a$)$ What do you understand by Latin square design $($LSD$)$; give two instances in which the design could be utilized. $(7$marks$)$

b$)$ Given the following LSD with $6$ treatments;

A B C D E F

B C D E F A

C D E F A B

D E F A B C

E F A B C D

F A B C D E

Present a randomisation of the layout.

Use the $3-$digit random numbers. Start from row $2,$ column $5$ and move vertically downwards. For the Column $($Row$)$ randomisation, continue from where you ended the Row $($Column$)$ randomisation. $(13$marks$)$

c$)$ The effect of five different ingredients $($A$,$ B$,$ C$,$ D$,$ and E$)$ on the reaction time of a chemical process is being studied. Each batch of new material is only large enough to permit five runs to be made. Furthermore, each run requires approximately $1$ and a half hours, so only five runs can be made in a day. The experimenter decides to run the experiment as a Latin Square so that day and batch effects may be systematically controlled. Analyze the data from the experiment $($use $\backslash$alpha $=0\mathrm{.}05)$ and draw conclusion. $(30$marks$)$

DAY

Batch $12345$

$1$ A$=8$ B$=7$ D$=1$ C$=7$ E$=3$

$2$ C$=11$ E$=2$ A$=7$ D$=3$ B$=8$

$3$ B$=4$ A$=9$ C$=10$ E$=1$ D$=5$

$4$ D$=6$ C$=8$ E$=6$ B$=6$ A$=10$

$5$ E$=4$ D$=2$ B$=3$ A$=8$ C$=8$