Question

Sweet Unga Ltd. packs wheat flour in packets whose weights are normally distributed with mean equal to 5kg and standard deviation equal to 0.2 kg. The company classifies its packets as “under weight” when its weight falls below 4.7 kg; ”over weight” when above 5.1 kg; otherwise it is classified as “normal weight”. The customers are charged Ksh 1700, 2000 and 2100 for the various packets respectively.

A customer is supplied with consignment of 1000 packets of the wheat flour. How much would the customer be expected to pay to the company?

Answer 1

Let $X=$ weight of a packet in kg: $X\sim N(\mu, \sigma_X^2)$

Then

$$Z= {X-\mu \over b}\sim N(0, 1)$$

Given that $\mu=5\ kg, \sigma=0.2\ kg, n=1000.$

$$P(X<4.7)=P(Z<{4.7-5 \over 0.2/\sqrt{1000}})\approx$$ $$\approx P(Z<-47.434)\approx0$$

$$P(X>5.1)=1-P(X\leq5.1)=$$

$$=1-P(Z\leq{5.1-5 \over 0.2/\sqrt{1000}})\approx1-P(Z\leq15.811)\approx0$$

Then

$$P(\text{normal weight})\approx1$$

$$2000(1000)(1)=2000000$$

The customer would be expected to pay $2,000,000$ Ksh to the company.