Stat401D
Lab#9 Key
Spring 2016
Problem 1a - Response Pull Strength
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.981137
0.979423
2.288047
29.0328
25
Analysis of Variance
Source
Model
Error
C. Total
DF
2
22
24
Sum of Squares
5990.7712
115.1735
6105.9447
Mean Square
2995.39
5.24
F Ratio
572.1672
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
Wire Length
Die Height
Estimate
2.2637914
2.7442696
0.0125278
Std Error
1.060066
0.093524
0.002798
t Ratio
2.14
29.34
4.48
Prob>|t|
0.0441*
<.0001*
0.0002*
We reject the null hypothesis and conclude that at least one of Wire Length, Die Height are linearly
related to the pull strength. The R^2 value is very high (.9811), but this does not mean our model fits the
data – we should consider residual plots and /or fit higher-order models and test whether they are a
significant improvement to be sure.
The residual plots suggest that there is curvature in the response that our model does not account for.
Considering the plots of residuals by x, we can see that it is likely this curvature occurs in relation to the
wire length. We may also wish to test for an interaction, as seen in the plot below, which has quadratic
fit lines for wire length, drawn separately for different ranges of die height. There may be an interaction
between die height and wire length, and there may even be an interaction between a quadratic effect
for wire length and die height (since the parabolic fits are not all of the same concavity).
Observations 15, 17, and 18 have high influence. Observation 9, 15, 17 have a high studentized residual.
Observations 17, 18 have large hat values. A wire length of 20 is somewhat of an outlier in x2 (obs 17),
and observation 18 is an outlier in the joint distribution of x1 and x2, that is, it’s an unusual combination
of wire length and die height, irrespective of its pull strength.
Full Model - Response Pull Strength
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.989811
0.987129
1.809544
29.0328
25
Analysis of Variance
Source
Model
Error
C. Total
DF
5
19
24
Sum of Squares
6043.7302
62.2145
6105.9447
Mean Square
1208.75
3.27
F Ratio
369.1448
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
Estimate
2.6279459
Std Error
1.283573
t Ratio
2.05
Prob>|t|
0.0547
Term
Wire Length
Die Height
(Wire Length-8.24)*(Die Height-331.76)
(Wire Length-8.24)*(Wire Length-8.24)
(Wire Length-8.24)*(Wire Length-8.24)
*(Die Height-331.76)
Estimate
2.6011491
0.011937
0.0011534
0.0213504
Std Error
0.08579
0.003031
0.000459
0.019778
t Ratio
30.32
3.94
2.51
1.08
Prob>|t|
<.0001*
0.0009*
0.0212*
0.2939
0.0000031
7.943e-5
0.04
0.9693
Reduced Model - Response Pull Strength
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.986426
0.984486
1.986679
29.0328
25
Analysis of Variance
Source
Model
Error
C. Total
DF
3
21
24
Sum of Squares
6023.0600
82.8847
6105.9447
Mean Square
2007.69
3.95
F Ratio
508.6752
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
Wire Length
Die Height
(Wire Length-8.24)*(Wire Length-8.24)
Estimate
2.4605969
2.6618372
0.0103749
0.0428265
Std Error
0.923009
0.086168
0.002544
0.014973
t Ratio
2.67
30.89
4.08
2.86
Prob>|t|
0.0145*
<.0001*
0.0005*
0.0094*
We construct the F statistic as F = (SSEReduced-SSEFull)/2/MSEFull = (82.8847-62.2145)/2/(3.27) = 3.161 on 2,
19 degrees of freedom. The rejection region is F>3.52189, and our computed F statistic is not in the
rejection region. We fail to reject the null hypothesis (that B4=B5=0) and thus conclude that the reduced
model is preferable because it explains a similar amount of variation with fewer parameters.
Stat 401D
Lab #9
Spring 2016
Prob#2
(a)
Source
d.f .
SS
MS
F
Lubricant
3
47.84375
15.9479
5.5557
Error
28
80.37500
2.8705
C. Total
31
128.21875
Test
H 0 : µ=
µ=
µ=
µ4
1
2
3
H a : at least
at α = .05 .
vs.
Reject the null hypothesis
(b)
The sample means are :
y1 = 9.75 , y2 = 8.0 , y3 = 7.124 , y4 = 6.5
LSD=
2.048 × 2.8705 × 2 / 8 = 1.735
Blu-Tek
Z907
L43
SP51
6.5
7.125
8.0
------------------
9.75
From JMP:
LSMeans Differences Student's t
α=0.050
Level
1
2
3
4
t=2.04841
A
B
B
B
Least Sq Mean
9.7500000
8.0000000
7.1250000
6.5000000
Levels not connected by same letter are significantly different.
(c)
W=
3.87 × 2.8705 × 1/ 8 = 2.32
Blu-Tek
Z907
L43
SP51
6.5
7.125
8.0 9.75
---------------------------
p-value
<.0001
one inequality. Since p-value<.05,
From JMP:
LSMeans Differences Tukey HSD
α= 0.050 Q= 2.73031 <<<<<<<Note :This Q value is different from the value from the table. i.e. 3.87/ 2 =2.736
Level
1
2
3
4
A
A
B
B
B
Least Sq Mean
9.7500000
8.0000000
7.1250000
6.5000000
Levels not connected by same letter are significantly different.
(d)
Hypothesis
H 0 :1/ 2( µ1 + µ2 ) − 1/ 2( µ3 + µ4 ) =
0
Contrast :
=
1 1 −1 −1
1
Estimate:
ˆ 1 = 9.75 + 8.0 − 7.125 − 6.5 = 4.125
12 + 12 + 12 + 12
4
2.8705 = (1.694255)
=
  1.198
8
8
Standard Error
=
=
V (ˆ 1 )
Thus the t-statistic is:
=
tc
4.125
= 3.4432 and the percentile from the t-table is t.025,28 = 2.048
1.198
Thus R.R. is t > 2.048 ; Thus the null hypothesis is rejected at α = .05 since
3.4434 is in the rejection region.
From JMP output:
t-statistic = 3.4432
p-value = .0018
Reject null hypothesis at α = .05
Hand computations for the other three comparisons are not shown here but those
computations are similar to above and are summarized below.
Contrast
SP51
L43
Z907
Blu-Tek
Est
s.e.
t
p-value
i)
1
1
-1
-1
4.125
1.198
3.4432
.0018
ii)
1
-1
1
-1
2.375
1.198
1.98
.0573
iii)
1
-1
0
0
1.75
0.8471
2.0658
.0482
iv)
0
0
1
-1
0.625
0.8471
0.7378
.4668
The results from JMP are shown below.
Contrast
Test Detail
1
2
3
4
Estimate
Std
Error
t Ratio
Prob>|t|
SS
0.5
0.5
1
0
0.5
-0.5
-1
0
-0.5
0.5
0
1
-0.5
-0.5
0
-1
2.0625 1.1875
1.75 0.625
0.599 0.599 0.8471 0.8471
3.4432 1.9824 2.0658 0.7378
0.0018 0.0573 0.0482 0.4668
34.031 11.281 12.25 1.5625
Summary Statement
The data show that there is a significant difference between the average effects of petroleum distillate
based lubricants from those of synthetic oil based lubricants on the wear of the machine part. The
average wear appear to be lower for the synthetic oil based lubricants. There is no significant difference
between the average effects of silicon additives vs. the hi-tech detergent additives on machine part
wear. There is a significant difference between the average effects of silicon additive vs. the hi-tech
detergent additive when used with petroleum distillate base as opposed to synthetic oil base, when the
two additive effects were not significantly different.