that has a stan- dard normal distribution

.65 Let denote a random variable that has a stan- dard normal distribution. Determine each of the follow- ing probabilities:

a.   P(2.36)

b.   P(2.36)

c.    P(-1.23)

d.   P(1.14 3.35)

e.    P(-0.77 -0.55)

f.   P(> 2)

g.    P(> -3.38)

h.   P(4.98)

7.66 Let denote a random variable having a normal distribution with m = 0 and s = 1. Determine each of the  following probabilities:

a.  P(0.10)

b.   P(-0.10)

c.  P(0.40 0.85)

d.   P(-0.85 -0.40)

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e.   P(-0.40 0.85)

f.   P(> -1.25)

g.   P(-1.50 or > 2.50)

Q476

7.67 Let denote a variable that has a standard normal distribution. Determine the value z* to satisfy the follow- ing conditions:

a.   P(z z*) = .025

b.   P(z z*) = .01

c.   P(z z*) = .05

d.   P(z*) = .02

e.   P(z*) = .01

f.    P(z* or z*) = .20

7.68   Determine the value z* that

a.      Separates the largest 3% of all values from the others

b.       Separates the largest 1% of all values from the others

c.       Separates the smallest 4% of all values from the others

d.      Separates the smallest 10% of all values from the others

Q478

7.69   Determine the value of z* such that

a.       z* and z* separate the middle 95% of all values from the most extreme 5%

b.       z* and z* separate the middle 90% of all values from the most extreme 10%

c.       z* and z* separate the middle 98% of all values from the most extreme 2%

d.       z* and z* separate the middle 92% of all values from the most extreme 8%

7.Suppose that 5% of cereal boxes contain a prize and the other 95% contain the message, “Sorry, try again.” Consider the random variable x, where = num- ber of boxes purchased until a prize is found.

a.       What is the probability that at most two boxes must be purchased?

b.       What is the probability that exactly four boxes must be purchased?

c.       What is the probability that more than four boxes must be purchased?

Q479

7.70 Because P(.44) = .67, 67% of all values are less than .44, and .44 is the 67th percentile of the stan- dard normal distribution. Determine the value of each of the following percentiles for the standard normal distri- bution (Hint: If the cumulative area that you must look for does not appear in the table, use the closest entry):

a.      The 91st percentile (Hint: Lookforarea.9100.)

b.       The 77th percentile

c.       The 50th percentile

d.      The 9th percentile

e.       What is the relationship between the 70th percen- tile and the 30th percentile?

Q480

7.71 Consider the population of all 1-gallon cans of dusty rose paint manufactured by a particular paint com- pany. Suppose that a normal distribution with mean m = 5 ml and standard deviation s = 0.2 ml is a reasonable model for the distribution of the variable = amount of

red dye in the paint mixture. Use the normal distribution model to calculate the following probabilities:

a.  P(5.0)                    b. P(5.4)

c.   P(5.4)                   d.  P(4.6 5.2)

e.  P(> 4.5)                    f.  P(> 4.0)

Q481

7.72 Consider babies born in the “normal” range of 37–43 weeks gestational age. The paper referenced in Example 7.27 (“Fetal Growth Parameters and Birth Weight: Their Relationship to Neonatal Body Compo- sition,” Ultrasound in Obstetrics and Gynecology [2009]: 441–446) suggests that a normal distribution with mean m = 3500 grams and standard deviation s = 600 grams is a reasonable model for the probability dis- tribution of the continuous numerical variable = birth weight of a randomly selected full-term baby.

a.      What is the probability that the birth weight of a randomly selected full-term baby exceeds 4000 g? is between 3000 and 4000 g?

b.       What is the probability that the birth weight of a randomly selected full-term baby is either less than 2000 g or greater than 5000 g?

c.       What is the probability that the birth weight of a randomly selected full-term baby exceeds 7 pounds? (Hint: 1 lb = 453.59 g.)

d.      How would you characterize the most extreme 0.1% of all full-term baby birth weights?

e.       If is a random variable with a normal distribution and is a numerical constant (# 0), then ax also has a normal distribution. Use this formula to determine the distribution of full-term baby birth weight expressed in pounds (shape, mean, and stan- dard deviation), and then recalculate the probability

from Part (c). How does this compare to your previ- ous answer?

