3 At a vineyard, the process used to fill bottles with wine is subject to variation. The contents of bottles are independently Normally distributed with mean \(\mu = 751.4 \mathrm { ml }\) and standard deviation \(\sigma = 2.5 \mathrm { ml }\).

1. Find the probability that a randomly selected bottle contains at least 750 ml .
2. A case of wine consists of 6 bottles. Find the probability that all 6 bottles in a case contain at least 750 ml .
3. Find the probability that, in a random sample of 25 cases, there are at least 2 cases in which all 6 bottles contain at least 750 ml . It is decided to increase the proportion of bottles which contain at least 750 ml to \(98 \%\).
4. This can be done by changing the value of \(\mu\), but retaining the original value of \(\sigma\). Find the required value of \(\mu\).
5. An alternative is to change the value of \(\sigma\), but retain the original value of \(\mu\). Find the required value of \(\sigma\).
6. Comment briefly on which method might be easier to implement and which might be preferable to the vineyard owners.