Tapsin secreton \([U/(kg/hr)]\)

\(\leq\ 50\)

\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 1.7 \\ \hline 2 & 2.0 \\ \hline 3 & 2.0 \\ \hline 4 & 2.2 \\ \hline 5 & 4.0 \\ \hline 6 & 4.0 \\ \hline 7 & 5.0 \\ \hline 8 & 6.7 \\ \hline 9 & 7.8 \\ \hline \end{array}\)

\(51\ -\ 1000\)

\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 1.4 \\ \hline 2 & 2.4 \\ \hline 3 & 2.4 \\ \hline 4 & 3.3 \\ \hline 5 & 4.4 \\ \hline 6 & 4.7 \\ \hline 7 & 6.7 \\ \hline 8 & 7.9 \\ \hline 9 & 9.5 \\ \hline 10 & 11.7 \\ \hline \end{array}\)

\(>\ 1000\)

\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 2.9 \\ \hline 2 & 3.8 \\ \hline 3 & 4.4 \\ \hline 4 & 4.7 \\ \hline 5 & 5.5 \\ \hline 6 & 5.6 \\ \hline 7 & 7.4 \\ \hline 8 & 9.4 \\ \hline 9 & 10.3 \\ \hline \end{array}\)

Step 2

1) By using Kruskal-Wallis test to compare 3 groups we get, Combining score of all the three groups, arranging them into ascending order and assigning them ra

.

\(A=group\ 1\ \leq\ 50\)

\(B=group\ 2\ =51\ -\ 1000\)

\(C=group\ 3\ =\ \Rightarrow\ 1000\)

\(\begin{array}{|c|c|}\hline \text{Observation} & \text{Rank} & \text{Groups} \\ \hline 1.4 & 1 & B \\ \hline 1.7 & 2 & A \\ \hline 2 & 3.5 & A \\ \hline 2 & 3.5 & A \\ \hline 2.2 & 5 & A \\ \hline 2.4 & 6.5 & B \\ \hline 2.4 & 6.5 & B \\ \hline 2.9 & 8 & C \\ \hline 3.3 & 9 & B \\ \hline 3.8 & 10 & C \\ \hline 4 & 11.5 & A \\ \hline 4 & 11.5 & A \\ \hline 4.4 & 13.5 & B \\ \hline 4.4 & 13.5 & C \\ \hline 4.7 & 15.5 & B \\ \hline 4.7 & 15.5 & C \\ \hline 5 & 17.5 & A \\ \hline 5 & 17.5 & C \\ \hline 5.6 & 19 & C \\ \hline 6.7 & 20.5 & A \\ \hline 6.7 & 20.5 & B \\ \hline 7.4 & 22 & C \\ \hline 7.6 & 23 & B \\ \hline 7.8 & 24 & A \\ \hline 9.4 & 25 & C \\ \hline 9.5 & 26 & B \\ \hline 10.3 & 27 & C \\ \hline 11.7 & 28 & B \\ \hline \end{array}\)

\(n_{A} = 9\)

\(n_{B} = 10\)

\(n_{C} = 9\)

\(n = n_{A}\ +\ n_{B}\ +\ n_{C} = 9\ +\ 10\ +\ 9 = 28\)

\(R_{A} = \sum\ \text{of ra

for group}\ A = 2\ +\ 3.5\ +\ 3.5\ +\ 5\ +\ 11.5\ +\ 11.5\ +\ 17.5\ +\ 20.5\ +\ 24 = 99\)

\(R_{B} = \sum \text{of ra

for group} B = 1\ +\ 6.5\ +\ 6.5\ +\ 9\ +\ 13.5\ +\ 15.5\ +\ 20.5\ +\ 23\ +\ 26\ +\ 28 = 149.5\)

\(R_{C} = \sum \text{of ra

for group} C = 8\ +\ 10\ +\ 13.5\ +\ 15.5\ +\ 17.5\ +\ 19\ +\ 22\ +\ 25\ +\ 27 = 157.5\)

Hypothesis is given as:

\(H_{0}:\ \mu_{A}=\ \mu_{b}=\ \mu_{C}\) i.e. three groups are equally effective.

\(H_{1} :\ \text{at least two of the} \mu\) are different.

Kruskal-Wallis test statistics is given as:

\(H=\ \frac{12}{n(n\ +\ 1)}\left[\frac{R_{A}^{2}}{n_{A}}\ +\ \frac{R_{B}^{2}}{n_{B}}\ +\ \frac{R_{C}^{2}}{n_{C}}\right]-3(n\ +\ 1)\)

\(=\ \frac{12}{28\ \times\ (29)}\left[\frac{(99)^{2}}{9}\ +\ \frac{(145.5)^{2}}{10}\ +\ \frac{157.5^{2}}{9}\right]\ -\ 87\)

\(=\ \frac{12}{812}\left[\frac{9810}{9}\ +\ \frac{22350.25}{10}\ +\ \frac{24806.25}{9}\right]\ -\ 87\)

\(=\ \frac{12}{812}\left[1089\ +\ 2235.025\ +\ 2756.25\right]\ -\ 87\)

\(=\ \frac{12}{812}\left[6080.275\right]\ -\ 87\)

\(=0.01479\ \times\ [6080.275]\ -\ 87\)

\(= 89.927\ -\ 87\)

\(H = 2.927\)

\(df = k\ -\ 1 = 3\ -\ 1 = 2\)

The table of chi — square for 2 d.f. at 3% level of significance is = 5.991

The calculated value \(H = 2.921\) is smaller than table value

Conclusion : Accept \(H_{0}\). i.e. three groups are equally effective.

Step 3

Compare result with a parametric analysis of the data is given as:

By using excel we get anova:

Summary:

\(\begin{array}{|c|c|}\hline \text{Groups} & \text{Count} & \text{Sum} & \text{Average} & \text{Variance} \\ \hline A & 9 & 35.4 & 3.933333 & 4.9025 \\ \hline B & 10 & 54.1 & 5.41 & 11.43656 \\ \hline C & 9 & 53.5 & 5.944444 & 6.480278 \\ \hline \end{array}\)

ANOVA:

\(\begin{array}{|c|c|}\hline \text{Source of Variation} & \text{SS} & \text{df} & \text{MS} & \text{F} & \text{P-value} & \text{F-crit} \\ \hline \text{Between Groups} & 19.62735 & 2 & 9.813675 & 1.264706 & 0.29977 & 3.38519 \\ \hline \text{Within Groups} & 193.9912 & 25 & 7.59649\\ \hline \text{Total} & 23.6186 & 27 \\ \hline \end{array}\)

\(F\ -\ calculated\ value\ = 1.264706\ is\ less\ than\ F\ -\ table(critical\ value) = 3.38519.\)

Conclusion: Accept \(H_{0}.\) i.e. three groups are equally effective.

By comparing non parametric Kruskal-Wallis test to parametric analysis of data both test have same result.