Q482

7.73 Emissions of nitrogen oxides, which are major constituents of smog, can be modeled using a normal distribution. Let denote the amount of this pollutant emitted by a randomly selected vehicle (in parts per bil- lion). The distribution of can be described by a normal distribution with m = 1.6 and s = 0.4. Suppose that the EPA wants to offer some sort of incentive to get the worst polluters off the road. What emission levels constitute the worst 10% of the vehicles?

Q483

7.74 The paper referenced in Example 7.30 (“Estimat- ing Waste Transfer Station Delays Using GPS,” Waste Management [2008]: 1742–1750) describing processing times for garbage trucks also provided information on processing times at a second facility. At this second facil-

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ity, the mean total processing time was 9.9 minutes and the standard deviation of the processing times was

6.2 minutes. Explain why a normal distribution with mean 9.9 and standard deviation 6.2 would not be an appropriate model for the probability distribution of the variable = total processing time of a randomly selected truck entering this facility.

Q484

7.75 The size of the left upper chamber of the heart is one measure of cardiovascular health. When the upper left chamber is enlarged, the risk of heart problems is increased. The paper “Left Atrial Size Increases with Body Mass Index in Children” (International Journal of Cardiology [2009]: 1–7) described a study in which the left atrial size was measured for a large number of children age 5 to 15 years. Based on this data, the au- thors concluded that for healthy children, left atrial di- ameter was approximately normally distributed with a mean of 26.4 mm and a standard deviation of 4.2 mm.

a.       Approximately what proportion of healthy children has left atrial diameters less than 24 mm?

b.       Approximately what proportion of healthy children has left atrial diameters greater than 32 mm?

c.       Approximately what proportion of healthy children has left atrial diameters between 25 and 30 mm?

d.       For healthy children, what is the value for which only about 20% have a larger left atrial diameter?

Q485

7.76 The paper referenced in the previous exercise also included data on left atrial diameter for children who were considered overweight. For these children, left atrial diameter was approximately normally distributed with a mean of 28 mm and a standard deviation of 4.7 mm.

a.      Approximately what proportion of overweight chil- dren has left atrial diameters less than 25 mm?

b.       Approximately what proportion of overweight chil- dren has left atrial diameters greater than 32 mm?

c.      Approximately what proportion of overweight chil- dren has left atrial diameters between 25 and 30 mm?

d.      What proportion of overweight children has left atrial diameters greater than the mean for healthy children?

Q486

7.77 According to the paper “Commuters’ Exposure to Particulate Matter and Carbon Monoxide in Hanoi, Vietnam” (Transportation Research [2008]: 206–211), the carbon monoxide exposure of someone riding a mo- torbike for 5 km on a highway in Hanoi is approximately normally distributed with a mean of 18.6 ppm. Suppose that the standard deviation of carbon monoxide exposure is 5.7 ppm. Approximately what proportion of those who ride a motorbike for 5 km on a Hanoi highway will

experience a carbon monoxide exposure of more than 20 ppm? More than 25 ppm?

Q487

7.78 A machine that cuts corks for wine bottles oper- ates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with mean 3 cm and standard deviation

0.1 cm. The specifications call for corks with diameters between 2.9 and 3.1 cm. A cork not meeting the specifi- cations is considered defective. (A cork that is too small leaks and causes the wine to deteriorate; a cork that is too large doesn’t fit in the bottle.) What proportion of corks produced by this machine are defective?

Q488

7.79 Refer to Exercise 7.78. Suppose that there are two machines available for cutting corks. The machine described in the preceding problem produces corks with diameters that are approximately normally distributed with mean 3 cm and standard deviation 0.1 cm. The second machine produces corks with diameters that are approximately normally distributed with mean 3.05 cm and standard deviation 0.01 cm. Which machine would you recommend? (Hint: Which machine would produce fewer defective corks?)

