Answer to Question #94104 in Statistics and Probability for Mina

Question

1. The weights of bags of red gravel may be modeled by a normal distribution with mean 25.8 kg and standard deviation 0.5 kg.
(a) Determine the probability that a randomly selected bag of gravel will weigh:
i. less than 25 kg (3pts)
ii. between 25.5 kg and 26.5 kg (4pts)
(b) Determine, to two decimal places, the weight exceeded by 75% of bags. (4pts)

Answer 1

a)i)

$$P(x<25)=P\left(z<\frac{25-25.8}{0.5}\right)=P\left(z<-1.6\right)$$

Using z-table:

$$P(x<25)=0.0548$$

ii)

$$P(25.5<x<26.5)=P\left(\frac{25.5-25.8}{0.5}<z<\frac{26.5-25.8}{0.5}\right)$$

$$P(25.5<x<26.5)=P\left(-0.6<z<1.4\right)=P\left(z<1.4\right)-P\left(z<-0.6\right)$$

Using z-table:

$$P(25.5<x<26.5)=0.9192-0.2743=0.6449$$

b)

$$P(x>X)=0.75\to P(x<X)=1-0.75=0.25$$

Using z-table:

$$P\left(z<-0.67\right)=0.2514,P\left(z<-0.68\right)=0.2483$$

Thus

$$Z=-0.68+\frac{0.25-0.2483}{0.2514-0.2483}(-0.67-(-0.68))=-0.6745$$

So,

$$X=25.8-0.6745(0.5)=25.46\ kg$$