EXERCISES Section 4.3 (28–58)

28

Let $Z$ be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.

a. $P\left(0\le Z\le 2.17\right)$ b. $P\left(0\le Z\le 1\right)$ c. $P\left(-2.50\le Z\le 0\right)$ d. $P\left(-2.50\le Z\le 2.50\right)$ e. $P\left(Z\le 1.37\right)$ f. $P\left(-1.75\le Z\right)$ g. $P\left(-1.50\le Z\le 2.00\right)$ h. $P\left(1.37\le Z\le 2.50\right)$ i. $P\left(1.50\le Z\right)$ j. $P\left(\left|Z\right|\le 2.50\right)$

29

In each case, determine the value of the constant $c$ that makes the probability statement correct.

a. $\Phi \left(c\right)=.9838$ b. $P\left(0\le Z\le c\right)=.291$ c. $P\left(c\le Z\right)=.121$ d. $P\left(-c\le Z\le c\right)=.668$ e. $P\left(c\le \left|Z\right|\right)=.016$

30

Find the following percentiles for the standard normal distribution. Interpolate where appropriate.

a. 91st b. 9th c. 75th d. $25\mathrm{th}$ e. 6th

31

Determine $z_{\alpha }$ for the following values of $\alpha$ :

a. $\alpha =.0055$ b. $\alpha =.09$ c. $\alpha =.663$

32

Suppose the force acting on a column that helps to support a building is a normally distributed random variable $X$ with mean value $15.0\mathrm{kips}$ and standard deviation 1.25 kips. Compute the following probabilities by standardizing and then using Table A.3.

a. $P\left(X\le 15\right)$ b. $P\left(X\le 17.5\right)$ c. $P\left(X\ge 10\right)$ d. $P\left(14\le X\le 18\right)$ e. $P\left(\left|X-15\right|\le 3\right)$

33

Mopeds (small motorcycles with an engine capacity below $50{\mathrm{cm}}^3$ ) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (J. of Automobile Engr., 2008: 1615-1623) described a rolling bench test for determining maximum vehicle speed. A normal distribution with mean value $46.8\mathrm{km}/\mathrm{h}$ and standard deviation $1.75\mathrm{km}/\mathrm{h}$ is postulated.

Consider randomly selecting a single such moped.

a. What is the probability that maximum speed is at most $50\mathrm{km}/\mathrm{h}$ ?

b. What is the probability that maximum speed is at least $48\mathrm{km}/\mathrm{h}$ ?

c. What is the probability that maximum speed differs from the mean value by at most 1.5 standard deviations?

34

The article “Reliability of Domestic-Waste Biofilm Reactors” (J. of Envir. Engr., 1995: 785-790) suggests that substrate concentration $\left(\mathrm{mg}/{\mathrm{cm}}^3\right)$ of influent to a reactor is normally distributed with $\mu =.30$ and $\sigma =.06$ .

a. What is the probability that the concentration exceeds .50 ?

b. What is the probability that the concentration is at most .20?

c. How would you characterize the largest $5\%$ of all concentration values?

35

In a road-paving process, asphalt mix is delivered to the hopper of the paver by trucks that haul the material from the batching plant. The article “Modeling of Simultaneously Continuous and Stochastic Construction Activities for Simulation” (J. of Construction Engr. and Mgmnt., 2013: 1037-1045) proposed a normal distribution with mean value $8.46\min$ and standard deviation $.913\min$ for the rv $X=$ truck haul time.

a. What is the probability that haul time will be at least 10 min? Will exceed 10 min?

b. What is the probability that haul time will exceed 15 min?

c. What is the probability that haul time will be between 8 and 10 min?

d. What value $c$ is such that $98\%$ of all haul times are in the interval from $8.46-c$ to $8.46+c$ ?

e. If four haul times are independently selected, what is the probability that at least one of them exceeds $10\min$ ?

