1. Okay, to find \(\displaystyle \mu\):

Let X= the number of mL of water in a carton

\(\displaystyle pr(X\geq 1,002)=\frac{225}{900}=0.25\)

This is given in the problem.

Take note that \(\displaystyle pr(X\geq 1,002)=0.25\) is the same as \(\displaystyle pr(X\leq 1,002)=0.75\)

Now standardize the distribution.

\(\displaystyle pr(X\leq 1,002)=0.75\)=\(\displaystyle pr(Z\leq \frac{1,002-\mu}{8})=0.675\) (Taken from z-table)

Now, solve for \(\displaystyle \mu\) and you should get \(\displaystyle \mu = 996.6\)

2. This is a simple binomial distribution:

X~binomial(n=3,p=0.25)

X= the number of cartons containing more than 1,002 mL of water.

\(\displaystyle {n\choose x}(p)^{x}(1-p)^{(n-x)}\)

\(\displaystyle pr(X=2)= {3\choose 2}(0.25)^{2}(0.75)\approx 0.1406\)

Hmm.. I now see that you gave the answers in parentheses. I don't understand why (1) is 0.674. It's approximately the same number I took from the z-table but the problem states that we're trying to find \(\displaystyle \mu\) not \(\displaystyle Z\). Pretty weird. Maybe someone can correct me on that.