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آمار مقدماتی و پیشرفته مدرس: دکتر بریم نژاد دانشیار واحد کرج PowerPoint PPT Presentation


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آمار مقدماتی و پیشرفته مدرس: دکتر بریم نژاد دانشیار واحد کرج. آمار چیست؟

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آمار مقدماتی و پیشرفته مدرس: دکتر بریم نژاد دانشیار واحد کرج

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:


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  • Statistics status .

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:

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- :

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:

. . ... . . 400 145 .

:

. (Statistical Process Control ) .


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()

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()


. 1 2


( )

( )


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:


(8 10 ...)




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:


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:



:


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Discrete or Continuous


( )

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-

0 1 2 3 4 5 6 7

-

0 1000


.

.

.


:



...

( ) .

11 .

2 3339 26


...

( ) .

250 1387 173

250 1387 138



  • ( )

  • ( )



  • .

  • x-bar .

  • outliers .


:


  • 50

  • .


.

: 2 8 3 4 1

: 12 3 4 8


:2 8 3 4 1 8

: 12 3 4 8 8

= (3+4)/2 = 3.5


  • .

  • .


Minitab:

Variable N Mean Median TrMean StDev SE Mean

Phone 139 121.6 60.0 88.1 217.7 18.5

Variable Minimum Maximum Q1 Q3

Phone 2.0 2000.0 30.0 120.0

N =



:

  • (unimodal) (multimodal)




Descriptive Statistics

Variable N Mean Median TrMean StDev SE Mean

GPA 92 3.0698 3.1200 3.0766 0.4851 0.0506

Variable Minimum Maximum Q1 Q3

GPA 2.0200 3.9800 2.6725 3.4675



Variable N Mean Median TrMean StDev

Males 84 70.048 70.000 70.092 3.030

Females 89 64.798 65.000 64.753 2.877

All 176 67.313 67.000 67.291 4.017

Variable SE Mean Min Max Q1 Q3

Males 0.331 63.0 76.0 68.0 72.0

Females 0.305 56.0 77.0 63.0 67.0

All 0.303 56.0 77.0 64.0 70.0





Descriptive Statistics

Variable N Mean Median TrMean StDev SE Mean

CDs 92 61.04 46.50 52.93 62.90 6.56

Variable Minimum Maximum Q1 Q3

CDs 0.00 400.00 21.50 83.00




Variable N Mean Median TrMean StDev SE Mean

grades 22 89.18 93.50 90.60 12.92 2.76

Variable Minimum Maximum Q1 Q3

grades 50.00 100.00 87.00 98.00


  • .

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.


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  • .



Descriptive Statistics

Variable N Mean Median TrMean StDev SE Mean

GPA 92 3.0698 3.1200 3.0766 0.4851 0.0506

Variable MinimumMaximum Q1 Q3

GPA 2.02003.9800 2.6725 3.4675

= 3.98 - 2.02 = 1.96


  • ( 75) ( 25)

  • IQR = Q3-Q1

  • .

  • .



Descriptive Statistics

Variable N Mean Median TrMean StDev SE Mean

GPA 92 3.0698 3.1200 3.0766 0.4851 0.0506

Variable Minimum Maximum Q1Q3

GPA 2.0200 3.9800 2.67253.4675

IQR = 3.4675 - 2.6725 = 0.795


  • .

  • .

  • .


  • 2 .

  • s2 .

  • .

  • . .

  • .


  • s .

  • .

  • .


(MPH)


Sex N Mean Median TrMean StDev SE Mean

female 126 91.23 90.00 90.83 11.32 1.01

male 100 06.79 110.00 105.62 17.39 1.74

Minimum Maximum Q1 Q3

female 65.00 120.00 85.00 98.25

male 75.00 162.00 95.00 118.75

Females: s = 11.32 mph and s2 = 11.322 = 128.1 mph2

Males: s = 17.39 mph and s2 = 17.392 = 302.5 mph2



Sex N Mean Median TrMean StDev SE Mean

female 126 152.05 150.00 151.39 18.86 1.68

male 100 177.98 183.33 176.04 28.98 2.90

Sex Minimum Maximum Q1 Q3

female 108.33 200.00 141.67 163.75

male 125.00 270.00 158.33 197.92

Females: s = 18.86 kph and s2 = 18.862 = 355.7 kph2

Males: s = 28.98 kph and s2 = 28.982 = 839.8 kph2


  • 100

  • .


