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Chapter 4: Q122E (page 269)

Shear strength of rock fractures. Understanding the characteristics

of rock masses, especially the nature of the fracturesis essential when building dams and power plants.The shear strength of rock fractures was investigated inEngineering Geology(May 12, 2010). The Joint RoughnessCoefficient (JRC) was used to measure shear strength.Civil engineers collected JRC data for over 750 rock fractures.The results (simulated from information provided in the article) are summarized in the accompanying SPSShistogram. Should the engineers use the normal probabilitydistribution to model the behavior of shear strength forrock fractures? Explain

Short Answer

Expert verified

Engineers should use the normal probability distribution to model the behavior of shear strength for rock fractures

Step by step solution

01

Given Information

The histograms for 750 rock fractures is given,

02

Explanation

From the above histogram, it is seen that the histogram looks like a normal probability curve. So, the engineers can use normal distribution to model the behavior of shear strength for rock fractures.

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Most popular questions from this chapter

Variable speed limit control for freeways. A common transportation problem in large cities is congestion on the freeways. In the Canadian Journal of Civil Engineering (January 2013), civil engineers investigated the use of variable speed limits (VSL) to control the congestion problem. A portion of an urban freeway was divided into three sections of equal length, and variable speed limits were posted (independently) in each section. Probability distributions of the optimal speed limits for the three sections were determined. For example, one possible set of distributions is as follows (probabilities in parentheses). Section 1: 30 mph (.05), 40 mph (.25), 50 mph (.25), 60 mph (.45); Section 2: 30 mph (.10), 40 mph (.25), 50 mph (.35), 60 mph (.30); Section 3: 30 mph (.15), 40 mph (.20), 50 mph (.30), 60 mph (.35).

  1. Verify that the properties of a discrete probability distribution are satisfied for Section 1 of the freeway.
  2. Repeat part a for Sections 2 and 3 of the freeway.
  3. Find the probability that a vehicle traveling at the speed limit in Section 1 will exceed 30 mph.
  4. Repeat part c for Sections 2 and 3 of the freeway.

Consider the probability distributions shown here:

  1. Use your intuition to find the mean for each distribution. How did you arrive at your choice?
  2. Which distribution appears to be more variable? Why?
  3. Calculateฮผโ€‰andโ€‰ฯƒ2 for each distribution. Compare these answers with your answers in parts a and b.

Errors in measuring truck weights. To help highway planners anticipate the need for road repairs and design futureconstruction projects, data are collected on the estimatedvolume and weight of truck traffic on specific roadways (Transportation Planning Handbook, 2016) using specialized โ€œweigh-in-motionโ€ equipment. In an experiment involving repeated weighing of a 27,907-pound truck, it wasfound that the weights recorded by the weigh-in-motionequipment were approximately normally distributed witha mean of 27,315 and a standard deviation of 628 pounds

(Minnesota Department of Transportation). It follows thatthe difference between the actual weight and recordedweight, the error of measurement, is normally distributedwith a mean of 592 pounds and a standard deviation of 628pounds.

a. What is the probability that the weigh-in-motion equipment understates the actual weight of the truck?

b. If a 27,907-pound truck was driven over the weigh-in-motion equipment 100 times, approximately howmany times would the equipment overstate the truckโ€™sweight?

c. What is the probability that the error in the weightrecorded by the weigh-in-motion equipment for a27,907-pound truck exceeds 400 pounds?

d. It is possible to adjust (or calibrate) the weigh-in-motion equipment to control the mean error of measurement. At what level should the mean error beset so the equipment will understate the weight of a27,907-pound truck 50% of the time? Only 40% of thetime

Suppose the random variable x is best described by a normal distribution with ฮผ=30 and ฯƒ=4. Find the z-score that corresponds to each of the following x values:

a.x=20b.x=30c.x=2.75d.x=15e.x=35f.x=25

Variable life insurance return rates. With a variable life insurance policy, the rate of return on the investment (i.e., the death benefit) varies from year to year. A study of these variable return rates was published in International Journal of Statistical Distributions (Vol. 1, 2015). A transformedratio of the return rates (x) for two consecutive years was shown to have a normal distribution, with ฮผ=1.5 and role="math" localid="1660283206727" ฯƒ=0.2. Use the standard normal table or statistical software to find the following probabilities.

a.P(1.3<x<1.6)

b. P(x>1.4)

c. P(x<1.5)
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