Mean and standard deviation with percentages

Question

(I am very new to this topic and I have tried many questions successfully. Except story questions like these, which confuse me. Can I just get hints that help me out instead of the answer? or like anything but the direct answer if possible)

a) The volume of fruit juice boxes are normally distributed with 1010ml and standard deviation 8ml.

i) Find the probability that a box that selected contains a volume more than 1004ml.

ii) Given that 8% of the boxes are rejected because their volumes are more than acceptable amount. Find the maximum possible volume, that a box should be acceptable.

iii) The labels on the boxes are stating it is of 1 litre fruit juice. How many numbers of boxes from the given 2000 boxes are expected to contain less than this volumes?

b) A machine producing boxes, where the boxes are normally distributed with standard deviation 5g. Given that 9% of these boxes contain more than 1015.7g, estimate the mean for the volume of these boxes.

Answer 1

(a) (i) if $X \sim \mathcal{N}(1010,8)$, what transformation will make $X$ into $Y \sim \mathcal{N}(0,1)$? And once you have the transformation, $\mathbb{P}[Y \leq y] = \Phi(y)$ is a standard tabulated function (the c.d.f. of the standard normal distribution).

(ii) What is $z$ such that $\mathbb{P}[X < z] = 92\%$? You will need the transformation from (i)...

(iii) Given $X \sim \mathcal{N}(1010,8)$, what is $\mathbb{P}[X<1000]$?

(b) If $X \sim \mathcal{N}(m,5)$, what is $p = \mathbb{P}[X > 1015.7]$ and if I tell you $p=0.09$ can you solve for $m$?

Answer 2

(i) This is the probability that the standard normal $Z$ is greater than $\frac{1004-1010}{8}$. The information may not be directly available in the tables. But by symmetry this is the probability that $Z$ is less than $\dfrac{1010-1004}{8}$.

(ii) If you look at the standard normal table, you will find that if $\Pr(Z\gt a)=0.08$, then $a\approx 1.405$. So if the volume the question talks about is $V$, then $\dfrac{V-1010}{8}\approx 1.405$.

(iii) This is like a more mechanical version of (i)

(iv) This is very close to (ii).