Sex N Mean Median TrMean StDev SE Mean

female 126 91.23 90.00 90.83 11.32 1.01

male 100 106.79 110.00 105.62 17.39 1.74

Minimum Maximum Q1 Q3

female 65.00 120.00 85.00 98.25

male 75.00 162.00 95.00 118.75

Females: CV = (11.32/91.23) x 100 = 12.4

Males: CV = (17.39/106.79) x 100 = 16.3


Sex N Mean Median TrMean StDev SE Mean

female 126 152.05 150.00 151.39 18.86 1.68

male 100 177.98 183.33 176.04 28.98 2.90

Sex Minimum Maximum Q1 Q3

female 108.33 200.00 141.67 163.75

male 125.00 270.00 158.33 197.92

Females: CV = (18.86/152.05) x 100 = 12.4

Males: CV = (28.98/177.98) x 100 = 16.3


...


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  • . A, B, C,


  • 1 .

  • .

  • .

  • .

  • A P(A) .


:

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  • .

  • .

  • .



...



=


...


...


...

  • . :

    P(X > 120), P(X<100), P(110 < X < 120)

  • =

  • = 1

  • 0 .

    P(X=120) = 0


p.d.f


  • .

  • .

  • .

  • .



75


=

  • .

  • .

  • ( )

  • standardize .


...

  • x . z . :

    Z = (X- )/

  • Z . 0 1 .

  • z .


z


65 70


65


!

  • .

  • .

  • ! ( )



.



  • .

  • :


  • . .

  • : 20


  • .

  • ()

  • .


.


5

7/2

100 9/2 7/2

100


  • .

  • (: )

  • . .


()

  • :

    • .

    • .

  • .


  • the null hypothesis (H0)

  • and the alternative hypothesis (HA)

    • H0:

    • HA:


  • .

  • .


  • .

  • .


    • .

  • .


  • .

    • ( ) . ( ).

    • ( )


  • .

  • .

  • .


...




:

  • : .

  • : .

  • .


6/98

80 4/98 .

80


()

    • H0: = 98.6

    • HA: < 98.6

  • = 98.6 .

  • : 80 4/98 . 80 4/98 6/98


p-value

  • p-value .

  • p-value .

  • .

  • p-value ( 05/0 .


()

p-value MINITAB .

Test of mu = 98.6000 vs mu < 98.6000

The assumed sigma = 0.600

Variable N Mean StDev SE Mean Z P

Temp 80 98.4 0.67 0.0671 -2.80 0.0026

p-value p .


()

  • p-value 0026/0 6/98 80 4/98 .

  • :

  • 6/98 .


  • 6/98 .

  • 6/98 .

  • 6/98 .



5

7/2

100 9/2 7/2

100


p

100 9/2 7/2


P

H0: = = 2.7

HA: = > 2.7

100 9/2 6/0 P :


  • P . 9/2 7/2 .

  • . 7/2 .


  • H0: = 2.7 HA: > 2.7

    P .

  • Z = 3.33

    .

  • P 05/0 05/0 . . =0.05 .


6/98

80 4/98 .

80


p

80 4/98 6/98


P

H0: = = 98.6

HA: = < 98.6

80 4/98 6/0 P :


  • P 4/98 6/98 .

  • . 6/98 .


  • H0: = 98.6 HA: < 98.6

    P .

  • Z = -2.98

  • P 02/0 02/0 . = 0.02. .


20

17 16 .

64


P

64 17 23 20


P

H0: = = 20

HA: = # 20

64 17 16 P :

P-value = 0.067 2 = 0.134


  • P . 17 23 20 .

  • . 20 .


  • H0: = 20 HA: # $20

    P .

  • Z = -1.5

    .


  • n > 60 .


  • P .

  • .


START

Paired t test (samples must come

from normal populations):

Yes

Are the

two samples

dependent?

No

where df = n - 1

z test (normal distribution):

Testing Hypotheses Made about the Means of Two Populations

Do n1 and n2

both exceed

30?

Yes

No

No

Are both populations

normally distributed?

