Data Analysis for Decision Makers Formulae sheet

Sample Statistics Population Parameters

n N
1
x = ∑ xi
n
∑( X i − X) 2
µ=
1
∑x
N

i
∑( X i − µ )2
n i =1 s= i =1
N i =1 σ= i =1
n −1 N
Grouped Data Quartile Position Coefficient of Variation

i (n + 1) S
Qi = CV =   100%
4 X

Combinations Permutations Bayes’ Theorem

p(A|Bi )p(Bi )
n n! n P = n! p(Bi |A) =
C r =   =
∑
n k
r ( n − r )! p(A|B j )p(B j )
 r  ( n − r )! r! j=1

Probability Distributions:
Probability
 Expected Variance
Distribution Value

 
1   
  1  

Binomial
 
         
 
Hypergeometric
   
    1

  

Poisson  
!

    
Uniform
     
 2 12

Mean inter-
Inter-event variance
P(interarrival ≤ x) = event
Exponential 1
1 -  
1 " 
!  


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Density of the Normal Distribution Z-score Transformation
( *+, )
$    / 0
#   ) - . 
√&' '

Sampling distribution (mean): Sampling Distribution (proportion):

/1  02 /1  0 / 7$7 78 7
. 3
 4 6  ; "78  9 ; .≅ ;
'2
3 ;(+;
√5 9
5

1  .  !  1  . 1  < $  !  1  < $
' ' = =
Confidence Intervals: or
√ √ √ √

78 $ 78  78 $ 78 
6  . 9   6  . 9

Sample Size Determination Sample Size Determination

for the Mean for a Proportion
> ) ') > ) 7$7
   
?) ?)

Hypothesis Testing, one sample:

/1  0 /1  0 78 7
Mean:  4 ; @ ; Proportion: . ≅ ;
;(+;
√5 √5 9
5

Two Sample Tests:

Differences in Two Means:
( X 1 − X 2 ) − ( µ1 − µ 2 ) ( X 1 − X 2 ) − ( µ1 − µ2 ) (n1 − 1) S12 + (n2 − 1) S 22
Z= or t = where S p2 =
 σ 12 σ 22  1 1 ( n1 − 1) + (n2 − 1)
 +  S p2  + 
 n1 n2   n1 n2 

Separate-Variance t Test for the Difference in Two Means:

2
 s12 s22 
 + 
( X 1 − X 2 ) − ( µ1 − µ 2 )  n1 n2 
t= with ν = 2 2
 s12 s22   s12   s22 
 +     
 n1 n2   n1  +  n2 
n1 − 1 n2 − 1

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Z Test for the Mean Difference: t Test for the Mean Difference:
n n
D − µD D − µD
Z=
σD
, where ∑ Di ,
i =1
where t =
SD ∑D
i =1
i
D= D=
n n n n

and ∑(D − D) i
2

i=1
SD =
n −1

F Test for the difference in variances:

B
A  $C 
B

n∑ xy −
(∑ x ) ( ∑ y ) DEF /,H
Correlation Coefficient: r = =
'
.'J
n∑ x − (∑ x ) n∑ y − (∑ y)
2 2
2 2

2
Coefficient of Determination: CD = r

Time-Series models: y =t+s+r and y = t×s×r

n∑ xy − (∑ x ) (∑ y) a = y − bx
Least squares regression ŷ = a + bx b =
n∑ x − ( ∑ x )
2
2

N N
µ = E(X) = ∑ X i P( X = X i ) σ 2 = ∑ [X i − E(X)]2 P(X = X i )
i =1 i =1

N
σ XY = ∑ [ X i − E ( X )][(Yi − E (Y )] P( X = X i , Y = Yi )
i =1

E(P) = w E ( X ) + (1 − w) E (Y ) σ P = w 2σ 2X + (1 − w) 2 σ 2Y + 2w(1- w)σ XY

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 Data Analysis for Decision Makers
t Table
cum. prob 0.50 0.75 0.80 0.85 0.90 0.95 0.975 0.99 0.995 0.999 0.9995
one-tail 0.50 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 0.001 0.0005
two-tails 1.00 0.50 0.40 0.30 0.20 0.10 0.05 0.02 0.01 0.002 0.001
df
1 0.000 1.000 1.376 1.963 3.078 6.314 12.71 31.82 63.66 318.31 636.62
2 0.000 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 22.327 31.599
3 0.000 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 10.215 12.924
4 0.000 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 0.000 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 5.893 6.869
6 0.000 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 0.000 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 0.000 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 0.000 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 0.000 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 0.000 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 0.000 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 0.000 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 0.000 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 0.000 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 0.000 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 0.000 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 0.000 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 0.000 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 0.000 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 0.000 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.527 3.819
22 0.000 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 0.000 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.485 3.768
24 0.000 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 0.000 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.450 3.725
26 0.000 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.435 3.707
27 0.000 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.421 3.690
28 0.000 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.408 3.674
29 0.000 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.396 3.659
30 0.000 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.385 3.646
40 0.000 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.307 3.551
60 0.000 0.679 0.848 1.045 1.296 1.671 2.000 2.390 2.660 3.232 3.460
80 0.000 0.678 0.846 1.043 1.292 1.664 1.990 2.374 2.639 3.195 3.416
z 0.000 0.674 0.842 1.036 1.282 1.645 1.960 2.326 2.576 3.090 3.291
0% 50% 60% 70% 80% 90% 95% 98% 99% 99.8% 99.9%
Confidence Level
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F-Distribution (α=0.05 in the right tail)

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F-Distribution (α=0.01 in the right tail)

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