Answer to Question #92853 in Statistics and Probability for londman

Question

Packets of milk powder produced by a machine were found to have a normal distribution with a mean mass of 650g and a standard deviation of 10g.

Find the probability that a packet selected at random will have a mass between 620g and 655g
If 500 packets are selected at random, how many of them will have a mass of more than 660g?
It is found that 10% of packets of milk powder will have a mass of less than k Calculate k.

Answer 1

Suppose $X$ is a variable representing the mass of the packet of milk powder. $X$ is normally distributed with mean of 650 g and standard deviation of 10 g: 

$$X\sim N(650, 10^2)$$

If $X\sim N(\mu, \sigma^2)$, then $Z={X-\mu \over \sigma}\sim N(1, 0)$

The probability that a packet selected at random will have a mass between 620g and 655g 

$$P(620<X<655)=P({620-650 \over 10}<Z<{655-650 \over 10})=$$ $$=P(-3<Z<0.5)=F(0.5)-F(-3)=$$ $$=F(0.5)-(1-F(3))=0.691462-0.001350=$$ $$=0.690112$$

If 500 packets are selected at random, how many of them will have a mass of more than 660g? 

Given that $N=500.$

$$P(X>660)=P(Z>{660-650 \over 10})=1-F(1)=$$ $$=1-0.841345=0.158655$$ $$n=N\cdot P(X>660)=500\cdot 0.158655=79$$

It is found that 10% of packets of milk powder will have a mass of less than k .

$$P(X<k)=0.1$$ $$P(X<k)=P(Z<{k-650 \over 10})=0.1$$ $${k-650 \over 10}=-1.281552$$

$$k=650-10\cdot1.281552\approx637$$

The probability that a packet selected at random will have a mass between 620g and 655g  is $0.690112.$

If 500 packets are selected at random, then 79 of then will have a mass of more than 660g.

It is found that 10% of packets of milk powder will have a mass of less than k=637 g .