# Design of Experiments (DOE)

Problem #1.
The data correspond to yields (in tons per hectare) of a pasture with 3 levels of nitrogen fertilizers, the design was randomized, applied in 5 plots per treatment (blocks).
You want to know if the performance was different in the treatments.
Check the hypothesis with = 0.01%

Nitrogen Levels

1

2

3

Plot 1

14.823

25.151

32.605

Plot 2

14.676

25.401

32.46

Plot 3

14.72

25.131

32.256

Plot 4

14.514

25.031

32.669

Plot 5

15.065

25.277

32.111
Problem #2.
A product development engineer wants to maximize the tensile strength of a new synthetic fiber that will be used to make shirts. From experience, it appears that strength (or strength) is influenced by the% cotton present in the fiber.
It is also suspected that high cotton values negatively affect other quality qualities that are desired (for example: that the fiber can receive a permanent press treatment).
Faced with this situation, the engineer decides to take five samples for different levels of cotton and measure the strength of the fibers thus produced.

Cotton %

Observations (strengths of the 25 manufactured fibers)

Total

Average

15%

7

7

15

11

9

49

9.8

20%

12

17

12

18

18

77

15.4

25%

14

18

18

19

19

88

17.6

30%

19

25

22

19

23

108

21.6

35%

7

10

11

15

11

54

10.8

Total sum of the 25 Strength values obtained

376

15.04

Problem # 3 Full factorial example

Data Source

This example uses data from a NIST   high performance ceramics experiment

This data set was taken from an   experiment that was performed a few years ago at NIST by Said Jahanmir of the   Ceramics Division in the Material Science and Engineering Laboratory. The   original analysis was performed primarily by Lisa Gill of the Statistical   Engineering Division. Do the DOE set up and analysis and find the significant   factors affecting your response.
The   original data set was part of a high performance ceramics experiment with the   goal of characterizing the effect of grinding parameters on sintered   reaction-bonded silicon nitride, reaction bonded silicone nitride, and   sintered silicon nitride.

Description of Experiment:   Response and Factors

Response and factor variables

Purpose: To determine the effect   of machining factors on ceramic strength

Response variable = mean (over 15 repetitions) of the ceramic strength

Number of observations = 32 (a complete 25 factorial design)
Response Variable Y = Mean (over   15 reps) of Ceramic Strength

Factor 1 = Table Speed (2 levels: slow (.025 m/s) and fast (.125 m/s))

Factor 2 = Down Feed Rate (2 levels: slow (.05 mm) and fast (.125 mm))

Factor 3 = Wheel Grit (2 levels: 140/170 and 80/100)

Factor 4 = Direction (2 levels: longitudinal and transverse)

Factor 5 = Batch (2 levels: 1 and 2)

The data

The design matrix, with measured ceramic   strength responses, appears below. The actual randomized run order is given   in the last column.
speed rate grit direction batch strength   order
1 -1 -1 -1 -1 -1 680.45 17
2 1 -1 -1 -1 -1 722.48 30
3 -1 1 -1 -1 -1 702.14 14
4 1 1 -1 -1 -1 666.93 8
5 -1 -1 1 -1 -1 703.67 32
6 1 -1 1 -1 -1 642.14 20
7 -1 1 1 -1 -1 692.98 26
8 1 1 1 -1 -1 669.26 24
9 -1 -1 -1 1 -1 491.58 10
10 1 -1 -1 1 -1 475.52 16
11 -1 1 -1 1 -1 478.76 27
12 1 1 -1 1 -1 568.23 18
13 -1 -1 1 1 -1 444.72 3
14 1 -1 1 1 -1 410.37 19
15 -1 1 1 1 -1 428.51 31
16 1 1 1 1 -1 491.47 15
17 -1 -1 -1 -1 1 607.34 12
18 1 -1 -1 -1 1 620.80 1
19 -1 1 -1 -1 1 610.55 4
20 1 1 -1 -1 1 638.04 23
21 -1 -1 1 -1 1 585.19 2
22 1 -1 1 -1 1 586.17 28
23 -1 1 1 -1 1 601.67 11
24 1 1 1 -1 1 608.31 9
25 -1 -1 -1 1 1 442.90 25
26 1 -1 -1 1 1 434.41 21
27 -1 1 -1 1 1 417.66 6
28 1 1 -1 1 1 510.84 7
29 -1 -1 1 1 1 392.11 5
30 1 -1 1 1 1 343.22 13
31 -1 1 1 1 1 385.52 22
32 1 1 1 1 1 446.73 29
Problem #4.
Replicate the DOE in the following website. Write a 3-line report with your conclusions
https://asq.org/quality-resources/design-of-experiments

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