Name ______________________________________
Exam 3 (100 points) ID Number __________________________
Part I. Workout Problems. Show solution in support of your answers. Unsupported answers will not receive full
credit. (61 points)
1. A 2!”# fractional factorial involving factors A, B, C, D, E and F is to be run. Practitioners have these two sets of
generators in mind:
Design 1 Generators: E=ABD and F=ACD
Design 2 Generators: E=ABCD and F=ABD
a. Consider Design 1. Which treatments in this experiment will have both factors A and B at their high (+)
levels? [6 pts]
b. Consider Design 1. Derive its defining relation and determine its resolution. [8 pts]
c. The defining relation for Design 2 is I=CEF=ABDF=ABCDE. Which design (1 or 2) is better? Explain briefly
and give at least one reason for your choice. [3 pts]
2. A 2$”% fractional factorial was conducted to study the effects of four factors on the bond strength of an
integrated circuit mounted on metallized glass substrate. The four factors (and their levels) that engineers
identified as potentially important determiners of bond strength are listed in the table below.
Factor Levels
A – Adhesive Type D2A (−) vs. H-1-E (+)
B – Conductor Material Copper (−) vs. Nickel (+)
C – Cure Time at 90°C 90 min (−) vs. 120 min (+)
D – Deposition Material Tin (−) vs. Silver (+)
Let �& = main effect of A, �’= main effect of B, �( = main effect of C, �) = main effect of D, and � = interaction
effect. Summary statistics and the results of the Yates algorithm for computing fitted effects are given below.
Treatment Replication
Sample
Variance ��
Sample
Mean �+
Yates Algorithm
Cycle 1 Cycle 2 Cycle 3 Fitted Effect
(1) 5 2.452 73.48 157.36 314.54 650.84 81.355
ad 5 4.233 83.88 157.18 336.30 7.84 0.980
bd 5 0.647 81.58 166.60 4.42 2.92 0.365
ab 5 26.711 75.60 169.70 3.42 2.08 0.260
cd 5 0.503 87.06 10.40 −0.18 21.76 2.720
ac 5 8.562 79.54 −5.98 3.10 −1.00 −0.125
bc 5 1.982 79.38 −7.52 −16.38 3.28 0.410
abcd 5 3.977 90.32 10.94 18.46 34.84 4.355
a. The replications and the sample variances of the 8 treatment combinations are given in the 2nd and 3rd
columns, respectively, in the table above. Compute �(0.05) for judging if a fitted effect is statistically
significant at the � = 0.05 level. Note that the sum of the variances is 49.067. [8 pts]
b. The generator and defining relation were D=ABC and I=ABCD, respectively. If you have no answer in (a), use
�(�. ��) = �. ���.
i. Based on your answer in (a), is the fitted effect 0.980 statistically significant? [2 pts]
Select one: NO YES
ii. What sum of effects does the fitted effect 0.980 estimate? Your answer should be a sum of
subscripted/superscripted Greek letters (e.g., �# + �##
+,). [4 pts]
3. The diameter � of a tree at breast height (in cm, relatively easy to measure) is used to predict the height � of a
tree (in m, difficult to measure). Summary data on � = 36 white spruce trees (in British Columbia) are given
below.
B� = 655.1, B�# = 12711.47, B� = 644.7, B�# = 11824.45,
B�� = 12112.34, �– = 790.4697, ��� = �.. = 278.9475, �̅= 18.1972, �G = 17.9083.
a. Do some calculations to show that the least-squares line is �H = 9.1468 + 0.4815�. [10 pts]
b. Compute the sample correlation � between � and �. Give a quick interpretation. [6 pts]
Interpretation:
c. Construct an interval with 95% confidence for the height of a new spruce tree with a breast height diameter �
= 19 cm. Plug in numbers in a formula and do not simplify. Use � = 36, �̅= 18.1972, �– = 790.4697,
�# = ��� = 2.815. [8 pts]
Problem 3 (continued).
d. A scatterplot of the data and ��� values for the linear and quadratic model fits are given below. Also, the tota
l sum of squares for either model is ��� = 1824.45. Which of the two models provides a better description o
f the data? Explain briefly. In your explanation, use both graphical AND numeric results [6 pts]
Part II. Multiple Choice. Circle the letter of the correct/best answer. (39 points)
1. Which of the following statements is NOT true?
A. The simple linear regression model is � = �/ + �%� + � where the � is a random variable that is normally
distributed with mean 0 and variance �#.
B. In simple linear regression, the independent variable � is also referred to as the predictor or explanatory
variable.
C. The goal of least-squares regression is to find the curve that maximizes the sum of the squared distances
between the curve and the data points.
D. A first step in a regression analysis involving two variables is to construct a scatter plot.
2. In fitting � = �/ + �%� + � through data, (1.7, 2.5) is a 90% confidence interval for �%. What is a 90%
confidence interval for the mean change in � when we reduce � by 0.65.
A. (−1.625, −1.105)
B. (1.05, 1.85)
C. (1.105, 1.625)
D. (2.35, 3.15)
3. Which of the following is/are TRUE about the correlation coefficient � between � and �?
A. For the simple linear regression, 100% × �# = �# where �# is the coefficient of determination (in %).
B. A correlation of � = −0.87 is weaker than a correlation of � = 0.25.
C. The correlation � is a measure of the strength of the linear relationship between � and �.
D. If � = −0.1, and we convert � (in inches) to centimeters (1 in = 2.54 cm), then the correlation becomes
2.54 × (−0.1) = −0.254.
E. Both (A) and (C).
Model ���
� = �/ + �%� + � 95.703
� = �/ + �%� + �#�# + � 63.007
5 10 15 20 25 30
8 10 12 14 16 18 20 22
Breast-Height Diameter x
Height y
4. Is � = �/ ⋅ �%
0 intrinsically linear? If yes, what is appropriate transformation to obtain a linear model?
Recall: log(��) = log(�) + log(�) , log(�1) = � ⋅ log(�)
A. No.
B. Yes, log(�) = log(�/) + log(�%) ⋅ �
C. Yes, log(�) = log(�/) + �% ⋅ log (�)
D. Yes, log(�) = log(�/) + �% ⋅ �
For Problems 5 to 8: A study investigated the effects of �% = Seal Temperature, �# = Cooling Bar Temperature, and
�2 = % Polyethylene Additive on the seal strength �. The three models in column of the table below were fit to the
data.
There were � = 20 observations, and the total sum of squares (for all 3 models) is ��� = 82.17 (total df = 19).
5. What is ��� for Model (1)?
A. 30.96
B. 51.21
C. 21.36
D. 60.81
6. What is �34′
# for Model (2)?
A. 49.42%
B. 76.66%
C. 23.34%
D. 84.03%
7. What is the F statistic for testing �/:{�% = �# = ⋯ = �5 = 0} versus �3: {�/ is false.} with model (3).
A. 6.59
B. 9.69
C. 3.23
D. 5.36
8. In the fit of Model (2), we get �^
6 = −0.5 and �78! = 0.3552 and find that the P-value is 0.1827 for testing
�/: �6 = 0 versus �3: �6 ≠ 0. What are the � test statistic and conclusion at � = 0.10 significance level?
A. � = −1.41. There is NO significant interaction between �% and �2.
B. � = 1.41. The predictor �6 has NO significant effect on the response �.
C. � = −0.84. There is NO significant interaction between �% and �2.
D. � = −1.41. There is significant interaction between �% and �2.
Model �� ����
� ���
(1) � = �/ + �%�% + �#�# + �2�2 + � 37.68% 25.99% ?
(2) � = �/ + �%�% + �2�2 + �$�%
# + �“? fractional factorial studies?
A. The loss of information and ambiguity (confounding) can be held to a minimum by careful planning and
wise analysis.
B. A loss of information is usually expected because we are unable to observe responses at all of the 2>
factor combinations.
C. If two effects are aliased or confounded together, it means that we can discuss their significance together
but not apart from each other.
D. None of the above.
10. A fitted multiple regression model is �H = 10 − 4�% + 3�#. If �% is decreased by 2, while holding �# fixed, then
then we can expect �
A. to increase by 8
B. to decrease by 6
C. to increase by 6
D. to decrease by 8
E. remain the same
11. Suppose that the least-squares line is �H = −2.12 + 15.75�. If the � test statistic for testing �/: �% = 0
against �3: �% ≠ 0 is � = 2.1 (from the ANOVA table), what is the � test statistic for testing the same
hypotheses?
A. � = 1.45
B. � = −4.41
C. � = −1.45
D. � = 4.41
12. Which of the following statements is true?
A. Model 1 with more predictor terms may not necessarily be a better than Model 2 with fewer predictor
terms even though Model 1’s coefficient of multiple determination �# is larger.
B. To balance the cost of using more parameters against the gain in the coefficient of multiple determination
�#, many statisticians use �34′
# = {the adjusted �#}.
C. An objective of regression analysis is to find a model that is simple (relatively few parameters) and provides
a good fit to the data.
D. All of the above.
13. A study investigated the effects of three explanatory variables �%, �#, and �2 on the response �. The model � =
�/ + �%�% + �#�# + �2�2 + � provided a good �# value. Which of the following is NOT appropriate in assessing
the (statistical) significance of the relationship between �2 and �?
