Question

The Army is concerned about the permeability of their parachutes. An Army researcher believes that temperature, fabric denier and air pressure all play a role in the permeability. Using the accompanying data, answer the following:

a) Name the factors and levels

b) Analyze the data making sure you state your hypothesis(es), any assumptions and

conclusions.

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Answer 1

Answer:

Ans. a) There are three factors each at three levels.

Factor 1) Temperature which is at 3 levels namely i) 8 degrees, ii) 50 degrees and iii) 75 degrees.

Factor 2) Pressure which is at 3 levels namely i) 17.2, ii) 34.4 and iii) 103.4

Factor 3) Fabric which is at 3 levels namely i) 420-D, ii) 630-D and iii) 840-D.

Ans. b)

Now based on 3³ factorial design, divide the 54 observations of permeability in to two Blocks as,

Temp.	Pressure	Fabric	Block I	Block II
8	17.2	420-D	73	80
8	17.2	630-D	35	433
8	17.2	840-D	125	111
8	34.4	420-D	157	155
8	34.4	630-D	91	98
8	34.4	840-D	234	233
8	103.4	420-D	332	322
8	103.4	630-D	288	271
8	103.4	840-D	477	464
50	17.2	420-D	52	51
50	17.2	630-D	16	12
50	17.2	840-D	96	100
50	34.4	420-D	125	118
50	34.4	630-D	72	78
50	34.4	840-D	149	155
50	103.4	420-D	281	264
50	103.4	630-D	169	173
50	103.4	840-D	338	350
75	17.2	420-D	37	31
75	17.2	630-D	30	41
75	17.2	840-D	102	98
75	34.4	420-D	95	106
75	34.4	630-D	91	100
75	34.4	840-D	170	160
75	103.4	420-D	276	281
75	103.4	630-D	213	211
75	103.4	840-D	307	311

Here assume that, Main factors Temp. as 'A' is at level 8 degrees as level '1', at level 50 degrees as level '2' & at level 75 degrees as level '3'. Main factor Pressure as 'B' is at level 17.2 as level '1', at level 34.4 as level '2' & at level 103.4 as level '3'. Main factor Fabric as 'C' is at level 420-D as level '1', at level 630-D as level '2' & at level 840-D as level '3'.

Analysis by using MINITAB

Hypothesis:
1) H_R: There is no significance difference between different block effects. V/s H_R': At least two block effects differ significantly.

2) H_A: Main effect A is not significant. V/s H_A': Main effect A is significant.

3) H_B: Main effect B is not significant. V/s H_B': Main effect B is significant.

4) HC: Main effect C is not significant. V/s H_C': Main effect C is significant.

5) H_AB: Interaction effect AB is not significant. V/s H_AB': Interaction effect AB is significant.

6) H_AC: Interaction effect AC is not significant. V/s H_AC': Interaction effect AC is significant.

7) H_BC: Interaction effect AC is not significant. V/s H_AC': Interaction effect AC is significant.

8) H_ABC: Interaction effect ABC is not significant. V/s H_ABC': Interaction effect ABC is significant.

To conduct a 3³ Factorial Experiment, follow these steps:  
Step 1. Click on Stat   $\rightarrow$ DOE $\rightarrow$ Factorial $\rightarrow$ Create Factorial Design   $\rightarrow$ Click on General full factorial design. Enter Number of factors as 3.
Step 2. Click on Designs, then following window will appear. Enter Number of Levels as 3 for factors A, B &C. Enter Number of replicates as 2. Click on Block on replicates.
Step 3. Click on OK $\rightarrow$ OK.

Then the 3³ factorial design is created. On worksheet following columns (say) appear.

StdOrder	RunOrder	PtType	Blocks	A	B	C
53	1	1	2	3	3	2
40	2	1	2	2	2	1
29	3	1	2	1	1	2
41	4	1	2	2	2	2
45	5	1	2	2	3	3
35	6	1	2	1	3	2
32	7	1	2	1	2	2
43	8	1	2	2	3	1
38	9	1	2	2	1	2
39	10	1	2	2	1	3
48	11	1	2	3	1	3
33	12	1	2	1	2	3
47	13	1	2	3	1	2
36	14	1	2	1	3	3
30	15	1	2	1	1	3
44	16	1	2	2	3	2
28	17	1	2	1	1	1
49	18	1	2	3	2	1
54	19	1	2	3	3	3
42	20	1	2	2	2	3
31	21	1	2	1	2	1
34	22	1	2	1	3	1
37	23	1	2	2	1	1
52	24	1	2	3	3	1
46	25	1	2	3	1	1
50	26	1	2	3	2	2
51	27	1	2	3	2	3
20	28	1	1	3	1	2
10	29	1	1	2	1	1
21	30	1	1	3	1	3
15	31	1	1	2	2	3
24	32	1	1	3	2	3
13	33	1	1	2	2	1
14	34	1	1	2	2	2
22	35	1	1	3	2	1
2	36	1	1	1	1	2
16	37	1	1	2	3	1
18	38	1	1	2	3	3
8	39	1	1	1	3	2
7	40	1	1	1	3	1
17	41	1	1	2	3	2
25	42	1	1	3	3	1
11	43	1	1	2	1	2
9	44	1	1	1	3	3
4	45	1	1	1	2	1
26	46	1	1	3	3	2
12	47	1	1	2	1	3
5	48	1	1	1	2	2
19	49	1	1	3	1	1
3	50	1	1	1	1	3
23	51	1	1	3	2	2
6	52	1	1	1	2	3
27	53	1	1	3	3	3
1	54	1	1	1	1	1

In this worksheet add another column Response (Values) and enter values like this:

Step 4. Click on Stat $\rightarrow$ DOE $\rightarrow$ Factorial $\rightarrow$ Analyze Factorial Design. Then enter Response in Responses box.

Step 5. Then click OK.

Output of MINITAB:

Analysis of Variance for Response, using Adjusted SS for Tests

Source	DF	Seq SS	Adj SS	Adj MS	F	P-value
Blocks	1	2618	2618	2618	0.88	0.358
A	2	67553	67553	33777	11.32	0.000
B	2	442111	442111	221056	74.09	0.000
C	2	72361	72361	36181	12.13	0.000
A*B	4	6213	6213	1553	0.52	0.721
A*C	4	9696	9696	2424	0.81	0.529
B*C	4	34928	34928	8732	2.93	0.040
A*B*C	8	33487	33487	4186	1.40	0.242
Error	26	77574	77574	2984
Total	53	746540

S = 54.6225, R-Sq = 89.61%, R-Sq (adj) = 78.82%

Here at 5% level of significance, For Blocks p-value > 0.05, therefore we accept H_R. For main effect A p-value < 0.05, therefore we reject H_A. For main effect B p-value < 0.05, therefore we reject H_B. For main effect C p-value < 0.05, therefore we reject H_C. For interaction effect AB p-value > 0.05, therefore we accept H_AB. For interaction effect AC p-value > 0.05, therefore we accept H_AC. For interaction effect BC p-value < 0.05, therefore we reject H_BC. For interaction effect ABC p-value > 0.05, therefore we accept H_ABC.

Conclusion:

1) There is no significance difference between different block effects.

2) Main effect A is significant.

3) Main effect B is significant.

4) Main effect C is significant.

5) Interaction effect AB is not significant.

6) Interaction effect AC is not significant.

7) Interaction effect BC is significant.

8) Interaction effect ABC is not significant.