  , ν₁ df   χ²₁  ν₂ df   χ²₂, 
   ν₁ + ν₂ df   -   :
           
5. -   ()
-    -  (MGF)  ,     
           ν df   χ²  MGF :
      ,          
6.    
 : -  
  ,   = √(8/ν)  - ν
 ,  
 :  ""   χ²   ,  = 12/ν  ,
- df  ,            
   
               
                  
   20   A, 30   B  50  C     , 
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    -     ,       
          
 , χ²   :
1.      -         
 
2.    -            
    
3.         , χ²    
      
   
               , 
    (,   )       
  (   )         
  χ²    (  )   ,  , ,
                
,   -             
  

 , -         ,  
            ,     
              
  , , ,       
        
χ²               ,   
           ,    
       
-          ,      
,     ,     ,    
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4.    -          
:               
           ,      
,           :       ,
      ,      ?
         ,     (
" "   )      ,   ,  
     ,        
     :       ,  
    ,           
󷊆󷊇      
       X1,X2,…,Xn    μ\mu   σ2 
    
  :
  :
,    σ2\sigma^2  ,   zstatistic     :
       
   , σ2          S2S^2 
      t   :
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  (n – 1)      t    
󷈷󷈸󷈹󷈺󷈻󷈼     ?
     ,    S2, σ2      ,
 ztest   
      ,  σ\sigma  SS     
 
         ,      
   
,            : "   
  ,              
 "
󽀼󽀽󽁀󽁁󽀾󽁂󽀿󽁃  
      ,        
1.       :
2.  ,   -    :
3.      Z  S2  
4.  , tstatistic       :
  (n – 1)      t    
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󷇮󷇭     
           
1. 
           
             
2. 
   v>1  , t   0
      t   μ\mu   
3. 
v>2  , t   :
     1   ,     t     
  
- v→∞,  1    ,  t    
   
4.  ()
          
         
       ,       
5.    ()
       v=n−1   
 df  ,     
- df  ,       
6.    
- - :         
       ,   
7.    
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               
     
: onesample ttest, twosample ttest, paired ttest.
󼩺󼩻   
-                
  ,             , -
 : ",    -   ,      "
              
-     ,  -        
   
󹶓󹶔󹶕󹶖󹶗󹶘   
 :         ,  ,
      
  :        ,  t
z     
:           ,   -  
    ,        
󹶜󹶟󹶝󹶞󹶠󹶡󹶢󹶣󹶤󹶥󹶦󹶧    
1.        :
  n−1n-1     t    
2.    :
o     .
o  = 0 (df > 1  ).
o  = v/(v−2)v/(v-2) (df > 2  ).
o       
o       
o    df → ∞      
󷘹󷘴󷘵󷘶󷘷󷘸
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 ,           
    :        ,    
     
 ,  ,     ,     
              
     ,      :       
 
,        ,        
            
-
5. (a) 500     , 280            
             ?
(b) Z       
:                  
         ,      ,  
               ,
               ,   
                
          
     ,   500         ,
    :
280      
220     
,      : “            
?”     ,            
  ,           
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  ():                ?
  ,    , "280, 220    ,     "
              :  
  ,         ?
     Z    Z         
        ,       
 1:     
  ,      :
  (H₀):      
  (H₁):        
  ,        ,    50%    
   :
    = 500 × 0.5 = 250
    = 500 × 0.5 = 250
 2: 
 :
    = 280
    = 220
          = 280 – 250 = 30
,       30             
   ,    
 3:     Z    
   Z   :
:
p =   ( )
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P =   ( )
n =  
  :
 4: Z     
 ,   Z    5%        :
- :  Z ≈ ±1.96
 2.68 > 1.96,       
:             
   ,      ,        
  (): Z   
      Z             
 ,       Z       ,    
              
,             
Z ,        ,        
           Z      
     ,       
     :          ,   
   ,          
Z    
1.            
Easy2Siksha.com
       
Z                
      
:            
     
2.      
Z     
            ,      
 
:              80% 
  ?
3.   
Z               
  ,    -   ,   Z     
            
    ,      
4.    
           Z    
:               
 
 Z       ,       
5.        
     ,       
       Z     
:  ,             
   
6.    
Z        ,      
 ,        
:            
7.    
Easy2Siksha.com
       ,          Z
     
:           ,   
     
  ()    ()  
             Z     
 :
             Z    
 
Z              
            :  
,         
  , Z            , ,
,      -,     

      
  ,                
                 
     Z     
      
     ,        ,  
       
 ,   Z        ,        
 ,               
 
Z          -        
    
Easy2Siksha.com

:
1.        :500     ,    (280) 
    (220)        Z   
 ,               
           -   
 
2. Z  :Z        
 ,  ,  ,   ,  
          
Z       ,      ,     
 ,         ,     ,    
   , Z            
6.     ,       :