A. a � test of �/: �2 = 0 versus �3: �2 ≠ 0
B. a prediction interval
C. a confidence interval for �2
D. the sample correlation between �2 and �
E. a comparison of �34′
# values for � = �/ + �%�% + �#�# + �2�2 + � and � = �/ + �%�% + �#�# + �
F-1
Stat 423, Stat 523 Formulas
Chapter 7 Sections 7.1, 7.2, 7.3
We take a random sample X1, …, Xn from N(µ,s2)
Two-Sided 100(1-a)% Confidence Intervals for µ
Requirements Confidence Interval
Normal, s known
Normal, s unknown
Chapter 8 Sections 8.1, 8.2, 8.4
Steps in Testing Hypotheses
1. null hypothesis H0 and alternative hypothesis Ha
H0: µ = µ0 Ha: µ > µ0, µ µ0 z ³ za t ³ ta,n-1
µ µ0 1 – P(Z £ z) 1-pnorm(z) P(tn-1 ³ t) 1-pt(t,n-1)
µ D0 z ³ za 1 – P(Z £ z) 1-pnorm(z)
µ1 – µ2 30, n>30)
Replace s1 and s2 in Case I with standard deviations s1 and s2.
ïî
ï
í
ì
= = =
= = =
Sample 2 : mean y, standard deviation s , sample size n
Sample 1 : mean x, standard deviation s , sample size m
2
1
ï
î
ï
í
ì
µ – µ ¹ D
µ – µ D
1 2 0
1 2 0
1 2 0
~ N(0,1)
m n
(x y) z 2
2
2
1
0
s
+
s
– – D =
m n (x y) z
2
2
2
1
2
s
+
s – ± a
, m n (x y) z
2
2
2
1
÷
÷
ø
ö
ç
ç
è
æ
+ ¥ s
+
s – – a m n ,(x y) z
2
2
2
1
÷
÷
ø
ö
ç
ç
è
æ s
+
s – ¥ – + a
9A
9B
F-3
Section 9.2 t Test and Confidence Interval
• Normal populations, s1 and s2 are unknown, and sample sizes are small.
Case III t-based Procedures
, round down to the nearest integer.
t Test:
1. H0: µ1 – µ2 = D0 vs. Ha: 2. Test statistic:
3. Rejection region and P-value
Ha Rejection Region P-value P-value in R
µ1 – µ2 > D0 t ³ ta, n P(tn ³ t) 1-pt(t,n)
µ1 – µ2 D
1 2 0
1 2 0
1 2 0
2
2
2
1
0 ~ t
n
s
m
s
x y t n
+
– – D = !
n
s
m
s
(x y) t
2
2 2
1 – ± a 2,n +
, n
s
m
s
(x y) t
2
2
2
1 , ÷
÷
ø
ö
ç
ç
è
æ
– – a n + + ¥ n
s
m
s ,(x y) t
2
2
2
1 , ÷
÷
ø
ö
ç
ç
è
æ
– ¥ – + a n +
9C
F-4
Chapter 10 Section 10.1 Single-Factor ANOVA (Equal Sample Sizes)
• I = total number of treatments
• J = common number of replications of each treatment
• µi = mean of treatment i (for i = 1, 2, …, I)
• Xij = random variable that represents the measurement from the jth EU under
treatment i (for i = 1, …, I and j = 1, …, J)
The One-Way Fixed Model: Xij = µi + Îij
where Îi1, Îi2, …, ÎiJ are iid N(0,s2).
Definition Sums of Squares (SS)
Treatment i average: Grand Average:
• Total SS = Treatment i standard deviation = si
• Treatment (Among) SS =
• Error (Within) SS =
Þ ,
————————————————————————————–
Alternative (Working) Formulas
Let , .
•
•
• SSE = SST – SSTr
Remarks:
• is called a residual and eij estimates Îij.
• SST = SSTr + SSE Þ SSE = SST – SSTr.
————————————————————————————–
ANOVA table:
J
x
x
J
j 1
ij
i.
å= = IJ
x
x
I
i 1
J
j 1
ij
..
å å= = =
å å ( ) = =
= –
I
i 1
J
j 1
2 SST xij x..
å ( ) =
= –
I
i 1
2 SSTr J xi. x..
å å ( ) = =
= –
I
i 1
J
j 1
2 SSE xij xi.
( ) /( 1)
1
. 2 = å – –
=
s x x J
J
j
i ij i å=
= –
I
i 1
2 SSE (J 1) si MeanSquaredError MSE s /I
I
i 1
2
i ÷
÷
ø
ö
ç
ç
è
æ = = å=
å å å = = =
= =
J
j 1
i. ij
I
i 1
J
j 1
.. ij x x , x x IJ
x correction factor CF
2
.. = =
SST x CF
I
i 1
J
j 1
2 = å å ij –
= =
CF
J
x
SSTr
I
i 1
2
i.
= –
å
=
eij = xij – xi.
Source of
Variation
degrees of
freedom (df)
Sum of
Squares (SS)
Mean Square
(MS)
Test
Statistic F P-value P-value in R
Treatments
(Among) I-1 SSTr
1-pf(F,I-1,I(J-1))
Error
(Within)
I(J-1) SSE
Total IJ-1 SST
I 1
SSTr
MSTr – = MSE
MSTr
F = P(F F) I-1,I(J -1) >
I(J 1)
SSE
MSE – =
10A
10B
10C
F-5
When H0: µ1 = µ2 = … = µI is true, ~ FI-1,I(J-1).
Hypothesis Testing
H0: µ1 = µ2 = … = µI vs. Ha: H0 is false
• F-statistic:
• P-value: P-value = (In R: 1-pf(F,I-1,I*(J-1)))
• rejection region: RR = {F > Fa,I-1,I(J-1)}
————————————————————————————-
Section 10.2 Multiple Comparison in ANOVA (Equal Treatment Reps J)
Tukey’s Procedure for Simultaneous 100(1-a)% CIs for µi-µj:
T Method for Significant Differences
1. Compute .
2. List the sample means in increasing order.
3. Underline groups of means that do not differ by more than w.
————————————————————————————–
Contrast where .
Hypothesis Test (Equal Sample Sizes J)
1. H0: C = c0 vs. Ha:
2. Test Statistic
3. Rejection Region and P-value
Ha Rejection Region P-value P-value in R
C > c0 t ³ ta, I(J-1) P(tI(J-1) ³ t) 1-pt(t,I*(J-1))
C
( ) J
MSE xi – xj ± Qa,I,I(J-1)
J
MSE w = Qa,I,I(J -1)
xi.
å
=
= µ
I
i 1
C ci i c 0
I
i 1
å i =
=
ï
î
ï
í
ì
¹
0
0
0
C c
C c
C c
I(J 1) I
i 1
2
i
0 ~ t
J
MSE c
Cˆ c
t –
=
å
– =
å=
= ´ I
i 1
2
i
2
c
Cˆ SS(C) J ~ F1,I(J 1) MSE
SS(C) F = –
RR = {F > Fa,1,I(J -1)
}
PF F ( 1, ( 1) I J – > )
10D
10E
10F
In R: qtukey(1-a,I,I*(J-1))
F-6
100(1-a)% CIs for Contrast C (Equal Sample Sizes)
2-sided:
1-sided:
————————————————————————————-
Section 10.3 ANOVA for Unequal Sample Sizes
Ji = sample size for treatment i, n = SJi (total sample size).
Treatment i total: , Treatment i average:
Grand Average:
• Total Sum of Squares:
• Treatment Sum of Squares:
• Error Sum of Squares: SSE = SST – SSTr Treatment i standard deviation = si
ANOVA table:
Source df SS MS F P-value P-value in R
Treatments
I-1 SSTr
1-pf(F,I-1,n-I)
Error n-I SSE
Total n-1 SST
• Reject Region = {F ³ Fa,I-1,n-I}
————————————————————————————–
T Method for Significant Differences (Unequal Treatment Reps)
1. Compute for all pairs i,j where i¹j.
2. List the sample means in increasing order.
3. Underline and if they do not differ by more than wij.
J
MSE c
Cˆ t
I
i 1
2
i
,I(J 1) 2
å
= a – ±
J
MSE c
, , , Cˆ t
J
MSE c
Cˆ t
I
i 1
2
i
,I(J 1)
I
i 1
2
i
,I(J 1)
÷
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
ç
è
æ
– ¥ +
÷
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
ç
è
æ
– + ¥
å å
= a – = a –
å
=
=
Ji
j 1
xi. xij
i
i. i. J
x x =
n
x
x
I
i 1
J
j 1
ij
..
i
å å= = =
å å ( ) = =
= –
I
i 1
J
j 1
2
ij ..
i
SST x x n
x x
2
.. I
i 1
J
j 1
2
ij
i
= å å –
= =
å ( ) =
= –
I
i 1
2 SSTr Ji xi. x.. n
x
J
x 2
.. I
i 1 i
2
i. = å –
=
å=
= –
I
i 1
2 SSE (Ji 1)si
I 1
SSTr
MSTr – = MSE
MSTr
F = P(F F) I-1,n -I >
n I
SSE
MSE – =
J
1
J
1
2
MSE w Q
i j
ij ,I,n I ÷
÷
ø
ö
ç
ç
è
æ = a – +
xi.
xi. xj.