Q489

7.80 A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the variable = actual capacity of a randomly selected tank has a distri- bution that is well approximated by a normal curve with mean 15.0 gallons and standard deviation 0.1 gallon.

a.      What is the probability that a randomly selected tank will hold at most 14.8 gallons?

b.       What is the probability that a randomly selected tank will hold between 14.7 and 15.1 gallons?

c.       If two such tanks are independently selected, what is the probability that both hold at most 15 gallons?

Q490

7.82 Suppose that the distribution of typing speed in words per minute (wpm) for experienced typists using a new type of split keyboard can be approximated by a normal curve with mean 60 wpm and standard deviation 15 wpm (“The Effects of Split Keyboard Geometry on Upper body Postures,” Ergonomics [2009]:  104–111).

a. What is the probability that a randomly selected typ- ist’s speed is at most 60 wpm? less than 60 wpm?

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Q491

7.83 The authors of the paper “Development of Nutritionally At-Risk Young Children is Predicted by Malaria, Anemia, and Stunting in Pemba, Zanzibar” (The Journal of Nutrition [2009]:763–772) studied fac- tors that might be related to dietary deficiencies in chil- dren. Children were observed for a length of time and the time spent in various activities was recorded.  One

a square root transformation to create a distribution that was more approximately normal. Data consistent with summary quantities in the paper for 15 children are given in the accompanying table. Normal scores for a samples size of 15 are also given.

    Fussing Time         Normal Score    

variable of interest was the length of time (in minutes) a0.05-1.739
child spent fussing. The authors comment that the dis-0.10-1.245
tribution of fussing times was skewed and that they used0.15-0.946

(continued)

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7.7 Checking for Normality and Normalizing Transformations       469

Fussing TimeNormal Score
0.40-0.714
0.70-0.515
1.05-0.333
1.95-0.165
2.150.000
3.700.165
3.900.335
4.500.515
6.000.714
8.000.946
11.001.245
14.001.739

a.       Construct a normal probability plot for the fussing time data. Does the plot look linear? Do you agree with the authors of the paper that the fussing time distribution is not normal?

b.       Transform the data by taking the square root of each data value. Construct a normal probability plot for the square root transformed data. How does this normal probability plot compare to the one from Part (a) for the untransformed data?

Q492

7.84 The paper “Risk Behavior, Decision Making, and Music Genre in Adolescent Males” (Marshall Uni- versity, May 2009) examined the effect of type of music playing and performance on a risky, decision-making task.

a. Participants in the study responded to a question- naire that was used to assign a risk behavior score. Risk behavior scores (read from a graph that ap- peared in the paper) for 15 participants follow. Use these data to construct a normal probability plot (the normal scores for a sample of size 15 appear in the previous exercise).

c. The author of the paper states that he believes that it is reasonable to consider both risk behavior scores and PANAS scores to be approximately normally distributed. Do the normal probability plots from Parts (a) and (b) support this conclusion? Explain.

Q493

7.85 Measures of nerve conductivity are used in the diagnosis of certain medical conditions. The paper “Ef- fects of Age, Gender, Height, and Weight on Late Responses and Nerve Conduction Study Parameters” (Acta Neurologica Taiwanica [2009]: 242–249) de- scribes a study in which the ulnar nerve was stimulated in healthy patients and the amplitude and velocity of the response was measured. Representative data (consistent with summary quantities and descriptions given in the paper) for 30 patients for the variable = response ve- locity (m/s) are given in the accompanying table. Also given are values of the log and square root of x.

102105113120125127134135
139141144145149150160 
36 40 45 47 48 49 50 52 53 54 56 59 61 62 70      

b. Participants also completed a positive and negative affect scale (PANAS) designed to measure emotional response to music. PANAS values (read from a graph that appeared in the paper) for 15 participants fol- low. Use these data to construct a normal probability plot (the normal scores for a sample of size 15 appear in the previous exercise).

x log(x) sqrt(x) 60.1 1.78 7.75 48.7 1.69 6.98 51.7 1.71 7.19 52.9 1.72 7.27 50.5 1.70 7.11 58.5 1.77 7.65 53.6 1.73 7.32 60.3 1.78 7.77 64.5 1.81 8.03 50.4 1.70 7.10 56.5 1.75 7.52 55.5 1.74 7.45 53.0 1.72 7.28 50.5 1.70 7.11 54.0 1.73 7.35 53.6 1.73 7.32 55.2 1.74 7.43 57.9 1.76 7.61 61.5 1.79 7.84 58.0 1.76 7.62 57.6 1.76 7.59 67.1 1.83 8.19 56.2 1.75 7.50 53.8 1.73 7.33 55.7 1.75 7.46 52.9 1.72 7.27 54.0 1.73 7.35 52.6 1.72 7.25 61.8 1.79 7.86    