Use nonparametric methods

Yes

Reject

After applying the F

test, what do we conclude

about ?

separate variances t test

(samples must come from

normal populations)

Fail to reject

Pooled variances t test (samples must

come from normal populations):

where

and


.

Nominal scale . ( )


2. ordinal scale:

. .


3. Interval Scale

.

20 10 10 .


4. Ratio Scale

. - .



.

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.


1) 2) 3) .

1) 2) .


:

.

( ) . .

- .


. ( ) ( ) .

.


1. : .

.

.


.

.

10 ( 5 )

. 50 .

.

( )


2. (T)

. . .


3. phi

.

. .


4. Pearson s coefficient contingency

C . .


5. :

.



1. kendall s rank correlation coefficient

. . 1- 1+ .


2. Gamma coefficient

. .


3. Spearman Rank Correlation Coefficient

(... 3 2 1) . .

rs 1+ 1- .



Pearson Correlation Coefficient

  • r . 1+ 1- .

  • .


:

.

- .

.

- .


( ) ( ) .


t F . . t F .


t F :

.

( )

.

. .


- t: .

( )


- F( ANOVA)

  • .

    ( )

  • : F . (Scheffe test) LSD Tukey Duncan . .


.

.

(ANOVA) .

j . .


:

  • ANOVA .

  • A B C A B B C A C.

  • t t .

  • ( .


:One-way Analysis of Variance () .

:

Two way Analysis of Variance

.


( ...)


. :


  • .

  • .

  • . .



1. :

.


:

  • (Two related)

    ( )


1. ( ) . (Ho) ( ) .

2. 1000 .


3. Wilcoxon Test

.

.

: .


4. Fridman Test

  • F F .

  • .

  • : 30 1 ( ) 5 ( ) .


5.

: .

:

- ( )

- ( )

- ( )


6. - Mann Whitney Test

.

: 30 1 5 .


7. - Kolmogrov Smirnov Test

20 5 50 . - . .


8. - Kruskal Wallis Test

. .

: 90 . 1 ( ) 5 ( ) .

Ho: .

.


:Median test

. .

:

40 ( ) .



  • .

  • .


:


.

  • (Dependenc Technique) (Interodependence Technique) .

  • .

  • . .


  • .

  • * .

  • .

  • .

  • .

  • .

    • (1 0) .


.

  • :

  • : ( ) .

  • .



    • :

      • ( )



    • .

    • SAS, SPSS, S-plus, R, MATLAB,


    • :

    • :

    • .


    • .


    . .


    1877 . .

    : (Regress)


    • :

    • ( ) ( ) ( ) .



    • . () . .

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    ()

    () ( ).


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    • .

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    • .

    • .

    • .

    • ( ) .


    Ordinary Least Square (OLS)

    ( ).





    .


    :

    • ( )

    • () ( )


    • OLS .

    • :

    • . . .


    :

    1: ui

    ui Xi .


    u .

    2: u


    3: () Ui

    Y X .


    y X .


    4: Ui , Xi

    x u ( ) y X u . y . X u X u u X u X u u X u Y .


    5: ( )

    .


    .

    . :

    • YiXi ui


    .

    .




    : -

    2 (BLUE) 2 :

    • . Y .

    • ( ).


    The Gauss-Markov Theorem: are the best linear unbiased estimators (BLUE).


    -

    BLUE .


    r2( ) r2

    :

    • r .

    • 1+ 1- .

    • x y rxy y x (ryx) .

    • .

    • x y r = 0 (h )

    • . hY=X2 r .

    • r .



    r2 r r2 r . r (R=) .


    R2

    • R2 R2

    • R2 .

    • .

    • R2 .

    • .

    • R2 R2 .


    :


    ui

    • (OLS) ui .

    • ui .


    • OLS .

    • ui .


    • ui uj


    .


    OLS

    • :

    • (N-2) .

    • .

    • .

      (BLUE) .


    :


    :

    .


    : .

    : .

    .


    • :

    • 1: (x) .

    • 2: ui .

    • 3: .

    • 4: . ui .

    • 5: (x) .

    • 6: ui 1 2 .

    • 7: .


    (OLS) BLUE .

    • 1 4 6 :

    • 1: : .

    • 4: : x u x .

    • 6: u: . .


    .

    Multicollinearity

    x

    .

    x . () BLUE .