36

Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper “Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition” (Weed Technology, 2005: 1030-1036) investigated the effects of herbicide formulation on spray atomization. A figure in the paper suggested the normal distribution with mean $1050\mu \mathrm{m}$ and standard deviation $150\mu \mathrm{m}$ was a reasonable model for droplet size for water (the “control treatment”) sprayed through a $760\mathrm{ml}/\min$ nozzle.

a. What is the probability that the size of a single droplet is less than $1500\mu \mathrm{m}$ ? At least $1000\mu \mathrm{m}$ ?

b. What is the probability that the size of a single droplet is between 1000 and $1500\mu \mathrm{m}$ ?

c. How would you characterize the smallest $2\%$ of all droplets?

d. If the sizes of five independently selected droplets are measured, what is the probability that exactly two of them exceed $1500\mu \mathrm{m}$ ?

37

Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5 (information in the article “Mathematical Model of Chloride Concentration in Human Blood,” J. of Med. Engr. and Tech., 2006: 25-30, including a normal probability plot as described in Section 4.6, supports this assumption).

a. What is the probability that chloride concentration equals 105? Is less than 105? Is at most 105?

b. What is the probability that chloride concentration differs from the mean by more than 1 standard deviation? Does this probability depend on the values of $\mu$ and $\sigma$ ?

c. How would you characterize the most extreme $.1\%$ of chloride concentration values?

38

There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean $3\mathrm{cm}$ and standard deviation $.1\mathrm{cm}$ . The second machine produces corks with diameters that have a normal distribution with mean $3.04\mathrm{cm}$ and standard deviation $.02\mathrm{cm}$ . Acceptable corks have diameters between $2.9\mathrm{cm}$ and $3.1\mathrm{cm}$ . Which machine is more likely to produce an acceptable cork?

39

The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value $30\mathrm{mm}$ and standard deviation $7.8\mathrm{mm}$

• suggested in the article “Reliability Evaluation of Corroding Pipelines Considering Multiple Failure Modes and Time-Dependent Internal Pressure” (J. of Infrastructure Systems, 2011: 216-224)

a. What is the probability that defect length is at most $20\mathrm{mm}$ ? Less than $20\mathrm{mm}$ ?

b. What is the 75th percentile of the defect length distribution-that is, the value that separates the smallest $75\%$ of all lengths from the largest $25\%$ ?

c. What is the 15th percentile of the defect length distribution?

d. What values separate the middle $80\%$ of the defect length distribution from the smallest $10\%$ and the largest $10\%$ ?

40

The article “Monte Carlo Simulation-Tool for Better Understanding of LRFD” (J. of Structural Engr., 1993: 1586-1599) suggests that yield strength (ksi) for

A36 grade steel is normally distributed with $\mu =43$ and $\sigma =4.5$ .

a. What is the probability that yield strength is at most 40? Greater than 60?

b. What yield strength value separates the strongest $75\%$ from the others?

41

The automatic opening device of a military cargo parachute has been designed to open when the parachute is $200\mathrm{m}$ above the ground. Suppose opening altitude actually has a normal distribution with mean value $200\mathrm{m}$ and standard deviation $30\mathrm{m}$ . Equipment damage will occur if the parachute opens at an altitude of less than $100\mathrm{m}$ . What is the probability that there is equipment damage to the payload of at least one of five independently dropped parachutes?

42

The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean $\mu$ , the actual temperature of the medium, and standard deviation $\sigma$ . What would the value of $\sigma$ have to be to ensure that $95\%$ of all readings are within ${.1}^{\circ }$ of $\mu$ ?