     62.8            1.80               7.92       

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a.       Construct a histogram of the untransformed data. Does the distribution of appear to be approxi- mately normal? Explain.

b.       Construct a histogram of the log transformed data. Is this histogram more symmetric than the histo- gram of the untransformed data?

c.       Construct a histogram of the square root trans- formed data. Does either of the two transformations (square root or log) result in a histogram that is more nearly normal in shape?

Q494

7.86 Macular degeneration is the most common cause of blindness in people older than 60 years. One vari- able thought to be related to a type of inflammation as- sociated with this disease is level of a substance called sol- uble Fas ligand (sFasL) in the blood. The accompanying

pear more nearly linear than the plot for the untrans- formed data?

d. Compute the correlation coefficient for the (normal score, transformed x) pairs. Compare this value to the critical value from Table 7.2 to determine if it is reasonable to consider the distribution of trans- formed sFasL levels to be approximately normal.

Q495

7.87 The following normal probability plot was con- structed using part of the data appearing in the paper “Trace Metals in Sea Scallops” (Environmental Con- centration and Toxicology 19:  1326–1334).

Observation

+

–                                                                                                                                              *

–  

table contains representative data on = sFasL level for                         +

–  
*  

10 patients with age-related macular degeneration. These                     –

data are consistent with summary quantities and descrip- tions of the data given in the paper “Associations of

+

–                                                                                                        *

–                                                                                             *

–                                                                                  *

+    *                         *        *      *    *

Plasma-Soluble Fas Ligand with Aging and Age-Related

Macular Degeneration” (Investigative Ophthalmology & Visual Science [2008]: 1345–1349). The authors of  the

–      +—–+—–+—–+—–+—–+

-1.60    -0.80     0.00     0.80    1.60    2.40

Normal

score

paper noted that the distribution of sFasL level was skewed and recommended a cube-root transformation. The cube- root values and the normal scores for a sample size of 10 are also given in the accompanying table.

       x            Cube Root of x            Normal Score    

The variable under study was the amount of cadmium in

North Atlantic scallops. Do the sample data suggest that the cadmium concentration distribution is not normal? Explain.

Q496

7.90 Example 7.33 examined rainfall data for Min- neapolis–St. Paul. The square-root transformation was used to obtain a distribution of values that was more

Number of Purchases

Number of Households (Frequency)

symmetric than the distribution of the original data. Another power transformation that has been suggested by meteorologists is the cube root: transformed value = (original value)1/3. The original values and their cube roots (the transformed values) are given in the following table:

OriginalTransformedOriginalTransformed
0.320.681.511.15
0.470.781.621.17
0.520.801.741.20
0.590.841.871.23
0.770.921.891.24
0.810.931.951.25
0.810.932.051.27
0.900.972.101.28
0.960.992.201.30
1.181.062.481.35
1.201.062.811.41
1.201.063.001.44
1.311.093.091.46
1.351.113.371.50

        1.43                 1.13                    4.75                 1.68            

Construct a histogram of the transformed data. Compare your histogram to those given in Figure 7.38. Which of the cube-root and square-root transformations appear to result in the more symmetric histogram?

Q497

2  

7.91 The article “The Distribution of Buying Fre- quency Rates” (Journal of Marketing Research [1980]: 210–216) reported the results of a 31-year study of tooth- paste purchases. The investigators conducted their re- search using a national sample of 2071 households and recorded the number of toothpaste purchases for each household participating in the study. The results are given in the following frequency distribution:

Number of Households

80 to 90                                            20

90 to 100                                          13

100 to 110                                            9

110 to 120                                            7

120 to 130                                            6

130 to 140                                            6

140 to 150                                            3

150 to 160                                            0

              160 to 170                                            2                      

a.       Draw a histogram for this frequency distribution. Would you describe the histogram as positively or negatively skewed?

b.       Does the square-root transformation result in a his- togram that is more symmetric than that of the original data? (Be careful! This one is a bit tricky because you don’t have the raw data; transforming the endpoints of the class intervals will result in class intervals that are not necessarily of equal widths, so the histogram of the transformed values will have to be drawn with this in mind.)