10G
10H
10I
F-7
F-8
Hypothesis Test with Contrasts (Unequal Sample Sizes)
1. H0: C = c0 vs. Ha:
2. Test Statistic
3. Rejection Region and P-value
Ha Rejection Region P-value P-value in R
C > c0 t ³ ta, n-I P(tn-I ³ t) 1-pt(t,n-I))
C
0
0
0
C c
C c
C c
n I
I
i 1 i
2
i
0 ~ t
J
c
MSE
Cˆ c
t –
=
å
– =
RR {F F }
F t ~ F
H : C 0 vs. H : C 0
,1,n I
1,n I
2
0 a
a –
–
= >
=
= ¹
å
=
a – ±
I
i 1 i
2
i
,n I 2 J
c
Cˆ t MSE
, J
c , Cˆ t MSE
J
c , Cˆ t MSE
I
i 1 i
2
i ,n I
I
i 1 i
2
i ,n I ÷
÷
÷
ø
ö
ç
ç
ç
è
æ
– + ¥ ÷
÷
÷
ø
ö
ç
ç
ç
è
æ
– ¥ + å å
=
a –
=
a –
. J
1
with
J
c
If C ,replace
j
I
i 1 i
2
i
j å=
= µ
ˆ MSE 2
s = r
MSTr MSE ˆ2
A
– s =
( ) 2
A
2
ij Vˆ X = sˆ + sˆ
2
A
2
2
A
ˆ ˆ
ˆ 100
s + s
s ´
10J
10K
F-9
Chapter 11 Formulas Set Section 11.1 Two-Factor ANOVA with No Replications
Notation
• A = 1st factor, I = number of levels of A
• B = 2nd factor, J = number of levels of B
• Xij = the measurement from the combination of the ith level of A and jth level of B
• xij = actual (observed) value of Xij
Two-Way Additive Fixed Model
Model equation and assumptions are
Xij = µ + ai + bj + Îij
where , and Îij’s are iid N(0,s2). The average response at the
level i of A and level j of B is
µij = E(Xij) = µ + ai + bj .
————————————————————————————–
Parameter Estimates
Factor A, level i total and average:
Factor B, level j total and average: ,
Grand Average:
Parameter Estimate
µ
ai
bj
= + + is the predicted or fitted value.
eij = xij – is a residual which estimates Îij.
0, 0
J
j 1
j
I
i 1
å i å = =
a = b =
J
x x x , x i. i.
J
j 1
i. = å ij =
=
I
x x x , x .j .j
I
i 1
.j = å ij =
=
IJ
x
x
I
i 1
J
j 1
ij
..
å å= = =
x.. µ
ˆ =
i xi. x.. aˆ = –
j x.j x.. ˆb = –
xij ˆ µ
ˆ i aˆ j
ˆb
xij ˆ
11A
11B
F-10
Hypothesis Tests
• Factor A: H0: a1 = a2 = … = aI = 0 vs. Ha: at least one ai ¹ 0
• Factor B: H0: b1 = b2 = … = bJ = 0 vs. Ha: at least one bj ¹ 0
Sums of Squares df
IJ-1
I-1
J-1
(I-1)(J-1)
ANOVA Table
Source df SS MS F P-value P-value in R
Factor
A I-1 SSA 1-pf(F,I-1,(I-1)*(J-1))
Factor
B
J-1 SSB
1-pf(F,J-1,(I-1)*(J-1))
Error (I-1)(J1)
SSE
Total IJ-1 SST
, SSE = SST – SSA – SSB
• Factor A: RR = {F=MSA/MSE > Fa,I-1,(I-1)(J-1)}
• Factor B: RR = {F=MSB/MSE > Fa,J-1,(I-1)(J-1)}
————————————————————————————-
T Method for Significant Differences
Compute
.
Apply
• wA to
or
• wB to
Block designs: ANOVA, T Method the same as above with Factor A = Blocks.
Two-Way Additive Random Model: Xij = µ + Ai + Bj + Îij
å å ( ) å å = = = =
= – = –
I
i 1
2
.. J
j 1
2
ij
I
i 1
J
j 1
2
ij .. IJ
x SST x x x
( ) IJ
x x J
1
SSA J x x
2
.. I
i 1
2
i.
I
i 1
2 = å i. – .. = å –
= =
( ) IJ
x x I
1
SSB I x x
2
.. J
j 1
2
.j
J
j 1
2 = å .j – .. = å –
= =
å å ( ) = =
= – – +
I
i 1
J
j 1
2 SSE xij xi. x.j x..
I 1
SSA
MSA – = MSE
MSA
F = P(F F) I-1,(I-1)(J -1) >
J 1
SSB
MSB – = MSE
MSB
F = P(F F) J -1,(I-1)(J -1) >
(I 1)(J 1)
SSE
MSE – – =
å å= =
=
I
i 1
J
j 1
2 SSE eij
å å = =
b – a = s + – = s = s +
J
j 1
2
j
2
I
i 1
2
i
2 2
J 1
I , E(MSB)
I 1
J
E(MSE) , E(MSA)
for factor B comparisons
I
MSE w Q
for factor A comparisons
J
MSE w Q
B ,J,(I 1)(J 1)
A ,I,(I 1)(J 1)
a – –
a – –
=
=
x1., x2.,…, xI.
x.1, x.2,…, x.J
11C
11D
F-11
Two-Way Additive Random Model: Xij = µ + Ai + Bj + Îij
where the Ai’s are iid N(0,sA2), the Bj’s are iid N(0,sB2), and Îij’s are iid N(0,s2).
tests H0: sA2 = 0 vs. Ha: sA2 ¹ 0.
tests H0: sB2 = 0 vs. Ha: sB2 ¹ 0.
Estimates:
total variance = .
—
Two-Way Additive Mixed Model: Xij = µ + Ai + bj + Îij
where the Ai’s are iid N(0,sA2), Sbj=0, and Îij’s are iid N(0,s2).
tests H0: sA2 = 0 vs. Ha: sA2 ¹ 0.
tests H0: b1 = b2 = … = bJ = 0 vs. Ha: at least one bj ¹ 0.
Estimates: , total variance =
————————————————————————————–
Section 11.2 Two-Way ANOVA with Replications
Two-Way Interaction Fixed Effects Model
Xijk = kth observation for level i of A and level j of B.
Xijk = µ + ai + bj + gij + Îijk
for i=1, …, I, j=1, …,J, k=1, …, K and where
, for all i, for all j,
and Îij’s are iid N(0,s2). The mean response at the level i of A and level j of B is
µij = E(Xij) = µ + ai + bj + gij .
————————————————————————————–
Estimates
, ,
, ,
Parameter Estimate
fitted value
residual
µ
ai
bj
gij
2
B
2 2
A
2 2 E(MSE) = s , E(MSA) = s + Js , E(MSB) = s + Is
MSE
MSA
F =
MSE
MSB
F =
I
MSB MSE , ˆ J
MSA MSE ˆ MSE, ˆ 2
B
2
A
2 – s = – s = s =
( ) 2
B
2
A
2
ij Vˆ X = sˆ + sˆ + sˆ
å
=
b – = s = s + s = s +
J
j 1
2
j
2 2
A
2 2
J 1
I
E(MSE) , E(MSA) J , E(MSB)
MSE
MSA
F =
MSE
MSB
F =
J
MSA MSE ˆ MSE, ˆ2
A
2 – s = s = ( ) 2
A
2
ij Vˆ X = sˆ + sˆ
0, 0
J
j 1
j
I
i 1
å i å = =
a = b = 0
J
j 1
å ij
=
g = 0
I
i 1
å ij
=
g =
å å å = = =
=
I
i 1
J
j 1
K
k 1
x… xijk IJK
x x … … = K
x
x
K
k 1
ijk
ij.
å
= =
J
x
x x , x
J
j 1
ij.
i..
J
j 1
K
k 1
i.. ijk
å
å å =
= =
= = I
x
x x , x
I
j 1
ij.
.j.
I
i 1
K
k 1
.j. ijk
å
å å =
= =
= =
ij i j ij xij. xˆ = µ
ˆ + aˆ + b
ˆ + ˆg =
ijk ijk xij e = x – ˆ
x… µ
ˆ =
i xi.. x… aˆ = –
j x.j. x… ˆb = –
ij xij. xi.. x.j. x… ˆg = – – +
11E
11F
11G
F-12
Hypothesis Tests
• Factor A: H0: a1 = a2 = … = aI = 0 vs. Ha: at least one ai ¹ 0
• Factor B: H0: b1 = b2 = … = bJ = 0 vs. Ha: at least one bj ¹ 0
• Interaction: H0: gij = 0 for all i,j vs. Ha: at least one gij ¹ 0
ANOVA Table (Two-Way Interaction Fixed Model)
Source df SS MS F P-value P-value in R
A
I-1 SSA 1-pf(F,I-1,I*J*(K-1))
B J-1 SSB 1-pf(F,J-1,I*J*(K-1))
Interaction (I1)(J-1)
SSAB 1-pf(F,(I-1)*(J-1),I*J*(K-1))
Error IJ(K-1) SSE
Total IJK-1 SST
• Factor A: RR = {F=MSA/MSE > Fa,I-1,IJ{K-1)}
• Factor B: RR = {F=MSB/MSE > Fa,J-1,IJ(K-1)}
• Interaction: RR = {F=MSAB/MSE > Fa,(I-1)(J-1),IJ(K-1)}
————————————————————————————–
T Method for Factor Levels (Use only when interactions are not significant.)
Note that I=# of A levels, J=# of B levels, K=# of replications.
Section 11.3 Three-Factor Fixed Effects ANOVA
Xijkl = µ + ai + bj + dk + gABij + gACik + gBCjk +gijk + Îijk
for i=1, …, I, j=1, …,J, k=1, …, K, l=1, …, L, where Îijk’s are iid N(0,s2) and
the sum of parameters over any subscript is 0:
= = = =
= = = = .
The mean response at level i of A, j of B and k of C is
µijk = µ + ai + bj + dk + gABij + gACik + gBCjk +gijk .
