5.30 5.31 5.32 5.33 5.34 Suppose the random variable x is best described by a nor mal distribution with p. = 25 and 0' = 5. Find the z-score that corresponds to each of the following x values: a. x = 25 b.x=30 c. x=37.5 d.x=10 e.x=50 f. x=32 Suppose x is a normally distributed random variable with p. = 11 and o' 2 2. Find each of the following: a. P(10 E .1: 5 12) . P(6 i: x E 10) P(13 <_: x . p :1 e suppose a: is a normally distributed random variable with and find value x0 of the such that a. .5 b. .133. .025 c.> x0) = .10 d. P(x > x0) = .95 Refer to Exercise 5.32. Find x0 such that a. 10% of the values of x are less than x0. I). 80% of the values of x are less than x0. (1. 1% of the values of x are greater than x0. rape-pa:- Suppose x is a normally distributed random variable with mean 100 and standard deviation 8. Draw a rough graph of the distribution of x. Locate ,u and the interval ,u. i 20' on the graph. Find the following probabilities: P(,u."2o'xE,u.+2cr) . P(x 2 [.L + 20') P(x 5 92) P(92 5 x E 116) P(92 E x 5 96) P(76 E x ii 124) weepcs 5.36 The random variable x has a normal distribution with y. = 3003ndo' = 30. a. Find the probability that .1: assumes a value more than two standard deviations from its mean. More than three standard deviations from 1.1.. b. Find the probability that x assumes a value within one standard deviation of its mean. Within two standard de- viations of ,u.. c. Find the value of x that represents the 80th percentile of this distribution.The 10th percentile. 5.39 Deep mixing of soil. Deep mixing is a ground improvement method developed for soft soils like clay, silt. and peat. Swedish civil engineers investigated the properties of soil improved by deep mixing with lime-cement columns in the journal Giorisk (Vol. 7'. 2013). The mixed soil was tested by advancing a cylindrical rod with a cone tip down into the soil. During penetration. the cone penetrometer mea- sures the cone tip resistance (megapascals, MPa). The re- searchers established that tip resistance for the deep mixed soil followed a normal distribution with p. = 2.2 MPa and o' = .9 MP. a. Find the probability that the tip resistance will fall between 1.3 and 4.0 MPa. I]. Find the probability that the tip resistance will exceed 1.0 lePa. c. Find a value of tip resistance. T. such that 35 $3 of all soil samples have tip resistance values that exceed T