Q498

7.92 The paper “Temperature and the Northern Distributions of Wintering Birds” (Ecology [1991]: 2274–2285) gave the following body masses (in grams) for 50 different bird species:

7.7  10.1   21.6     8.6    12.0    11.4    16.6     9.4

11.59.08.220.248.521.626.16.2
19.121.028.110.631.66.75.068.8
23.919.820.16.099.619.816.59.0
448.021.317.436.934.041.015.912.5
10.231.021.511.932.59.893.910.9
19.614.5      

a.      Construct a stem-and-leaf display in which 448.0 is listed separately beside the display as an outlier on the high side, the stem of an observation is the tens digit, the leaf is the ones digit, and the tenths digit is suppressed (e.g., 21.5 has stem 2 and leaf 1). What

Number of Purchases

(Frequency)

do you perceive as the most prominent feature of the

10 to 20                                         904

20 to 30                                         500

30 to 40                                         258

40 to 50                                         167

50 to 60                                           94

60 to 70                                           56

70 to 80                                           26

(continued)

display?

b.       Draw a histogram based on class intervals 5 to 10, 10 to 15, 15 to 20, 20 to 25, 25 to 30, 30 to 40, 40 to 50, 50 to 100, and 100 to 500. Is a transformation of the data desirable? Explain.

c.       Use a calculator or statistical computer package to calculate logarithms of these observations, and con-

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472            Chapter 7    Random Variables and Probability Distributions

struct a histogram. Is the log transformation success- ful in producing a more symmetric distribution?

d.      Consider transformed value =            1           and

!original value

construct a histogram of the transformed data. Does the histogram appear to resemble a normal curve?

Q499

7.93   The following figure appeared in the   paper

EDTA-Extractable Copper, Zinc, and Manganese  in

Soils of the Canterbury Plains” (New Zealand Journal of Agricultural Research [1984]: 207–217). A large num- ber of topsoil samples were analyzed for manganese (Mn), zinc (Zn), and copper (Cu), and the resulting data were summarized using histograms. The investigators transformed each data set using logarithms in an effort to obtain more symmetric distributions of values. Do you think the transformations were successful?   Explain.

Untransformed data

30                                                   Cu

20

10

0.5           1.0           1.5          2.0

–0.6                       0.0

30                                                   Zn

20

10

0.5           1.0           1.5          2.0

–0.6                      0.0

30                                                  Mn

20

10

15            30            45            60

1.0                          2.0

μg/g                                                                          log10(μg/g)

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Q500

7.94 Let denote the IQ for an individual selected at random from a certain population. The value of must be a whole number. Suppose that the distribution of can be approximated by a normal distribution with mean value 100 and standard deviation 15. Approximate the following probabilities:

a. P(= 100) b.  P(110)

c. P(110) (Hint: 110 is the same as 109.) d.   P(75 125)

7.95 Suppose that the distribution of the number of items produced by an assembly line during an 8-hour shift can be approximated by a normal distribution with mean value 150 and standard deviation 10.

a.       What is the probability that the number of items produced is at most 120?

b.       What is the probability that at least 125 items are produced?

c.       What is the probability that between 135 and 160 (inclusive) items are produced?

Q501

7.96 The number of vehicles leaving a turnpike at a certain exit during a particular time period has approxi- mately a normal distribution with mean value 500 and standard deviation 75. What is the probability that the number of cars exiting during this period is

a.      At least 650?

b.       Strictly between 400 and 550? (Strictly means that the values 400 and 550 are not included.)

c.      Between 400 and 550 (inclusive)?

7.97   Suppose that has a binomial distribution with = 50 and = .6, so that m = np = 30 and s = !np 11 2 p2

= 3.4641. Calculate the following probabilities using the normal approximation with the continuity correction:

a.   P(= 30)

b.    P(= 25)

c.    P(25)

d.   P(25 40)

e. P(25 40) (Hint: 25 , 40 is the same as 26 39.)

