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Unformatted text preview: Missing Values In A Randemized Blecggépﬁggﬁtm_ Texﬁ) "Eats . «‘7 Randomized Cempiete Biesk Design for the Hardness
Testing Experiment with One Missing Vaiue Specimen {Bieeki Tyee et Tip 1 2 3 4
‘i —2 =t 1 5
2 4 2 x 4
3 3 t 0 2
4 2 ‘i 5 7 335., + by}, " 3’", , (547)
(aTIXbl) ><
II Fer the data in Table 57 we find that ‘yz' 2 i, y3 = 6, and y” = 17.
Therefore, from Equation 5—17, ’
4(1) + 4(6) 17 122 X y” (36)
The usual analysis of variance may now be performed, using y23 =
1.22, and reducing the error degrees of freedom by one. The analysis
of variance is shown in Table 58. Compare the results of this
approximate analysis with the results obtained for the full data set
(Table 56).
Tablg§;§:e8?éi‘iAﬁproximate Analysis of Variance for Example 41 with one
Missing Value Source of Sum of Degrees of Mean Variation Squares Freedom Square FO
Type of tip 39.98 3 13.33 17.312a
Specimens (blocks) 79.53 3 26.51
Error 6.22 8 0.78
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%? SMs‘ylréiae/hr rcwmw Mag—Q. 6%} CL CﬁWM COMPOSJ‘ (£32? (C(13)) MOoQaE gym 3K arms. Nozzle Type (A) 10122“ Syrup Loss Data for Example 10——1 (units are cubic centimeters ~ 70) ' 1 2 3
Pressure (in psi) Speed on RPM) (B)
(C) 100 120 140 100 120 140 100 120 140
10 ~35 ~45 ~40 17 ~65 20 ~39 ~55 15
~25 ~60 15 24 ~58 4 ~35 ~67 ~30
15 110 ~10 80 55 ~55 110 90 ~28 110
75 30 54 120 ~44 44 113 ~26 135
20 4 ~40 31 ~23 ~64 ~20 ~30 ~61 54
5 ~30 36 ~5 ~62 ~31 ~55 ~52 4
W
Examptelﬂel A machine is used to ﬁll 5gallon metal containers with soft drink syrup. The variable
of interest is the amount of syrup loss due to frothing. Three factors are thought to
inﬂuence frothing: the nozzle design (A), the ﬁlling speed (B), and the operating
pressure (C). Three nozzles, three ﬁlling speeds, and three pressures are chosen and
two replicates of a 33 factorial experiment are run. The coded data are shown in Table 10—2. The analysis of variance for the syrup loss data is shown in Table 10—3. The
sums of squares have been computed by the usual methods. We see that the ﬁlling
speed and operating pressure are statistically signiﬁcant. All three two—factor interac—
tions are also signiﬁcant. The two—factor interactions are analyzed graphically in
Figure 10—4. The middle level of speed gives the best performance, nozzle types 2
and 3, and either the low (10 psi) or high (20 psi) pressure seem most effective in
reducing syrup loss. Table 10—3 Analysis of Variance for Syrup Loss Data Source of
Variation A, nozzle
B, speed
C, pressure
AB AC BC ABC
Error
Total Sum of
Squares 993.77
61,190.33
69,105.33 6,300.90
7,513.90
12,854.34
4,628.76
11,515.50
174,102.83 Degrees of
Freedom 2 OODAhNN 27
53 I
Mean Square F0 P—Value
496.89 1.17 0.3256
30,595.17 71.74 <0.0001
34,552.67 81.01 <0.0001
1,575.22 3.69 0.0383
1,878.47 4.40 0.0222
3,213.58 7.53 0.0025
578.60 1.36 0.2737 426.50 M ”ml U2 lv uu‘ y—m ,M. 10—1 The at Factorial Design 443 400 400 2 200 1’ 200
53 $3
8 53
"E3 s
cc: 0 o 0
x x
<2 <2 —200 ~200 400 ~400 1 2 3 ‘l 2 3
Nozzle type (A) Nozzle type (Al
(a) lb) 600 400 200 B x C cell totals —200  400
100 120 140 Speed in rpm (B) (C) Figure 10—4. Two—factor interactions for Example 101. Example 10—1 illustrates a situation where the three—level design often ﬁnds
some application; one or more of the factors is qualitative, naturally taking on
three levels, and the remaining factors are quantitative. In this example, suppose
that there are only three nozzle designs that are of interest. This is clearly, then,
a qualitative factor that requires three levels. The ﬁlling speed and the operating
pressure are quantitative factors. Therefore, we could ﬁt a quadratic model such
as Equation 10—1 in the two factors speed and pressure at each level of the nozzle factor. ...
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 Spring '08
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