43

Vehicle speed on a particular bridge in China can be modeled as normally distributed (“Fatigue Reliability Assessment for Long-Span Bridges under Combined Dynamic Loads from Winds and Vehicles,” J. of Bridge Engr., 2013: 735-747).

a. If $5\%$ of all vehicles travel less than $39.12\mathrm{m}/\mathrm{h}$ and $10\%$ travel more than $73.24\mathrm{m}/\mathrm{h}$ , what are the mean and standard deviation of vehicle speed? [Note: The resulting values should agree with those given in the cited article.]

b. What is the probability that a randomly selected vehicle’s speed is between 50 and $65\mathrm{m}/\mathrm{h}$ ?

c. What is the probability that a randomly selected vehicle’s speed exceeds the speed limit of $70\mathrm{m}/\mathrm{h}$ ?

44

If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is

a. Within 1.5 SDs of its mean value?

b. Farther than 2.5 SDs from its mean value?

c. Between 1 and 2 SDs from its mean value?

45

A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is .500 in. A bearing is acceptable if its diameter is within .004 in. of this target value. Suppose, however, that the setting has changed during the course of production, so that the bearings have normally distributed diameters with mean value .499 in. and standard deviation .002 in. What percentage of the bearings produced will not be acceptable?

46

The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation 3.

a. If a specimen is acceptable only if its hardness is between 67 and 75 , what is the probability that a randomly chosen specimen has an acceptable hardness?

b. If the acceptable range of hardness is $\left(70-c,70+c\right)$ , for what value of $c$ would $95\%$ of all specimens have acceptable hardness?

c. If the acceptable range is as in part (a) and the hardness of each of ten randomly selected specimens is indepen-dently determined, what is the expected number of acceptable specimens among the ten?

d. What is the probability that at most eight of ten independently selected specimens have a hardness of less than 73.84? [Hint: $Y=$ the number among the ten specimens with hardness less than 73.84 is a binomial variable; what is $p$ ?]

47

The weight distribution of parcels sent in a certain manner is normal with mean value $12\mathrm{lb}$ and standard deviation $3.5\mathrm{lb}$ . The parcel service wishes to establish a weight value $c$ beyond which there will be a surcharge. What value of $c$ is such that $99\%$ of all parcels are at least $1\mathrm{lb}$ under the surcharge weight?

48

Suppose Appendix Table A. 3 contained $\Phi \left(z\right)$ only for $z\ge 0$ . Explain how you could still compute

a. $P\left(-1.72\le Z\le -.55\right)$

b. $P\left(-1.72\le Z\le .55\right)$

Is it necessary to tabulate $\Phi \left(z\right)$ for $z$ negative? What property of the standard normal curve justifies your answer?

49

Consider babies born in the “normal” range of 37-43 weeks gestational age. Extensive data supports the assumption that for such babies born in the United States, birth weight is normally distributed with mean $3432\mathrm{g}$ and standard deviation $482\mathrm{g}$ .

• The article “Are Babies Normal?” (The American Statistician, 1999: 298-302) analyzed data from a particular year; for a sensible choice of class intervals, a histogram did not look at all normal, but after further investigations it was determined that this was due to some hospitals measuring weight in grams and others measuring to the nearest ounce and then converting to grams. A modified choice of class intervals that allowed for this gave a histogram that was well described by a normal distribution.

a. What is the probability that the birth weight of a randomly selected baby of this type exceeds $4000\mathrm{g}$ ? Is between 3000 and $4000\mathrm{g}$ ?

b. What is the probability that the birth weight of a randomly selected baby of this type is either less than $2000\mathrm{g}$ or greater than $5000\mathrm{g}$ ?

c. What is the probability that the birth weight of a randomly selected baby of this type exceeds $7\mathrm{lb}$ ?

d. How would you characterize the most extreme $.1\%$ of all birth weights?

e. If $X$ is a random variable with a normal distribution and $a$ is a numerical constant $\left(a\ne 0\right)$ , then $Y=aX$ also has a normal distribution. Use this to determine the distribution of birth weight expressed in pounds (shape, mean, and standard deviation), and then recalculate the probability from part (c). How does this compare to your previous answer?

