Question

Problem 1:

A research laboratory was developing a new compound for the relief of severe cases of hay fever. In an experiment with 36 volunteers, the amounts of the two active ingredients (A & B) in the compound were varied at three levels each. Randomization was used in assigning four volunteers to each of the nine treatments. The data on hours of relief can be found in the following .csv file: Fever.csv

1.1) State the Null and Alternate Hypothesis for conducting one-way ANOVA for both the variables ‘A’ and ‘B’ individually.

1.6) Mention the business implications of performing ANOVA for this particular case study.

A	B	Volunteer	Relief
1	1	1	2.4
1	1	2	2.7
1	1	3	2.3
1	1	4	2.5
1	2	1	4.6
1	2	2	4.2
1	2	3	4.9
1	2	4	4.7
1	3	1	4.8
1	3	2	4.5
1	3	3	4.4
1	3	4	4.6
2	1	1	5.8
2	1	2	5.2
2	1	3	5.5
2	1	4	5.3
2	2	1	8.9
2	2	2	9.1
2	2	3	8.7
2	2	4	9
2	3	1	9.1
2	3	2	9.3
2	3	3	8.7
2	3	4	9.4
3	1	1	6.1
3	1	2	5.7
3	1	3	5.9
3	1	4	6.2
3	2	1	9.9
3	2	2	10.5
3	2	3	10.6
3	2	4	10.1
3	3	1	13.5
3	3	2	13
3	3	3	13.3
3	3	4	13.2

Answer 1

Null Hypothesis: the two active ingredients (A & B) are same

Alternative Hypothesis: the two active ingredients (A & B) are not same

we conduct factorial experiment

and we perform in minitab software

Factorial Fit: Relief versus A, B, Volunteer

* NOTE * This design has some botched runs. It will be analyzed using a
         regression approach.

Estimated Effects and Coefficients for Relief (coded units)

Term             Effect      Coef SE Coef      T      P
Constant                  7.18333   0.1477 48.63 0.000
A               5.95000   2.97500   0.1809 16.44 0.000
B               4.35000   2.17500   0.1809 12.02 0.000
Volunteer      -0.00667 -0.00333   0.1982 -0.02 0.987
A*B             2.58750   1.29375   0.2216   5.84 0.000
A*Volunteer     0.02000   0.01000   0.2427   0.04 0.967
B*Volunteer    -0.01000 -0.00500   0.2427 -0.02 0.984
A*B*Volunteer -0.03750 -0.01875   0.2973 -0.06 0.950

S = 0.886292    PRESS = 34.4507
R-Sq = 94.13%   R-Sq(pred) = 90.81%   R-Sq(adj) = 92.66%

Analysis of Variance for Relief (coded units)

Source              DF   Seq SS   Adj SS   Adj MS       F      P
Main Effects         3 325.950 325.950 108.650 138.32 0.000
A                  1 212.415 212.415 212.415 270.42 0.000
B                  1 113.535 113.535 113.535 144.54 0.000
Volunteer          1    0.000    0.000    0.000    0.00 0.987
2-Way Interactions   3   26.782   26.782    8.927   11.37 0.000
A*B                1   26.781   26.781   26.781   34.09 0.000
A*Volunteer        1    0.001    0.001    0.001    0.00 0.967
B*Volunteer        1    0.000    0.000    0.000    0.00 0.984
3-Way Interactions   1    0.003    0.003    0.003    0.00 0.950
A*B*Volunteer      1    0.003    0.003    0.003    0.00 0.950
Residual Error      28   21.994   21.994    0.786
Total               35 374.730

Unusual Observations for Relief

Obs StdOrder Relief     Fit SE Fit Residual St Resid
17        17 8.9000 7.1867 0.2472    1.7133      2.01R
18        18 9.1000 7.1844 0.1618    1.9156      2.20R
20        20 9.0000 7.1800 0.2472    1.8200      2.14R

R denotes an observation with a large standardized residual.

Estimated Coefficients for Relief using data in uncoded units

Term                Coef
Constant         2.20556
A                0.30833
B               -0.46667
Volunteer      -0.058889
A*B              1.32500
A*Volunteer     0.031667
B*Volunteer     0.021667
A*B*Volunteer -0.012500

* NOTE * Some factors have more than 2 levels, no alias table was printed.

Residual Plots for Relief