Normal distribution - What should be value of $x$ if the company wants no more than $0.5\%$ of the packets to be underweight?

Question

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.

Answer 1

Set

$$\mathbb{P}[Y<500]\leq 0.5\%$$

$$\mathbb{P}\Bigg[Z<\frac{500-x}{1.8}\Bigg]\leq 0.5\%$$

$$\frac{500-x}{1.8}=-2.58$$

$$x=504.64$$

Answer 2

You are expected to use the z-score table. $x$ is the mean of the distribution. You want less than $5\%$ of the area to be less than $500$ grams. How many standard deviations does that mean $x$ has to be above $500$ grams?