Normal distribution and algebra problem

Question

Bags of cement are labeled $25 \operatorname{kg}$. The bags are filled by machine and the actual weights are normally distributed with mean 26.0 kg and standard deviation $0.50 \operatorname{kg}$. It is decided to purchase a more accurate machine for filling the bags. The new mean is $25.5$ and the standard deviation is $.255$. The cost of the new machine is $\$5000$. Cement sells for $\$0.80$ per $\operatorname{kg}$. Compared to the cost of operating with a $26 \operatorname{kg}$ mean, how many bags must be filled in order to recover the cost of the new equipment?

I'm unsure of why the answer is $12,500 --- 5000 / (0.5 \cdot 0.8 )$

Why isn't it $20.4$ because of $5000 / (25.5 \cdot 0.8)$? I know that the question wants to compare the new mean with the old, but I'm not quite clear on what that means?

Answer 1

You know that the old machine is underselling each bag by an average of $\mu _1 - 25 = 1 {\rm kg}$, and the new machine is underselling each bag by an average of $\mu _2 - 25 = 0.5{\rm kg}$.

Hence the average savings per bag by switching to the new machine is $1 {\rm kg} - 0.5 {\rm kg} = 0.5 {\rm kg}$.

Since the cement sells for \$0.80 a kilo, this is an average savings of $0.5 {\rm kg} \cdot 0.8 {\rm \$/kg} = {\rm \$ 0.40}$ per bag.

Hence to recoup the full investment, the company needs to sell $$n = \frac{\rm \$5,000}{\rm \$ 0.40} = {\rm 12,500\; bags}$$ to recoup the investment.

Answer 2

On average the weight of a bag is only 0,5 kg (=25.5-25) over 25 kg with the new maschine-instead of 1 kg (=26-25) . The equation is $\Delta e=p \cdot \Delta w\cdot q$.

$\Delta e$= difference of earnings=$5,000

$\Delta w$=difference of the weights per bag=1 kg/bag-0.5 kg/bag=0.5 kg/bag

q=amount of bags

p=price of a bag of cement=$0.8/kg

You have to solve the equation for q.

Thus on average you have sell 12,500 bags to recover the costs. In this calculation the concept of the expected value has been used.

Remark:

It is not sure, that you recover the costs. With the sample size of 12,500 bags, there is a possibility, that you do not recover the costs, because the weights of the bags are random variables. It is also possible, that your additional earnings are higher than $5,000.