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Deep mixing is a ground improvement method developed for soft soils like​ clay, silt, and peat. Civil engineers investigated the properties of soil improved by deep mixing with​ lime-cement columns. The mixed soil was tested by advancing a cylindrical rod with a cone tip down into the soil. During​ penetration, the cone penetrometer measures the cone tip resistance​ (megapascals, MPa). The researchers established that tip resistance for the deep mixed soil followed a normal distribution with muμequals=2.72.7 MPa and sigmaσequals=0.80.8 MPa. Complete parts a through c.

a. Find the probability that the tip resistance will fall between 1.3 and 4.0 MPa.
P (1.3 < x < 4.0) = ____

b. Find the probability that the tip resistance will exceed 1.0 MPa.
P (x > 1) = ____

c. Find a value of tip​ resistance, T, such that 36​% of all soil samples have tip resistance values that exceed T.
T = ____

Answer :

Answer:

a)  0.9078

b) 0.9832

c) T = 2.9867

Step-by-step explanation:

Given:-

T  ~ N (2.7 , 0.8 )

Find:

Find the probability that the tip resistance will fall between 1.3 and 4.0 MPa.

Find the probability that the tip resistance will exceed 1.0 MPa.

Find a value of tip​ resistance, T, such that 36​% of all soil samples have tip resistance values that exceed T

Solution:

a) To find the P ( 1.3 < x < 4.0 ) we will first compute the z-score values:

        P (1.3 < x < 4.0 ) = P ( (1.3 - 2.7)/0.8 < Z < (4.0 - 2.7)/0.8)

        P (1.3 < x < 4.0 ) = P ( -1.75 < Z < 1.625)

- Using the Z-table values we can evaluate the probability:

       P (1.3 < x < 4.0 ) = P ( -1.75 < Z < 1.625) = 0.9078

b) To find the P ( x > 1 ) we will first compute the z-score values:

        P ( x > 1.0 ) = P ( Z > (1.0 - 2.7)/0.8)

        P ( x > 1.0 ) = P ( Z > -2.125)

- Using the Z-table values we can evaluate the probability:

       P ( x > 1.0 ) = P ( Z > -2.125) = 0.9832

c) To find the P ( x > T ) we will first compute the z-score values:

       P ( x > T ) = P ( Z > (T - 2.7)/0.8) = 0.36

  - We will search for the z value with the condition above from Z-table.

       P ( Z > (T - 2.7)/0.8) = 0.36,  Z = 0.358375

  - Hence, we calculate for T:

        T = 0.358375*0.8 + 2.7

        T = 2.9867

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