Question

The weight of topsoil sold in a week is normally distributed with a mean of 1600 tons and a standard deviation of 32 tons. (a) What percentage of weeks will sales exceed 1648 tons? (Round your answer to two decimal places.) % (b) What percentage of weeks will sales be less than 1568 tons? (Round your answer to two decimal places.) % (c) What percentage of weeks will sales be between 1536 and 1648 tons? (Round your answer to two decimal places.)

Answer 1

Solution:

We are given

Mean = 1600

SD = 32

Part a

We have to find P(X>1648) = 1 – P(X<1648)

Z = (X – mean) / SD

Z = (1648 - 1600)/32

Z = 1.5

P(Z<1.5) = P(X<1648) = 0.933193

(by using z-table or excel)

P(X>1648) = 1 – P(X<1648)

P(X>1648) = 1 – 0.933193

P(X>1648) = 0.066807

Required probability = 0.066807

Required percentage = 6.68%

Part b

Here, we have to find P(X<1568)

Z = (X – mean) / SD

Z = (1568 - 1600)/32

Z =-1

P(Z<-1) = P(X<1568) = 0.158655

(by using z-table or excel)

Required percentage = 15.87%

Part c

Here, we have to find P(1536<X<1648)

P(1536<X<1648) = P(X<1648) – P(X<1536)

P(X<1648) = 0.933193

(from part a)

Now find P(X<1536)

Z = (X – mean) / SD

Z = (1536 - 1600)/32

Z =-2

P(Z<-2) = P(X<1536) = 0.02275

(by using z-table)

P(1536<X<1648) = P(X<1648) – P(X<1536)

P(1536<X<1648) = 0.933193 - 0.02275

P(1536<X<1648) = 0.910443

Required percentage = 91.04%