3
A medical device manufacturer produces vascular grafts using a randomized block design (RBD) with 4 extrusion pressure levels and 6 resin batches (blocks). The percentage of non-defective tubes is tabulated below with one missing observation $x$:
The value of $x$ by employing missing plot technique in RBD setup is:
| Pressure | R1 | R2 | R3 | R4 | R5 | R6 | Treatment Total |
|---|---|---|---|---|---|---|---|
| T1 | 90.3 | 89.2 | 98.2 | 93.9 | 87.4 | 97.9 | 556.9 |
| T2 | 92.5 | 89.5 | 90.6 | 94.7 | 87.0 | 95.8 | 550.1 |
| T3 | 85.5 | 90.8 | 89.6 | 86.2 | 88.0 | 93.4 | 533.5 |
| T4 | 82.5 | 89.5 | 85.6 | 87.4 | 78.9 | $x$ | $423.9 + x$ |
| Block Total | 350.8 | 359.0 | 364.0 | 362.2 | 341.3 | $287.1 + x$ | $2064.4 + x$ |
EXPLANATION
Linear Estimation, Gauss-Markov Theorem & Bas
- Yates' Missing Plot Technique in Randomized Block Design (RBD):
- Number of treatments ($p$) = 4
- Number of blocks ($q$) = 6
- Let $x$ be the single missing observation in treatment 4, block 6.
- From the table of known values:
- Total of known observations in treatment 4: $T' = 423.9$
- Total of known observations in block 6: $B' = 287.1$
- Grand total of all known observations: $G' = 2064.4$
- Yates' formula for estimating a single missing value in RBD:
- Substituting the values:
- Conclusion: The value of $x$ is 90.25 (Option 4).
Correct option: (4) 90.25
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