A cereal machine is sending $x$ grams of cereal into each packet. The value of
$x$ is programmed into the machine and the packets are then filled
with the weight of cereal in each being normally distributed with:
- mean of $x$ grams and
- standard deviation of $1.8$ grams
The machine is used to fill packets that will be labelled as
containing $500$ grams. What should be the value of $x$ if the company
wants no more than $0.5\%$ of the packets to be underweight?
I'm not quite sure how to start this question but here's what I am thinking so far. Do we have to set up some sort of simultaneous equations here since we are only given an unknown mean value of $x$? Does this involve using the standard deviation of $1.8$ grams? Any starting hints would be much appreciated.
Best Answer
Set
$$\mathbb{P}[Y<500]\leq 0.5\%$$
$$\mathbb{P}\Bigg[Z<\frac{500-x}{1.8}\Bigg]\leq 0.5\%$$
$$\frac{500-x}{1.8}=-2.58$$
$$x=504.64$$