I 1
SSA
MSA – = MSE
MSA
F = P(F F) I-1,IJ(K -1) >
J 1
SSB
MSB – = MSE
MSB
F = P(F F) J -1,IJ(K -1) >
(I 1)(J 1)
SSAB
MSAB – – = MSE
MSAB
F = P(F F) (I-1)(J -1),IJ(K -1) >
IJ(K 1)
SSE
MSE – =
ï
ï
î
ï
ï
í
ì
=
=
a –
a –
B ,J,IJ(K 1) .1. .2. .J.
A ,I,IJ(K 1) 1.. 2.. I..
apply to x , x ,…, x IK
MSE w Q
apply to x , x ,…, x JK
MSE w Q
å a = å b = å d =
= = =
K
k 1
k
J
j 1
j
I
i 1
i
I
i 1
AB å ij
=
g
J
j 1
AB å ij
=
g
I
i 1
AC å ik
=
g å
=
g
K
k 1
AC
ik
J
j 1
BC å jk
=
g
K
k 1
BC å jk
=
g
I
i 1
å ijk
=
g
J
j 1
å ijk
=
g 0
K
k 1
å ijk
=
g =
11I
11J
11H
F-13
Test of Hypotheses
• Factor A: H0: a1 = a2 = … = aI = 0 vs. Ha: at least one ai ¹ 0
• Factor B: H0: b1 = b2 = … = bJ = 0 vs. Ha: at least one bj ¹ 0
• Factor C: H0: d1 = d2 = … = dK = 0 vs. Ha: at least one dk ¹ 0
• AB Interaction: H0: all gABij = 0 vs. Ha: at least one gABij ¹ 0
• AC Interaction: H0: all gACik = 0 vs. Ha: at least one gACik ¹ 0
• BC Interaction: H0: all gBCjk = 0 vs. Ha: at least one gBCjk ¹ 0
• ABC Interaction: H0: all gijk = 0 vs. Ha: at least one gijk ¹ 0
Assume that there are L observations from each ABC level combination (balanced data).
Total sample size is IJKL.
ANOVA Table (3 Factors Fixed Effects Model)
Source df SS MS F P-value*
A
I-1 SSA
B J-1 SSB
C K-1 SSC
AB
Interaction
(I-1)(J-1) SSAB
AC
Interaction
(I-1)(K-1) SSAC
BC
Interaction
(J-1)(K-1) SSBC
ABC
Interaction
(I-1)
´(J-1)(K-1)
SSABC
Error IJK(L-1) SSE * In R, 1-pf(F,m,n) gives P(Fm,n > F).
Total IJKL-1 SST
! There should be at least L=2 observations per treatment to test for all interactions. If L=1,
there is no MSE and, hence, no F-test of interactions. !
• Factor A: RR = {F=MSA/MSE > Fa,I-1,IJK{L-1)}
• Factor B: RR = {F=MSB/MSE > Fa,J-1,IJK(L-1)}
• Factor C: RR = {F=MSC/MSE > Fa,K-1,IJK(L-1)}
• AB Interaction: RR = {F=MSAB/MSE > Fa,(I-1)(J-1),IJK(L-1)}
• AC Interaction: RR = {F=MSAC/MSE > Fa,(I-1)(K-1),IJK(L-1)}
• BC Interaction: RR = {F=MSBC/MSE > Fa,(J-1)(K-1),IJK(L-1)}
• ABC Interaction: RR = {F=MSABC/MSE > Fa,(I-1)(J-1)(K-1),IJK(L-1)}
T Method for Factor Levels (use when no interaction is significant)
where {total reps per level} = JKL for factor A
= IKL for factor B
= IJL for factor C
Coefficient of Determination: , Adjusted R2:
I 1
SSA
MSA – = MSE
MSA
F = P(F F) I-1,IJK(L-1) >
J 1
SSB
MSB – = MSE
MSB
F = P(F F) J -1,IJK(L -1) >
K 1
SSC
MSC – = MSE
MSC
F = P(F F) K -1,IJK(L -1) >
(I 1)(J 1)
SSAB
MSAB – – = MSE
MSAB
F = P(F F) (I-1)(J -1),IJK(L-1) >
(I 1)(K 1)
SSAC
MSAC – – = MSE
MSAC
F = P(F F) (I-1)(K -1),IJK(L-1) >
(J 1)(K 1)
SSBC
MSBC – – = MSE
MSBC
F = P(F F) (J -1)(K -1),IJK(L-1) >
(I 1)(J 1)(K 1)
SSABC
MSBC – – – = MSE
MSABC
F = P(F F) (I-1)(J -1)(K -1),IJK(L -1) >
IJK(L 1)
SSE
MSE – =
total reps per level
MSE w = Qa,{# of factor levels},{MSE df} ´
SST
SSE
R = 1 – 2
SST
SSE
Radj ÷ ´
ø
ö ç
è
æ = – error df
total df 1 2
11K
F-14
Latin Squares Design
Model Assumptions
Xij(k) = µ + ai + bj + dk + eij(k)
where and eij(k)’s are iid N(0,s2).
N = # of factor levels (note that N=I=J=K)
, , ,
, , ,
Sums of Squares (Latin Squares Design)
Sums of Squares df
N2-1
N-1
N-1
N-1
(N-1)(N-2)
Note: SSE = SST – SSA – SSB – SSC
T Method for Factor Levels: For all factors, use .
————————————————————————————–
Section 11.4 2p Factorial Experiments, Factor Effects, Yates Algorithm
23 Factorial Model: Xijkl = µ + ai + bj + dk + gABij + gACik + gBCjk +gijk + Îijkl
for i=1,2, j=1,2, k=1,2, l=1, …, L
Estimates
•
• Fitted main effects of factors A, B and C
• Fitted 2-way interactions
• Fitted 3-way interactions
å ai = å bj = å dk = 0
= å
j
xi.. xij(k) = åi
x.j. xij(k) = åi,j
x..k xij(k) = åi,j
x… xij(k)
N
x x i.. i.. = N
x x .j.
.j. = N
x x ..k ..k = 2
… … N
x x =
( ) 2
2
… N
i 1
N
j 1
2
ij(k)
N
i 1
N
j 1
2 ij(k) … N
x SST = å å x – x = å å x –
= = = =
( ) 2
2
… N
i 1
2
i..
N
i 1
2 i.. … N
x x N
1
SSA = Nå x – x = å –
= =
( ) 2
2
… N
j 1
2
.j.
N
j 1
2 .j. … N
x x N
1
SSB = Nå x – x = å –
= =
( ) 2
2
… N
k 1
2
..k
N
k 1
2 ..k … N
x x N
1
SSC = Nå x – x = å –
= =
( )
2 N
i 1
N
j 1
SSE å å xij(k) xi.. x.j. x..k 2x…
= =
= – – – +
N
MSE w = Qa,N, {MSE df} ´
x…. µ
ˆ =
i i… …. j .j.. …. k x..k. x…. ˆ x x ˆ aˆ = x – x b = – d = –
.jk. .j.. ..k. …. BC
jk
i.k. i… ..k. …. AC
ik
ij.. i… .j.. …. AB
ij
ˆ x x x x
ˆ x x x x
ˆ x x x x
g = – – +
g = – – +
g = – – +
ijk xijk. xij.. xi.k. x.jk. xi… x.j.. x..k. x…. ˆg = – – – + + + –
11L
11M
F-15
Yates Algorithm
1. List sample means (xbars) in Yates standard order.
• Start with (1) then a.
• “Multiply by b” the previous treatments to get b and ab.
• “Multiply by c” the previous treatments to get c, ac, bc, abc.
etc.
There should be 2p treatments in the list.
2. The next column is obtained by adding the numbers in the previous column in pairs
and subtracting in pairs (2nd minus 1st).
Repeat this process p times.
3. Divide the pth new column by 2p. The results are the overall mean and fitted
effects (with all factors at the 2nd level).
Reverse the sign of the fitted effect if you change an odd number of subscripts.
————————————————————————————–
Section 11.4 Fractional Factorial Studies
A. Choice of 1/2q Fraction of a 2p Factorial
1. Pick any p-q factors and list all their level combinations using -‘s and +’s.
2. Pick q different groups of these “first” factors and multiply the signs of the
members of each group. Use the q products to determine the levels of the
remaining q factors.
B. Determining the “Alias Structure” of the 1/2q Fraction
Multiplication Rules:
• A*A = B*B = … = I
• I*A = A, I*B = B, etc.
1. Take the q generators and apply multiplication so that I is on the left-hand-side
of the equation.
2. Multiply (LHS x LHS and RHS x RHS) the new equations in pairs, then in triples,
then in sets of four, etc.
(2q – 1) factor products are equivalent to I. Factor effects are aliased in 2p-q
groups of 2q members.
C. Analyzing a 2p-q Fractional Factorial
1. Initially ignore the “last” q factors and treat the data as a full factorial in
the “first” p-q factors. Estimate the factor effects (e.g. using formulas in
Section 11.4 or by Yates algorithm = p-q cycles and divide last cycle by 2p-q) and
judge their statistical significance.
a. (with replication)
• Compute
or get them from the ANOVA table.
• Compute .
A 100(1-a)% CI for an effect is .
Note that an effect is judged not statistically significant at the a level
if .
b. If no replication, do a normal probability plot of fitted effects (exclude ).
2. Interpret the estimates in the light of the alias structure.
df sum of (sample size – 1)
sum of (sample size – 1)
sum of [(sample size – 1) corresponding s ] MSE
2
=
´ =
sum of reciprocals of all sample sizes
2
1
r( ) t MSE ,df p-q 2
a = a ´ ´
effect r( ) ^
± a
effect r( ) ^
.
