A clinic monitors the amount, $X$ milligrams per litre, of a certain chemical in the blood stream of patients. For patients who are taking drug $A$, it has been found that the mean value of $X$ is $0.336$. $A$ random sample of $100$ patients taking a new drug, $B$, was selected and the values of $X$ were found. 

The results are summarised below.

$n = 100$, $\sum x  = 43.5$, $\sum {{x^2}}  = 31.56$.

a) Test at the 1% significance level whether the mean amount of the chemical in the blood stream of patients taking drug $B$ is different from that of patients taking drug $A$.

b) For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical in the blood stream of patients taking drug $B$? Justify your answer.