    - OLS

    - : ( )

    - r: t .

    - R2 .

    - OLS


    -

    • . .

    • .

      -

      1. R2 t

      2.

      3.

      4.

      5. (Eigenvalue) (Condition Index)

    SAS .


    :

    .

    • ()

    • ( )

    • ( )


    Heteroscedasticity ui .

    () () .


    OLS

    • BLUE (GLS) .

    • t F .


    • : ei2 .

    • x .

    2. : .

    3. -

    4.

    5.

    6.


    • .

    • : ( ) ( ) .

    ( )


    OLS

    • GLS BLUE OLS . OLS .

    • OLS ( )

    • (OLS GLS)


    OLS

    • .

    • R2 .

    • t F .


    D.W1. 2. 3. .4. .5. .

    :

    • -

    • -

    • - : DW


    • . .

    • :

      • .

      • 5 .


    • : () .

    • : .

    • :

    • : () .



    • :

      • ( )

      • .

      • .

      • .

      • .

      • . ( ) ( ) ( )


      • ( F)

      • ( )

      • Reset


    ( )


    • ( ...) ( ...) .

    • . (Dummy Variable) .

    • .


    • .

    • .

    • (ACOV) .

      : m m-1 ( )


    ( 653)

    [1] [2] [3] [4]


    • . () .

    • . .

    • . OLS . .

    • . .


    • TorbatjamKhaf Taybad . Torbatjam Khaf Taybad .

    • . . .


    • Roshan Sardary Gaskojen . S68 Model . S68 . Abideym .

    • Omr Tarikh Saat .

    • Omr Tarikh 84 Saat 12 .

      F t .


    • . .

    • 0 1 .


    :

    • linear probability model (LPM)

    • (Logit)

    • (Probit)



    • . ( ) ( ) . ( x y .

    • () () . (OLS) . .


    • OLS .

    • :

      • (2SLS)

      • (3SLS)

      • (I3SLS)

        • (LIML)

        • (FIML)


    .

    .

    .


    • Sewell Wright .

      Formulated in series of papers published in 1918, 1921, 1934, 1960

    • .

    • .


    • .

    • . .


    • = +


    . :


    :

    - : . :

    : .

    : .


    ( 1380):

    1- : .

    2- : .


    .



    • .

    • .

      • .


    • .

    • x y

      x1 y

    • x y .



    • .

    • :

      • (P1) .

      • (P5) (P6) (P2) (P4) (P6).

      • (P3) (P4) (P6)

      • (P6) .


    • e1 (P7) .

    • e2 (P8) .

    • e3 (P9) .

    • .


    • () . ( ) . .


    • ( ) .

    • e1 + () x1 =

    • e2 + () x3 + () x2+() x1 =

    • e3 + () x2+() x1 =


    ( ) .

    • e1 + () x1 =

    • e2 + () x3 + () x2+() x1 =

    • e3 + () x2+() x1 =

      (1) 2P: (2) 1P 2P 3P (3) 5P 4P . .


    • . .

    • .


    • . .


    • (08/0 -) . :

    • 27/0 = (47/0 57/0)

    • 16/0 = (58/0 28/0)

    • 03/0 = (47/0 22/0 28/0)

    • 46/0 = 03/0 + 16/0 + 27/0 .

    • 38/0 = 46/0 + 08/0 .

    • .


    • . . (P5 P4) : (P1)

    • (P3) (P2) (P3). r :

    e1 + ( ) x2 +() x1 + a=

    e2 + () x3 + ( ) x2 +() x1 =


    • . . :

    • (p3)(p1) + (p5) =

    • (p3)(p2) + (p4) =

    • p3=



      • X1 x4

      • X1 x4 4

      • X4 x1 1

      • 1 4 .

    • ...

      • OLS . OLS .


    2SLS .


    Factor Analysis



    • :

      • ( ).

      • .

      • .


    • : (1963) ( ) .

    • : . . . .


    • (exploratory) (confirmatory) .


    :

    • (field) . (1904) . . ( ).


    • . (1973) .


    • .

    • .

    • .


    • KMO 0 1 . KMO 5/0 . 5/0 69/0 . 7/0 .

    • Kaiser-Meyer-Olkin


    50 100 . . . . R .


    • .

    • .


  • Login