Coefficient of Determination: , SSR = SST-SSE
Adjusted R2 = where MST = SST/(n-1)
Yi 0 1xi i = b + b + e
( )
( ) n
x x
n
x y x y ˆ 2
2 i
i
i i
i i
1
å å
å å å
–
– b = x ˆ y ˆb0 = – b1
i 0 1xi ˆ ˆ yˆ = b + b
i i yi e = y – ˆ
å ( ) å = =
= – =
n
i 1
2
i
n
i 1
2 SSE yi yˆi e i i 0 i 1
2
i x y ˆ y ˆ SSE = å y – b å – b å
n 2
SSE
MSE ˆ2
– = s =
( ) ( )
( ) ( )
( )( ) n
x y
S x x y y x y
n
y
SST or S y y y
n
x S x x x
i i
xy i i i i
2
2 i
i
2
yy i
2
2 i
i
2 xx i
å å å å
å å å
å å å
= – – = –
= – = –
= – = –
xx yy
xy
S S
S r =
y
x 1
s
s ˆ r = b
SST
SSR
SST
SSE
R 1 2 = – =
MST
MSE 1 n 2
(n 1)R 1
R
2
2
adj = – –
– – =
12A
12B
12C
F-17
Section 12.3 Inference for b1
, .
100(1-a)% CI for b1 :
Hypothesis Test
1. H0: b1 = b10, Ha: b1 > b10, b1 b10 t ³ ta,n-2 1 – P(tn-2 £ t) 1-pt(t,n-2))
b1 F) 1-pf(F,1,n-2)
Error n-2 SSE MSE
Total n-1 SST
The rejection region is {F=MSR/MSE ³ Fa,1,n-2}
————————————————————————————–
Section 12.4 CI for Mean Response µy.x and Prediction Interval at x=x*
CI for Mean Response
At x=x*, the mean response is µy.x* = b0 + b1x*.
100(1-a)% CI for µy.x*:
where
Prediction Interval
100(1-a)% PI for a response Y at x=x*:
or
ˆ MSE 2 s =
xx
2
2
ˆ S
ˆ s 1
s = b
1
ˆ ,n 2 2 1 t s ˆ a – b b ± ×
1
ˆ
1 10
s
ˆ t
b
b – b =
yˆ ,n 2 2
yˆ ± t × s a –
( ) xx
2 *
yˆ S
x x
n
1
s s – = +
( ) xx
2 *
,n 2 2 S
x x
n
1
yˆ t s 1 – ± × + + a –
2
yˆ 2
,n 2 2
yˆ ± t × s + s a –
12D
12E
F-18
Section 13.1 More on Residuals
ith residual (random version) is
•
where
• Standardized residual where
Diagnostic Plots
1. ei
* (or ei) versus xi (no pattern)
2. ei
* (or ei) versus yi (no pattern)
3. (linear)
4. normal probability plot of ei
* (or ei) (linear)
Section 13.2 Transformed Variables
• intrinsically linear models – function of x and y that can be transformed
as
y’ = b0 + b1x’
where y’ = {function of y only} and x’ = {function of x only}
Sections 13.4, 13.3 Multiple and Polynomial Regression
Model: Y = b0 + b1x1 + b2x2 + … + bkxk + e
where the e’s are independently distributed N(0,s2)
Data: (x11 , x21, …, xk1, y1), (x12 , x22, …, xk2, y2), …, (x1n , x2n, …, xkn, yn)
(Least-squares criterion) Find that minimize
.
Fitted model/value:
Estimate for s2:
where .
• n-k-1 is the SSE or MSE df.
• is the jth residual
i i Yi E = Y – ˆ
( ) ( ) ( )
( ) i i
xx
2
2 i
i i
E ~ N(mean 0, variance V E
S
x x
n
1
E E 0,V E 1
= =
ú
ú
û
ù
ê
ê
ë
é – = = s – –
= å ( – )
2 Sxx xi x
ei
i i
i s
y y
e* – ˆ = ( )
xx
i
e S
x x
n s s i
2 1
1 – = – –
yi versus yi ˆ
0 0 1 1 k k
ˆ ,…,b ˆ ,b ˆ b = b = b = b
å [ ( )] =
= – + + +
n
j 1
2 SSE yj b0 b1x1j … bkxkj
j 0 1 1j kxkj ˆ x … ˆ ˆ yˆ = b + b + + b
j 0 1 1j kxkj ˆ x … ˆ ˆ yˆ = b + b + + b
j j yj e = y – ˆ
( )
n k 1
y yˆ
n k 1
SSE
s MSE
2
2 j j
– –
– = – – = = å
13A
13B
13C
F-19
Diagnostics: Assessing Model Fit to Data
1. Plots of Residuals
standardized residual
• Residual Plots. Plot ej
* versus x1j, x2j, …, xkj, .
• Normal Probability Plot of Residuals
2. Coefficient of Multiple Determination = R2
or
where and SSR = SST – SSE
3. Radj2 = Adjusted R2:
k = {number of predictor terms (x terms) in the model}
4. Mallows Cp:
k = number of x’s (predictors) in the smaller model, n = sample size
SSEk = {fitted/smaller model’s SSE}, = {MSE of the full model}
Smaller values of Cp and close to k+1 indicate better models.
Analysis of Variance and Regression
, , and SSR = SST-SSE.
Source df SS MS F P-value P-value in R
Regression k SSR MSR MSR/MSE P(Fk,n-k-1 ³ F) 1-pf(F,k,n-k-1)
Error n-k-1 SSE MSE
Total n-1 SST
Model-Utility Test:
H0: b1 = b2 = … = bk = 0 versus Ha: at least one of the b’s is not 0
Rejection Region = {F ³ Fa,k,n-k-1}
Inference for Model Coefficients
1. Confidence Intervals
100(1-a)% CI for bi:
where is an estimate of the standard deviation of .
2. Test of Hypothesis
H0: bi = 0 versus Ha: bi ¹ 0
Test statistic ; P-value = 2*P(tn-k-1 ≥ |t|), (in R) 2*(1-pt(abs(t),n-k-1))
RR = {t ³ ta/2,n-k-1 or t ≤ -ta/2,n-k-1}
j ej
j j
e
* j
j s
y yˆ
s
e
e – = =
yj ˆ
SST
SSE
R 1 2 = – SST
SSR
R2 =
( )
2
SST = å yj – y
SST
SSE
error df
total df 1 n (k 1)
(n 1)R k
R
2
2
adj = – ´ – +
– – =
2(k 1) n s
SSE
C 2
f
k
p = + + –
2 sf
( )
2
SST = å yj – y ( )
2
j yj SSE = å y – ˆ
i
ˆ ,n k 1
2
i t s ˆ b – – b ± a ×
i sˆb i
ˆb
i
ˆ
i
s
ˆ
t
b
b =
13E
13D
F-20
More Intervals
1. Confidence Intervals for Mean Response at (x1
*, x2
* , …, xk
*)
100(1-a)% CI for µy.x*:
where and is an estimate of the standard
deviation of .
2. Prediction Interval for New Observation
100(1-a)% PI for a new response Y at (x1
*, x2
* , …, xk
*):
where s2 = MSE.
N(0,1) 100pth Percentiles (p-Quantiles)
p 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
.0 -2.33 -2.05 -1.88 -1.75 -1.64 -1.55 -1.48 -1.41 -1.34
.1 -1.28 -1.23 -1.17 -1.13 -1.08 -1.04 -0.99 -0.95 -0.92 -0.88
.2 -0.84 -0.81 -0.77 -0.74 -0.71 -0.67 -0.64 -0.61 -0.58 -0.55
.3 -0.52 -0.50 -0.47 -0.44 -0.41 -0.39 -0.36 -0.33 -0.31 -0.28
.4 -0.25 -0.23 -0.20 -0.18 -0.15 -0.13 -0.10 -0.08 -0.05 -0.03
.5 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.20 0.23
.6 0.25 0.28 0.31 0.33 0.36 0.39 0.41 0.44 0.47 0.50
.7 0.52 0.55 0.58 0.61 0.64 0.67 0.71 0.74 0.77 0.81
.8 0.84 0.88 0.92 0.95 0.99 1.04 1.08 1.13 1.17 1.23
.9 1.28 1.34 1.41 1.48 1.55 1.64 1.75 1.88 2.05 2.33
yˆ ,n k 1 2
yˆ ± t × s a – –
* k k * 0 1 1 x ˆ x … ˆ ˆ yˆ = b + b + + b syˆ
yˆ
2
yˆ 2
,n k 1 2
yˆ ± t × s + s a – –
13F
F-21
Standard Normal Probabilities (Part 1)
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
-3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002
-3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003
-3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005
-3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007
-3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010
-2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014
-2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019
-2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026
-2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036
-2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048
-2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064
-2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084
-2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110
-2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143
-2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183
-1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233
-1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294
-1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367
-1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455
-1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559
-1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681
-1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823
-1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985
-1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170
-1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379
-0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611
-0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867
-0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148
-0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451
-0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776
-0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121
-0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483
-0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859
-0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247
-0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641
F-22
Standard Normal Probabilities (Part 2)
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
+0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
+0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
+0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
+0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
+0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
+0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
+0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
+0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
+0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8079 .8106 .8133
+0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
+1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
+1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
+1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
+1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
+1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
+1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
+1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
+1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
+1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
+1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
+2.0 .9773 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
+2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
+2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
+2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
+2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
+2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
+2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
+2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
+2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
+2.9 .9981 .9982 .9983 .9983 .9984 .9984 .9985 .9985 .9986 .9986
+3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
+3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
+3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
+3.3 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9996 .9997
+3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998
F-23
Table A.5 Critical Values for the t Distribution
n
α
0.1 0.05 0.025 0.01 0.005 0.001 0.0005
1 3.078 6.314 12.706 31.821 63.657 318.309 636.619
2 1.886 2.920 4.303 6.965 9.925 22.327 31.599
3 1.638 2.353 3.182 4.541 5.841 10.215 12.924
4 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 5.893 6.869
6 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 1.323 1.721 2.080 2.518 2.831 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 1.319 1.714 2.069 2.500 2.807 3.485 3.768
24 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.450 3.725
26 1.315 1.706 2.056 2.479 2.779 3.435 3.707
27 1.314 1.703 2.052 2.473 2.771 3.421 3.690
28 1.313 1.701 2.048 2.467 2.763 3.408 3.674
29 1.311 1.699 2.045 2.462 2.756 3.396 3.659
30 1.310 1.697 2.042 2.457 2.750 3.385 3.646
32 1.309 1.694 2.037 2.449 2.738 3.365 3.622
34 1.307 1.691 2.032 2.441 2.728 3.348 3.601
36 1.306 1.688 2.028 2.434 2.719 3.333 3.582
38 1.304 1.686 2.024 2.429 2.712 3.319 3.566
40 1.303 1.684 2.021 2.423 2.704 3.307 3.551
50 1.299 1.676 2.009 2.403 2.678 3.261 3.496
60 1.296 1.671 2.000 2.390 2.660 3.232 3.460
120 1.289 1.658 1.980 2.358 2.617 3.160 3.373
Inf 1.282 1.645 1.960 2.326 2.576 3.090 3.291
F-24
Table A.9 Critical Values for the F Distributions (part 1)
denom.