Q502

7.98 Symptom validity tests (SVTs) are sometimes used to confirm diagnosis of psychiatric disorders. The paper “Developing a Symptom Validity Test for Post- Traumatic Stress Disorder: Application of the Bino- mial Distribution” (Journal of Anxiety Disorders [2008]: 1297–1302) investigated the use of SVTs in the

diagnosis of post-traumatic stress disorder. One SVT proposed is a 60-item test (called the MENT test), where each item has only a correct or incorrect response. The MENT test is designed so that responses to the individ- ual questions can be considered independent of one an- other. For this reason, the authors of the paper believe that the score on the MENT test can be viewed as a bi- nomial random variable with = 60. The MENT test is designed to help in distinguishing fictitious claims of post-traumatic stress disorder. The items on the MENT test are written so that the correct response to an item should be relatively obvious, even to people suffering from stress disorders. Researchers have found that a pa- tient with a fictitious claim of stress disorder will try to “fake” the test, and that the probability of a correct re- sponse to an item for these patients is .7 (compared to

.96 for other patients). The authors used a normal ap- proximation to the binomial distribution with = 60 and p= .70 to compute various probabilities of interest, where = number of correct responses on the MENT test for a patient who is trying to fake the test.

a.      Verify that it is appropriate to use a normal approxi- mation to the binomial distribution in this situation.

b.       Approximate the following probabilities: i.       15 422

ii.       1, 422

iii.     1# 422

c.       Explain why the probabilities computed in Part (b) are not all equal.

d.       The authors computed the exact binomial probabil- ity of a score of 42 or less for someone who is not faking the test. Using = .96, they found

1# 422 5 .000000000013

Explain why the authors computed this probability using the binomial formula rather than using a nor- mal approximation.

e.       The authors propose that someone who scores 42 or less on the MENT exam is faking the test. Explain why this is reasonable, using some of the probabili- ties from Parts (b) and (d) as justification.

Q503

7.99 Studies have found that women diagnosed with cancer in one breast also sometimes have cancer in the other breast that was not initially detected by mammo- gram or physical examination (“MRI Evaluation of the Contralateral Breast in Women with Recently Diag- nosed Breast Cancer,” The New England Journal   of

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Medicine [2007]: 1295–1303). To determine if magnetic resonance imaging (MRI) could detect missed tumors in the other breast, 969 women diagnosed with cancer in one breast had an MRI exam. The MRI detected tumors in the other breast in 30 of these women.

a.  Use 5 30 5 .031 as an estimate of the probabil-

c.    P(x  > 35)

d. The probability that is farther than 2 standard de- viations from its mean value

Q504

7.102 Suppose that 65% of all registered voters in a certain area favor a 7-day waiting period before purchase of a handgun. Among 225 randomly selected registered

ity that

969

a woman

diagnosed with cancer in one

voters, what is the probability that

breast has an undetected tumor in the other breast. Consider a random sample of 500 women diagnosed with cancer in one breast. Explain why it is reason- able to think that the random variable = number in the sample who have an undetected tumor in the other breast has a binomial distribution with = 500 and = .031.

b.       Is it reasonable to use the normal distribution to ap- proximate probabilities for the random variable defined in Part (b)? Explain why or why not.

c.       Approximate the following probabilities: i.       1, 102

ii.   110 # # 252

iii.  1. 202

d. For each of the probabilities computed in Part (c), write a sentence interpreting the probability.

Q505

7.100 Seventy percent of the bicycles sold by a certain store are mountain bikes. Among 100 randomly selected bike purchases, what is the approximate probability that

a.      At most 75 are mountain bikes?

b.       Between 60 and 75 (inclusive) are mountain bikes?

c.       More than 80 are mountain bikes?

d.      At most 30 are not mountain bikes?

7.101 Suppose that 25% of the fire alarms in a large city are false alarms. Let denote the number of false alarms in a random sample of 100 alarms. Give approxi- mations to the following probabilities:

a.   P(20 30)

b.   P(20 30)

a.      At least 150 favor such a waiting period?

b.       More than 150 favor such a waiting period?

c.       Fewer than 125 favor such a waiting period?

