Answer to Question #262759 in Statistics and Probability for yyy

Question

A factory produces apple juice contained in a bottle of 1.5L. However, due to random fluctuations in the automatic bottling machine, the actual volume per bottle varies according to a normal distribution. It is observed that 10% of bottles are under 1.45L whereas 5% contain more than 1.55L. Calculate the mean and standard deviation of the volume distribution。

Answer 1

Let $X=$ the volume per bottle: $X\sim N(\mu, \sigma^2)$

$$P(X<1.45)=P(Z<\dfrac{1.45-\mu}{\sigma})=0.1$$

$$\dfrac{1.45-\mu}{\sigma}\approx-1.28155$$

$$P(X>1.55)=1-P(Z\leq\dfrac{1.55-\mu}{\sigma})=0.05$$

$$P(Z\leq\dfrac{1.55-\mu}{\sigma})=0.95$$

$$\dfrac{1.55-\mu}{\sigma}\approx1.64485$$

$$\dfrac{1.45-\mu}{1.55-\mu}=\dfrac{-1.28155}{1.64485}$$

$$2.3850325-1.64485\mu=-1.9864025+1.28155\mu$$

$$2.9264\mu=4.371435$$

$$\mu=1.493793$$

$$\sigma=\dfrac{1.45-1.493793}{-1.2815}$$

$$\sigma=0.034173$$

$$mean=\mu=1.4938L$$

$$\sigma=0.0342L$$