df
n2 α
n1 = numerator df
1 2 3 4 5 6 7 8 9
1 0.1 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86
1 0.05 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
1 0.01 4052.18 4999.50 5403.35 5624.58 5763.65 5858.99 5928.36 5981.07 6022.47
1 0.001 405284.07 499999.50 540379.20 562499.58 576404.56 585937.11 592873.29 598144.16 602283.99
2 0.1 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38
2 0.05 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
2 0.01 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39
2 0.001 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39
3 0.1 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24
3 0.05 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
3 0.01 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35
3 0.001 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86
4 0.1 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94
4 0.05 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
4 0.01 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66
4 0.001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47
5 0.1 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32
5 0.05 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
5 0.01 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16
5 0.001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24
6 0.1 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96
6 0.05 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
6 0.01 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98
6 0.001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 18.69
7 0.1 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72
7 0.05 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
7 0.01 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72
7 0.001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 14.33
8 0.1 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56
8 0.05 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
8 0.01 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91
8 0.001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.77
9 0.1 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44
9 0.05 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
9 0.01 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
9 0.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11
10 0.1 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35
10 0.05 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
10 0.01 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94
10 0.001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.96
11 0.1 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27
11 0.05 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90
11 0.01 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63
11 0.001 19.69 13.81 11.56 10.35 9.58 9.05 8.66 8.35 8.12
12 0.1 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21
12 0.05 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80
12 0.01 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39
12 0.001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.48
F-25
Table A.9 Critical Values for the F Distributions (part 2)
denom.
df
n2 α
n1 = numerator df
10 12 15 20 25 30 40 50 60 120 1000
1 0.1 60.19 60.71 61.22 61.74 62.05 62.26 62.53 62.69 62.79 63.06 63.30
1 0.05 241.88 243.91 245.95 248.01 249.26 250.10 251.14 251.77 252.20 253.25 254.19
1 0.01 6055.85 6106.32 6157.28 6208.73 6239.83 6260.65 6286.78 6302.52 6313.03 6339.39 6362.68
1 0.001 605620.97 610667.82 615763.66 620907.67 624016.83 626098.96 628712.03 630285.38 631336.56 633972.40 636301.21
2 0.1 9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.47 9.48 9.49
2 0.05 19.40 19.41 19.43 19.45 19.46 19.46 19.47 19.48 19.48 19.49 19.49
2 0.01 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.48 99.49 99.50
2 0.001 999.40 999.42 999.43 999.45 999.46 999.47 999.47 999.48 999.48 999.49 999.50
3 0.1 5.23 5.22 5.20 5.18 5.17 5.17 5.16 5.15 5.15 5.14 5.13
3 0.05 8.79 8.74 8.70 8.66 8.63 8.62 8.59 8.58 8.57 8.55 8.53
3 0.01 27.23 27.05 26.87 26.69 26.58 26.50 26.41 26.35 26.32 26.22 26.14
3 0.001 129.25 128.32 127.37 126.42 125.84 125.45 124.96 124.66 124.47 123.97 123.53
4 0.1 3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.80 3.79 3.78 3.76
4 0.05 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.70 5.69 5.66 5.63
4 0.01 14.55 14.37 14.20 14.02 13.91 13.84 13.75 13.69 13.65 13.56 13.47
4 0.001 48.05 47.41 46.76 46.10 45.70 45.43 45.09 44.88 44.75 44.40 44.09
5 0.1 3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.15 3.14 3.12 3.11
5 0.05 4.74 4.68 4.62 4.56 4.52 4.50 4.46 4.44 4.43 4.40 4.37
5 0.01 10.05 9.89 9.72 9.55 9.45 9.38 9.29 9.24 9.20 9.11 9.03
5 0.001 26.92 26.42 25.91 25.39 25.08 24.87 24.60 24.44 24.33 24.06 23.82
6 0.1 2.94 2.90 2.87 2.84 2.81 2.80 2.78 2.77 2.76 2.74 2.72
6 0.05 4.06 4.00 3.94 3.87 3.83 3.81 3.77 3.75 3.74 3.70 3.67
6 0.01 7.87 7.72 7.56 7.40 7.30 7.23 7.14 7.09 7.06 6.97 6.89
6 0.001 18.41 17.99 17.56 17.12 16.85 16.67 16.44 16.31 16.21 15.98 15.77
7 0.1 2.70 2.67 2.63 2.59 2.57 2.56 2.54 2.52 2.51 2.49 2.47
7 0.05 3.64 3.57 3.51 3.44 3.40 3.38 3.34 3.32 3.30 3.27 3.23
7 0.01 6.62 6.47 6.31 6.16 6.06 5.99 5.91 5.86 5.82 5.74 5.66
7 0.001 14.08 13.71 13.32 12.93 12.69 12.53 12.33 12.20 12.12 11.91 11.72
8 0.1 2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.35 2.34 2.32 2.30
8 0.05 3.35 3.28 3.22 3.15 3.11 3.08 3.04 3.02 3.01 2.97 2.93
8 0.01 5.81 5.67 5.52 5.36 5.26 5.20 5.12 5.07 5.03 4.95 4.87
8 0.001 11.54 11.19 10.84 10.48 10.26 10.11 9.92 9.80 9.73 9.53 9.36
9 0.1 2.42 2.38 2.34 2.30 2.27 2.25 2.23 2.22 2.21 2.18 2.16
9 0.05 3.14 3.07 3.01 2.94 2.89 2.86 2.83 2.80 2.79 2.75 2.71
9 0.01 5.26 5.11 4.96 4.81 4.71 4.65 4.57 4.52 4.48 4.40 4.32
9 0.001 9.89 9.57 9.24 8.90 8.69 8.55 8.37 8.26 8.19 8.00 7.84
10 0.1 2.32 2.28 2.24 2.20 2.17 2.16 2.13 2.12 2.11 2.08 2.06
10 0.05 2.98 2.91 2.85 2.77 2.73 2.70 2.66 2.64 2.62 2.58 2.54
10 0.01 4.85 4.71 4.56 4.41 4.31 4.25 4.17 4.12 4.08 4.00 3.92
10 0.001 8.75 8.45 8.13 7.80 7.60 7.47 7.30 7.19 7.12 6.94 6.78
11 0.1 2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.04 2.03 2.00 1.98
11 0.05 2.85 2.79 2.72 2.65 2.60 2.57 2.53 2.51 2.49 2.45 2.41
11 0.01 4.54 4.40 4.25 4.10 4.01 3.94 3.86 3.81 3.78 3.69 3.61
11 0.001 7.92 7.63 7.32 7.01 6.81 6.68 6.52 6.42 6.35 6.18 6.02
12 0.1 2.19 2.15 2.10 2.06 2.03 2.01 1.99 1.97 1.96 1.93 1.91
12 0.05 2.75 2.69 2.62 2.54 2.50 2.47 2.43 2.40 2.38 2.34 2.30
12 0.01 4.30 4.16 4.01 3.86 3.76 3.70 3.62 3.57 3.54 3.45 3.37
12 0.001 7.29 7.00 6.71 6.40 6.22 6.09 5.93 5.83 5.76 5.59 5.44
F-26
Table A.9 Critical Values for the F Distributions (part 3)
denom.
