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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

Explain why each of the following is or is not a valid probability distribution for a discrete random variable x:

a.

x0123
P(x).1.3.3.2

b.

x-2-10
P(x).25.50.25

c.

x4920
P(x)-3.4.3

d.

x2356
P(x).15.15.45.35

The binomial probability distribution is a family of probability distributions with every single distribution depending on the values of n and p. Assume that x is a binomial random variable with n = 4.

  1. Determine a value of p such that the probability distribution of x is symmetric.
  2. Determine a value of p such that the probability distribution of x is skewed to the right.
  3. Determine a value of p such that the probability distribution of x is skewed to the left.
  4. Graph each of the binomial distributions you obtained in parts a, b, and c. Locate the mean for each distribution on its graph.\
  5. In general, for what values of p will a binomial distribution be symmetric? Skewed to the right? Skewed to the left?

Tax returns audited by the IRS. According to the Internal Revenue Service (IRS), the chances of your tax return being audited are about 1 in 100 if your income is less than \(1 million and 9 in 100 if your income is \)1 million or more (IRS Enforcement and Services Statistics).

  1. What is the probability that a taxpayer with income less than \(1 million will be audited by the IRS with income \)1 million or more?
  2. If five taxpayers with incomes under \(1 million are randomly selected, what is the probability that exactly one will be audited? That more than one will be audited?
  3. Repeat part b, assuming that five taxpayers with incomes of \)1 million or more are randomly selected.
  4. If two taxpayers with incomes under \(1 million are randomly selected, and two with incomes more than \)1 million are randomly selected, what is the probability that none of these taxpayers will be audited by the IRS?
  5. What assumptions did you have to make in order to answer these questions using the methodology presented in this section?

Acceptance sampling of a product. An essential tool in the monitoring of the quality of a manufactured product is acceptance sampling. An acceptance sampling plan involves knowing the distribution of the life length of the item produced and determining how many items to inspect from the manufacturing process. The Journal of Applied Statistics (April 2010) demonstrated the use of the exponential distribution as a model for the life length x of an item (e.g., a bullet). The article also discussed the importance of using the median of the lifetime distribution as a measure of product quality since half of the items in a manufactured lot will have life lengths exceeding the median. For an exponential distribution with a mean 胃, give an expression for the median of the distribution. (Hint: Your answer will be a function of 胃.)

Suppose x is a normally distributed random variable with 渭= 11 and 蟽= 2. Find each of the following:

a)P(10鈮�蠂鈮�12)

b) P(6鈮�蠂鈮�10)

c)P(13鈮�蠂鈮�16)

d)P(7.8鈮�蠂鈮�12.6)

e)P(蠂鈮�13.24)

f)P(蠂鈮�7.62)
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