 
  
    
 
1
10
15
90
2
12
14
108
5%           
:    
   :
 1:
o  :1=10
o   :xˉ1=15
o       :1=90
 2:
o  :2=12
o   :xˉ2=14
o       :2=108
 5%              
 F-    , -   ,   -  
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    
            ,     
F- ,                
  (  ),          
          
       :
 1  :
 2  :
        :
v1=n1−1=9,v2=n2−1=11
Easy2Siksha.com
     
            ,  
     :
   :
≈1.0185
        ,           

-   ( )
   
 : 5%        
  :            
    (1010  9.829.82)   ,  F- 
  
 :     ,         

     --  
Easy2Siksha.com
  :5%  ,       
       
-           :
:
o        
   :
o            
  F- -     ;    ,
  -        
  ,     
   :
o                
    ;  SS  S2        
    :
o             ,  
  ( ,  )   
   ,     -       

   
    -   ""         
      ,     :       , 
 ? F-     ,     ,   , "
      5%        "   , 
       , "  "
   
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         :
         , - - (
    -    )      
           ,   
        ,       
      
 
  :           ;    
   
-
7.                
                
(   )            :
  1
  2
  3
  4
20
19
21
15
23
15
19
17
18
17
20
16
17
20
17
18
16
16
               
:        :      
         ,        
      ,          
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                :
          ? ,    
         ,      , 
    
,                 
   ,               ,
                 
                   
  ,      :
  1
  2
  3
  4
20
19
21
15
23
15
19
17
18
17
20
16
17
20
17
18
16
16
   :         ,        
 ?
    ,     ,   
(ANOVA)  ,     ANOVA           
                ,   
  
 1:    
             :
  (H₀):          
,
  (H₁):           
                
                 
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 2:   
      ,          :
 1: 20, 23, 18, 17
 2: 19, 15, 17, 20, 16
 3: 21, 19, 20, 17, 16
 4: 15, 17, 16, 18
                
,    ,        ANOVA   
 3:    
        ( )     
  :
1.   1 :
2.   2 :
3.   3 :
4.   4 :
 ,  ,    ,    :
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,      ,        
 4:   
          :
1. -  ():       
2.     (SSW):       
  
 4a:      (SSB)
 :
:
   :
 4b:      (SSW)
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 :
 5: -  
,       :
1.   :
2.  :
 , - :
 6: F        
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  F = 1.67
 F = 3.34
 1.67 < 3.34,         
 7:     
                
  ,        ,        
     
       ,          
 ,           ,       
           
 8:       
    :  ""       ,   
    ,        ( )      
      1 19.5     4 16.5    
               
 -           
 ,                 
    ,    
 9: ANOVA  
1.    ,             ""
 ,         
2.             ,  
  1       4     
3.               
                
  
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
 , ANOVA             
      ,   ,    , 
               : -
,    ""      ,         
      
  , ANOVA            
   ,       ,       
        ,      
  
8.    ?          
 ANOVA       
:  
                 
          -   (    
)  ,              , 
                 
  :-         
  :
 F  ,        
  :"      ?"        
;  ,  -  (,  )    
       :  (  ),  (   ),
 (  ),  (  )
     
-         :
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:          (, 
  - )     
   :   ,        
ANOVA       ,        
   :     ()  ,   
ANOVA       
    :         
     (-  ) - ANOVA      
 
-  (  ANOVA ):      ;  
 ,  /- ANOVA   
   :        
-   - ANOVA
            -  - 
    ; -               
  
- 
 :  (, k     )
 :
  :
o    ()
o    ()
  :      ()     
 : k       ?
- 
 :  (,      ),      
 ;      
 :
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  :
o  A   
o  B  
o   
o 
  :            
       
 :  A    ?  B    ?   
  ?
 
    :-  1  ; -  2 ( ) 
  : -     -         
         
 :-             
  
    :-        
   
  : ANOVA  
    
   :            
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   :         
   
      :           
    
  : 5%       (-)
   :
 1:    
  :
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 3:   (- )
 4: 
 :H0H_0   .
  :5%   ,            
 (10  9.82)   ; F-      
       
      :   -  (- 
  )   
      :      
ANOVA       
 -  ,      ,       
 MSW         F-     - 
           (  ,  t-   t-
     )
    
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 :-            
          
   :, ,   , ,  , 

-   - :
o :  ,  =  + 
o -:   +  ,  = A + B + A×B + 
    (   ):
o        
o        
o     ; , F ≈ 1.0185 →  
   
      : ANOVA          , 
              ,      
              ,     
 ( )        F-    ;  
      ,         
"                 
     ,     "