df
n2 α
n1 = numerator df
1 2 3 4 5 6 7 8 9
13 0.1 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16
13 0.05 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
13 0.01 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19
13 0.001 17.82 12.31 10.21 9.07 8.35 7.86 7.49 7.21 6.98
14 0.1 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12
14 0.05 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
14 0.01 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03
14 0.001 17.14 11.78 9.73 8.62 7.92 7.44 7.08 6.80 6.58
15 0.1 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09
15 0.05 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
15 0.01 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89
15 0.001 16.59 11.34 9.34 8.25 7.57 7.09 6.74 6.47 6.26
16 0.1 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06
16 0.05 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54
16 0.01 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78
16 0.001 16.12 10.97 9.01 7.94 7.27 6.80 6.46 6.19 5.98
17 0.1 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03
17 0.05 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49
17 0.01 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68
17 0.001 15.72 10.66 8.73 7.68 7.02 6.56 6.22 5.96 5.75
18 0.1 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00
18 0.05 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
18 0.01 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60
18 0.001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.56
19 0.1 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98
19 0.05 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
19 0.01 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52
19 0.001 15.08 10.16 8.28 7.27 6.62 6.18 5.85 5.59 5.39
20 0.1 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96
20 0.05 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
20 0.01 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46
20 0.001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 5.24
21 0.1 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95
21 0.05 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
21 0.01 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40
21 0.001 14.59 9.77 7.94 6.95 6.32 5.88 5.56 5.31 5.11
22 0.1 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93
22 0.05 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34
22 0.01 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35
22 0.001 14.38 9.61 7.80 6.81 6.19 5.76 5.44 5.19 4.99
23 0.1 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92
23 0.05 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32
23 0.01 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30
23 0.001 14.20 9.47 7.67 6.70 6.08 5.65 5.33 5.09 4.89
24 0.1 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91
24 0.05 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
24 0.01 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26
24 0.001 14.03 9.34 7.55 6.59 5.98 5.55 5.23 4.99 4.80
F-27
Table A.9 Critical Values for the F Distributions (part 4)
denom.
df
n2 α
n1 = numerator df
10 12 15 20 25 30 40 50 60 120 1000
13 0.1 2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.92 1.90 1.88 1.85
13 0.05 2.67 2.60 2.53 2.46 2.41 2.38 2.34 2.31 2.30 2.25 2.21
13 0.01 4.10 3.96 3.82 3.66 3.57 3.51 3.43 3.38 3.34 3.25 3.18
13 0.001 6.80 6.52 6.23 5.93 5.75 5.63 5.47 5.37 5.30 5.14 4.99
14 0.1 2.10 2.05 2.01 1.96 1.93 1.91 1.89 1.87 1.86 1.83 1.80
14 0.05 2.60 2.53 2.46 2.39 2.34 2.31 2.27 2.24 2.22 2.18 2.14
14 0.01 3.94 3.80 3.66 3.51 3.41 3.35 3.27 3.22 3.18 3.09 3.02
14 0.001 6.40 6.13 5.85 5.56 5.38 5.25 5.10 5.00 4.94 4.77 4.62
15 0.1 2.06 2.02 1.97 1.92 1.89 1.87 1.85 1.83 1.82 1.79 1.76
15 0.05 2.54 2.48 2.40 2.33 2.28 2.25 2.20 2.18 2.16 2.11 2.07
15 0.01 3.80 3.67 3.52 3.37 3.28 3.21 3.13 3.08 3.05 2.96 2.88
15 0.001 6.08 5.81 5.54 5.25 5.07 4.95 4.80 4.70 4.64 4.47 4.33
16 0.1 2.03 1.99 1.94 1.89 1.86 1.84 1.81 1.79 1.78 1.75 1.72
16 0.05 2.49 2.42 2.35 2.28 2.23 2.19 2.15 2.12 2.11 2.06 2.02
16 0.01 3.69 3.55 3.41 3.26 3.16 3.10 3.02 2.97 2.93 2.84 2.76
16 0.001 5.81 5.55 5.27 4.99 4.82 4.70 4.54 4.45 4.39 4.23 4.08
17 0.1 2.00 1.96 1.91 1.86 1.83 1.81 1.78 1.76 1.75 1.72 1.69
17 0.05 2.45 2.38 2.31 2.23 2.18 2.15 2.10 2.08 2.06 2.01 1.97
17 0.01 3.59 3.46 3.31 3.16 3.07 3.00 2.92 2.87 2.83 2.75 2.66
17 0.001 5.58 5.32 5.05 4.78 4.60 4.48 4.33 4.24 4.18 4.02 3.87
18 0.1 1.98 1.93 1.89 1.84 1.80 1.78 1.75 1.74 1.72 1.69 1.66
18 0.05 2.41 2.34 2.27 2.19 2.14 2.11 2.06 2.04 2.02 1.97 1.92
18 0.01 3.51 3.37 3.23 3.08 2.98 2.92 2.84 2.78 2.75 2.66 2.58
18 0.001 5.39 5.13 4.87 4.59 4.42 4.30 4.15 4.06 4.00 3.84 3.69
19 0.1 1.96 1.91 1.86 1.81 1.78 1.76 1.73 1.71 1.70 1.67 1.64
19 0.05 2.38 2.31 2.23 2.16 2.11 2.07 2.03 2.00 1.98 1.93 1.88
19 0.01 3.43 3.30 3.15 3.00 2.91 2.84 2.76 2.71 2.67 2.58 2.50
19 0.001 5.22 4.97 4.70 4.43 4.26 4.14 3.99 3.90 3.84 3.68 3.53
20 0.1 1.94 1.89 1.84 1.79 1.76 1.74 1.71 1.69 1.68 1.64 1.61
20 0.05 2.35 2.28 2.20 2.12 2.07 2.04 1.99 1.97 1.95 1.90 1.85
20 0.01 3.37 3.23 3.09 2.94 2.84 2.78 2.69 2.64 2.61 2.52 2.43
20 0.001 5.08 4.82 4.56 4.29 4.12 4.00 3.86 3.77 3.70 3.54 3.40
21 0.1 1.92 1.87 1.83 1.78 1.74 1.72 1.69 1.67 1.66 1.62 1.59
21 0.05 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.94 1.92 1.87 1.82
21 0.01 3.31 3.17 3.03 2.88 2.79 2.72 2.64 2.58 2.55 2.46 2.37
21 0.001 4.95 4.70 4.44 4.17 4.00 3.88 3.74 3.64 3.58 3.42 3.28
22 0.1 1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.65 1.64 1.60 1.57
22 0.05 2.30 2.23 2.15 2.07 2.02 1.98 1.94 1.91 1.89 1.84 1.79
22 0.01 3.26 3.12 2.98 2.83 2.73 2.67 2.58 2.53 2.50 2.40 2.32
22 0.001 4.83 4.58 4.33 4.06 3.89 3.78 3.63 3.54 3.48 3.32 3.17
23 0.1 1.89 1.84 1.80 1.74 1.71 1.69 1.66 1.64 1.62 1.59 1.55
23 0.05 2.27 2.20 2.13 2.05 2.00 1.96 1.91 1.88 1.86 1.81 1.76
23 0.01 3.21 3.07 2.93 2.78 2.69 2.62 2.54 2.48 2.45 2.35 2.27
23 0.001 4.73 4.48 4.23 3.96 3.79 3.68 3.53 3.44 3.38 3.22 3.08
24 0.1 1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.62 1.61 1.57 1.54
24 0.05 2.25 2.18 2.11 2.03 1.97 1.94 1.89 1.86 1.84 1.79 1.74
24 0.01 3.17 3.03 2.89 2.74 2.64 2.58 2.49 2.44 2.40 2.31 2.22
24 0.001 4.64 4.39 4.14 3.87 3.71 3.59 3.45 3.36 3.29 3.14 2.99
F-28
Table A.9 Critical Values for the F Distributions (part 5)
denom.
df
n2 α
n1 = numerator df
1 2 3 4 5 6 7 8 9
25 0.1 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89
25 0.05 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28
25 0.01 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22
25 0.001 13.88 9.22 7.45 6.49 5.89 5.46 5.15 4.91 4.71
26 0.1 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88
26 0.05 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27
26 0.01 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18
26 0.001 13.74 9.12 7.36 6.41 5.80 5.38 5.07 4.83 4.64
27 0.1 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87
27 0.05 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25
27 0.01 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15
27 0.001 13.61 9.02 7.27 6.33 5.73 5.31 5.00 4.76 4.57
28 0.1 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87
28 0.05 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24
28 0.01 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12
28 0.001 13.50 8.93 7.19 6.25 5.66 5.24 4.93 4.69 4.50
29 0.1 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86
29 0.05 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
29 0.01 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09
29 0.001 13.39 8.85 7.12 6.19 5.59 5.18 4.87 4.64 4.45
30 0.1 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85
30 0.05 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21
30 0.01 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07
30 0.001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.39
40 0.1 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79
40 0.05 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12
40 0.01 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89
40 0.001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 4.02
50 0.1 2.81 2.41 2.20 2.06 1.97 1.90 1.84 1.80 1.76
50 0.05 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07
50 0.01 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78
50 0.001 12.22 7.96 6.34 5.46 4.90 4.51 4.22 4.00 3.82
60 0.1 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74
60 0.05 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
60 0.01 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72
60 0.001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.69
100 0.1 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.69
100 0.05 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97
100 0.01 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59
100 0.001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.44
200 0.1 2.73 2.33 2.11 1.97 1.88 1.80 1.75 1.70 1.66
200 0.05 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93
200 0.01 6.76 4.71 3.88 3.41 3.11 2.89 2.73 2.60 2.50
200 0.001 11.15 7.15 5.63 4.81 4.29 3.92 3.65 3.43 3.26
1000 0.1 2.71 2.31 2.09 1.95 1.85 1.78 1.72 1.68 1.64
1000 0.05 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89
1000 0.01 6.66 4.63 3.80 3.34 3.04 2.82 2.66 2.53 2.43
1000 0.001 10.89 6.96 5.46 4.65 4.14 3.78 3.51 3.30 3.13
F-29
Table A.9 Critical Values for the F Distributions (part 6)
denom.