50

In response to concerns about nutritional contents of fast foods, McDonald’s has announced that it will use a new cooking oil for its french fries that will decrease substantially trans fatty acid levels and increase the amount of more beneficial polyunsaturated fat. The company claims that 97 out of 100 people cannot detect a difference in taste between the new and old oils. Assuming that this figure is correct (as a long-run proportion), what is the approximate probability that in a random sample of 1000 individuals who have purchased fries at McDonald’s,

a. At least 40 can taste the difference between the two oils?

b. At most $5\%$ can taste the difference between the two oils?

51

Chebyshev’s inequality, (see Exercise 44, Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number $k$ satisfying $k\ge 1,P\left(\left|X-\mu \right|\ge k\sigma \right)\le 1/k^2$ (see Exercise 44 in Chapter 3 for an interpretation). Obtain this probability in the case of a normal distribution for $k=1,2$ , and 3, and compare to the upper bound.

52

Let $X$ denote the number of flaws along a 100 -m reel of magnetic tape (an integer-valued variable). Suppose $X$ has approximately a normal distribution with $\mu =25$ and $\sigma =5$ . Use the continuity correction to calculate the probability that the number of flaws is

a. Between 20 and 30, inclusive.

b. At most 30. Less than 30.

53

Let $X$ have a binomial distribution with parameters $n=25$ and $p$ . Calculate each of the following probabilities using the normal approximation (with the continuity correction) for the cases $p=.5,.6$ , and .8 and compare to the exact probabilities calculated from Appendix Table A.1.

a. $P\left(15\le X\le 20\right)$

b. $P\left(X\le 15\right)$

c. $P\left(20\le X\right)$

54

Suppose that $10\%$ of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). Consider a random sample of 200 shafts, and let $X$ denote the number among these that are nonconforming and can be reworked. What is the (approximate) probability that $X$ is

a. At most 30 ?

b. Less than 30 ?

c. Between 15 and 25 (inclusive)?

55

Suppose only $75\%$ of all drivers in a certain state regularly wear a seat belt. A random sample of 500 drivers is selected. What is the probability that

a. Between 360 and 400 (inclusive) of the drivers in the sample regularly wear a seat belt?

b. Fewer than 400 of those in the sample regularly wear a seat belt?

56

Show that the relationship between a general normal percentile and the corresponding $z$ percentile is as stated in this section.

57

a. Show that if $X$ has a normal distribution with parameters $\mu$ and $\sigma$ , then $Y=aX+b$ (a linear function of $X$ ) also has a normal distribution. What are the parameters of the distribution of $Y$ [i.e., $E\left(Y\right)$ and $V\left(Y\right)]$ ? [Hint: Write the cdf of $Y,P\left(Y\le y\right)$ , as an integral involving the pdf of $X$ , and then differentiate with respect to $y$ to get the pdf of $Y$ .]

b. If, when measured in ${}^{\circ }\mathrm{C}$ , temperature is normally distributed with mean 115 and standard deviation 2, what can be said about the distribution of temperature measured in ${}^{\circ }\mathrm{F}$ ?

58

There is no nice formula for the standard normal cdf $\Phi \left(z\right)$ , but several good approximations have been published in articles. The following is from “Approximations for Hand Calculators Using Small Integer Coefficients” (Mathematics of Computation, 1977: 214-222). For $0<z\le 5.5$

$$P\left(Z\ge z\right)=1-\Phi \left(z\right)$$ $$\approx .5\exp \left\{-\left[\frac{\left(83z+351\right)z+562}{703/z+165}\right]\right\}$$

The relative error of this approximation is less than $.042\%$ . Use this to calculate approximations to the following probabilities, and compare whenever possible to the probabilities obtained from Appendix Table A.3.

a. $P\left(Z\ge 1\right)$ b. $P\left(Z<-3\right)$

c. $P\left(-4<Z<4\right)$ d. $P\left(Z>5\right)$