Question

The masses of articles produced in a particular workshop are normally distributed with mean A and standard deviation B. 5% of the article have a mass greater than 85g and 10% have a mas less than 25g. Find the values of A and B.

Answer 1

Let X = the random variable denoting the mass of an article

Then by the problem,

X ~ N(A, B²)

Therefore, we know,

Z = $\frac{X-A}{B}$ ~ N(0, 1)

Now, 5% of the article have a mass greater than 85g

i.e. the probability that an article having mass greater than 85g is 5%

i.e. P(X > 85) = 5%

i.e. P($\frac{X-A}{B}>\frac{85-A}{B}$) = 0.05

i.e. 1 - P($\frac{X-A}{B}\leq\frac{85-A}{B}$) = 0.05

i.e. P($Z\leq\frac{85-A}{B}$) = 1 - 0.05 = 0.95

i.e. $\Phi(\frac{85-A}{B})$ = 0.95 = $\Phi(1.64)$

i.e. $\frac{85-A}{B}=1.64$

i.e. A + 1.64B = 85 .......................(1)

Again, 10% of the article have a mas less than 25g

i.e. the probability that an article having mass less than 25g is 10%

i.e. P(X < 25) = 10%

i.e. P($\frac{X-A}{B}$ <$\frac{25-A}{B}$​) = 0.1

i.e. P(Z <$\frac{25-A}{B}$​) = 0.1

i.e. $\Phi(\frac{25-A}{B})$ = 0.1 = $\Phi(-1.28)$

i.e. $\frac{25-A}{B}$ = - 1.28

i.e. A - 1.28B = 25 ........................(2)

Solving (1) and (2) we get, A = 51.3, B = 20.55

Answer: The values of A and B are 51.3g and 20.55g respectively.