df
n2 α
n1 = numerator df
10 12 15 20 25 30 40 50 60 120 1000
25 0.1 1.87 1.82 1.77 1.72 1.68 1.66 1.63 1.61 1.59 1.56 1.52
25 0.05 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.84 1.82 1.77 1.72
25 0.01 3.13 2.99 2.85 2.70 2.60 2.54 2.45 2.40 2.36 2.27 2.18
25 0.001 4.56 4.31 4.06 3.79 3.63 3.52 3.37 3.28 3.22 3.06 2.91
26 0.1 1.86 1.81 1.76 1.71 1.67 1.65 1.61 1.59 1.58 1.54 1.51
26 0.05 2.22 2.15 2.07 1.99 1.94 1.90 1.85 1.82 1.80 1.75 1.70
26 0.01 3.09 2.96 2.81 2.66 2.57 2.50 2.42 2.36 2.33 2.23 2.14
26 0.001 4.48 4.24 3.99 3.72 3.56 3.44 3.30 3.21 3.15 2.99 2.84
27 0.1 1.85 1.80 1.75 1.70 1.66 1.64 1.60 1.58 1.57 1.53 1.50
27 0.05 2.20 2.13 2.06 1.97 1.92 1.88 1.84 1.81 1.79 1.73 1.68
27 0.01 3.06 2.93 2.78 2.63 2.54 2.47 2.38 2.33 2.29 2.20 2.11
27 0.001 4.41 4.17 3.92 3.66 3.49 3.38 3.23 3.14 3.08 2.92 2.78
28 0.1 1.84 1.79 1.74 1.69 1.65 1.63 1.59 1.57 1.56 1.52 1.48
28 0.05 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.79 1.77 1.71 1.66
28 0.01 3.03 2.90 2.75 2.60 2.51 2.44 2.35 2.30 2.26 2.17 2.08
28 0.001 4.35 4.11 3.86 3.60 3.43 3.32 3.18 3.09 3.02 2.86 2.72
29 0.1 1.83 1.78 1.73 1.68 1.64 1.62 1.58 1.56 1.55 1.51 1.47
29 0.05 2.18 2.10 2.03 1.94 1.89 1.85 1.81 1.77 1.75 1.70 1.65
29 0.01 3.00 2.87 2.73 2.57 2.48 2.41 2.33 2.27 2.23 2.14 2.05
29 0.001 4.29 4.05 3.80 3.54 3.38 3.27 3.12 3.03 2.97 2.81 2.66
30 0.1 1.82 1.77 1.72 1.67 1.63 1.61 1.57 1.55 1.54 1.50 1.46
30 0.05 2.16 2.09 2.01 1.93 1.88 1.84 1.79 1.76 1.74 1.68 1.63
30 0.01 2.98 2.84 2.70 2.55 2.45 2.39 2.30 2.25 2.21 2.11 2.02
30 0.001 4.24 4.00 3.75 3.49 3.33 3.22 3.07 2.98 2.92 2.76 2.61
40 0.1 1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.48 1.47 1.42 1.38
40 0.05 2.08 2.00 1.92 1.84 1.78 1.74 1.69 1.66 1.64 1.58 1.52
40 0.01 2.80 2.66 2.52 2.37 2.27 2.20 2.11 2.06 2.02 1.92 1.82
40 0.001 3.87 3.64 3.40 3.14 2.98 2.87 2.73 2.64 2.57 2.41 2.25
50 0.1 1.73 1.68 1.63 1.57 1.53 1.50 1.46 1.44 1.42 1.38 1.33
50 0.05 2.03 1.95 1.87 1.78 1.73 1.69 1.63 1.60 1.58 1.51 1.45
50 0.01 2.70 2.56 2.42 2.27 2.17 2.10 2.01 1.95 1.91 1.80 1.70
50 0.001 3.67 3.44 3.20 2.95 2.79 2.68 2.53 2.44 2.38 2.21 2.05
60 0.1 1.71 1.66 1.60 1.54 1.50 1.48 1.44 1.41 1.40 1.35 1.30
60 0.05 1.99 1.92 1.84 1.75 1.69 1.65 1.59 1.56 1.53 1.47 1.40
60 0.01 2.63 2.50 2.35 2.20 2.10 2.03 1.94 1.88 1.84 1.73 1.62
60 0.001 3.54 3.32 3.08 2.83 2.67 2.55 2.41 2.32 2.25 2.08 1.92
100 0.1 1.66 1.61 1.56 1.49 1.45 1.42 1.38 1.35 1.34 1.28 1.22
100 0.05 1.93 1.85 1.77 1.68 1.62 1.57 1.52 1.48 1.45 1.38 1.30
100 0.01 2.50 2.37 2.22 2.07 1.97 1.89 1.80 1.74 1.69 1.57 1.45
100 0.001 3.30 3.07 2.84 2.59 2.43 2.32 2.17 2.08 2.01 1.83 1.64
200 0.1 1.63 1.58 1.52 1.46 1.41 1.38 1.34 1.31 1.29 1.23 1.16
200 0.05 1.88 1.80 1.72 1.62 1.56 1.52 1.46 1.41 1.39 1.30 1.21
200 0.01 2.41 2.27 2.13 1.97 1.87 1.79 1.69 1.63 1.58 1.45 1.30
200 0.001 3.12 2.90 2.67 2.42 2.26 2.15 2.00 1.90 1.83 1.64 1.43
1000 0.1 1.61 1.55 1.49 1.43 1.38 1.35 1.30 1.27 1.25 1.18 1.08
1000 0.05 1.84 1.76 1.68 1.58 1.52 1.47 1.41 1.36 1.33 1.24 1.11
1000 0.01 2.34 2.20 2.06 1.90 1.79 1.72 1.61 1.54 1.50 1.35 1.16
1000 0.001 2.99 2.77 2.54 2.30 2.14 2.02 1.87 1.77 1.69 1.49 1.22
F-30
Table A.10 Critical Values for Studentized Range Distribution (Tukey’s Q)
n α
m
2 3 4 5 6 7 8 9 10 11 12
5 0.05 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.17 7.32
5 0.01 5.70 6.98 7.80 8.42 8.91 9.32 9.67 9.97 10.24 10.48 10.70
6 0.05 3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49 6.65 6.79
6 0.01 5.24 6.33 7.03 7.56 7.97 8.32 8.61 8.87 9.10 9.30 9.48
7 0.05 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.30 6.43
7 0.01 4.95 5.92 6.54 7.00 7.37 7.68 7.94 8.17 8.37 8.55 8.71
8 0.05 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.05 6.18
8 0.01 4.75 5.64 6.20 6.62 6.96 7.24 7.47 7.68 7.86 8.03 8.18
9 0.05 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 5.87 5.98
9 0.01 4.60 5.43 5.96 6.35 6.66 6.91 7.13 7.33 7.49 7.65 7.78
10 0.05 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.72 5.83
10 0.01 4.48 5.27 5.77 6.14 6.43 6.67 6.87 7.05 7.21 7.36 7.49
11 0.05 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 5.61 5.71
11 0.01 4.39 5.15 5.62 5.97 6.25 6.48 6.67 6.84 6.99 7.13 7.25
12 0.05 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 5.51 5.61
12 0.01 4.32 5.05 5.50 5.84 6.10 6.32 6.51 6.67 6.81 6.94 7.06
13 0.05 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 5.43 5.53
13 0.01 4.26 4.96 5.40 5.73 5.98 6.19 6.37 6.53 6.67 6.79 6.90
14 0.05 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 5.36 5.46
14 0.01 4.21 4.89 5.32 5.63 5.88 6.08 6.26 6.41 6.54 6.66 6.77
15 0.05 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 5.31 5.40
15 0.01 4.17 4.84 5.25 5.56 5.80 5.99 6.16 6.31 6.44 6.55 6.66
16 0.05 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.26 5.35
16 0.01 4.13 4.79 5.19 5.49 5.72 5.92 6.08 6.22 6.35 6.46 6.56
17 0.05 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11 5.21 5.31
17 0.01 4.10 4.74 5.14 5.43 5.66 5.85 6.01 6.15 6.27 6.38 6.48
18 0.05 2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07 5.17 5.27
18 0.01 4.07 4.70 5.09 5.38 5.60 5.79 5.94 6.08 6.20 6.31 6.41
19 0.05 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04 5.14 5.23
19 0.01 4.05 4.67 5.05 5.33 5.55 5.73 5.89 6.02 6.14 6.25 6.34
20 0.05 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.11 5.20
20 0.01 4.02 4.64 5.02 5.29 5.51 5.69 5.84 5.97 6.09 6.19 6.28
24 0.05 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 5.01 5.10
24 0.01 3.96 4.55 4.91 5.17 5.37 5.54 5.69 5.81 5.92 6.02 6.11
30 0.05 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82 4.92 5.00
30 0.01 3.89 4.45 4.80 5.05 5.24 5.40 5.54 5.65 5.76 5.85 5.93
40 0.05 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73 4.82 4.90
40 0.01 3.82 4.37 4.70 4.93 5.11 5.26 5.39 5.50 5.60 5.69 5.76
60 0.05 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.73 4.81
60 0.01 3.76 4.28 4.59 4.82 4.99 5.13 5.25 5.36 5.45 5.53 5.60
120 0.05 2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56 4.64 4.71
120 0.01 3.70 4.20 4.50 4.71 4.87 5.01 5.12 5.21 5.30 5.37 5.44
Inf 0.05 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55 4.62
Inf 0.01 3.64 4.12 4.40 4.60 4.76 4.88 4.99 5.08 5.16 5.23 5.29
The post A 2!”# fractional factorial involving factors A, B, C, D, E and F is to be run. Practitioners have these two sets of generators in mind appeared first on Versed Writers.