Answer to Question #281184 in Statistics and Probability for Amit

Question

Cereal boxes are filled at a factory. Through extensive sampling, it is found that their equipment fills the boxes to a mean of 502 grams, with a standard deviation of 3.1 grams.

10,000 boxes of cereal are produced each day. 

In a daily run, how many cereal boxes are expected to weigh between 500 and 504 grams?

Answer 1

Let $X=$ the weight of cereal box: $X\sim N(\mu, \sigma^2)$

Given $\mu=502\ g, \sigma=3.1\ g$

$$P(500<X<504)=P(X<504)-P(X\leq 500)$$

$$=P(Z<\dfrac{504-502}{3.1})-P(Z<\dfrac{500-502}{3.1})$$

$$\approx P(Z<0.64516)-P(Z<-0.64516)$$

$$\approx0.7405887-0.2594113$$

$$\approx0.481177$$

$$10000(0.481177)=4812$$

In a daily run, 4812 cereal boxes are expected to weigh between 500 and 504 grams.