Q506

7.103   Flashlight bulbs manufactured by a  certain company are sometimes  defective.

a.       If 5% of all such bulbs are defective, could the tech- niques of this section be used to approximate the probability that at least five of the bulbs in a random sample of size 50 are defective? If so, calculate this probability; if not, explain why not.

b.       Reconsider the question posed in Part (a) for the probability that at least 20 bulbs in a random sample of size 500 are defective.

Q507

7.104 A company that manufactures mufflers for cars offers a lifetime warranty on its products, provided that ownership of the car does not change. Suppose that only 20% of its mufflers are replaced under this warranty.

a.      In a random sample of 400 purchases, what is the ap- proximate probability that between 75 and 100 (in- clusive) mufflers are replaced under warranty?

b.       Among 400 randomly selected purchases, what is the probability that at most 70 mufflers are ulti- mately replaced under warranty?

c.       If you were told that fewer than 50 among 400 ran- domly selected purchases were ever replaced under warranty, would you question the 20% figure? Explain.

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Q508

7.105 There are approximately 40,000 travel agencies in the United States, of which 11,000 are members of the American Society of Travel Agents (booking a tour through an ASTA member increases the likelihood of a refund in the event of cancellation).

a. If is the number of ASTA members among 5000 randomly selected agencies, could you use the meth- ods of Section 7.8 to approximate P(1200 1400)? Why or why not?

ability that each of the 15 is able to purchase the type of drink desired? (Hint: Let denote the number among the 15 who want a diet drink. For which possible values of is everyone satisfied?)

Q509

7.107 A business has six customer service telephone lines.  Let  x  denote  the  number  of  lines  in   use at a specified time. Suppose that the probability distribu- tion of is as follows:

b.In a random sample of 100 agencies, what are thex0123456
 mean value and standard deviation of the number ofp(x).10.15.20.25.20.06.04

ASTA members?

c. If the sample size in Part (b) is doubled, does the standard deviation double?  Explain.

7.94 Let denote the IQ for an individual selected at random from a certain population. The value of must be a whole number. Suppose that the distribution of can be approximated by a normal distribution with mean value 100 and standard deviation 15. Approximate the following probabilities:

a. P(= 100) b.  P(110)

c. P(110) (Hint: 110 is the same as 109.) d.   P(75 125)

Q510

7.106 A soft-drink machine dispenses only regular Coke and Diet Coke. Sixty percent of all purchases from this machine are diet drinks. The machine currently has 10 cans of each type. If 15 customers want to purchase drinks before the machine is restocked, what is the prob-

Write each of the following events in terms of x, and then calculate the probability of each one:

a.       At most three lines are in use

b.       Fewer than three lines are in use

c.       At least three lines are in use

d.       Between two and five lines (inclusive) are in use

e.      Between two and four lines (inclusive) are not in use

f.       At least four lines are not in use





Standard normal distribution  





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Q511

7.109 A new battery’s voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two bat- teries, so batteries will be independently selected and tested until two acceptable ones have been found. Sup- pose that 80% of all batteries have acceptable voltages, and let denote the number of batteries that must be tested.

a.       What is p(2), that is, P(= 2)?

b.       What is p(3)? (Hint: There are two different out- comes that result in = 3.)

c.       In order to have = 5, what must be true of the fifth battery selected? List the four outcomes for which = 5, and then determine p(5).

d.       Use the pattern in your answers for Parts (a)–(c) to obtain a general formula for p(y).

Q512

7.110 A pizza company advertises that it puts 0.5 pounds of real mozzarella cheese on its medium pizzas. In fact, the amount of cheese on a randomly selected medium pizza  is  normally  distributed  with  a  mean  value of

0.5 pounds and a standard deviation of 0.025 pounds.

a.      What is the probability that the amount of cheese on a medium pizza is between 0.525 and 0.550 pounds?

b.       What is the probability that the amount of cheese on a medium pizza exceeds the mean value by more than 2 standard deviations?

c.       What is the probability that three randomly selected medium pizzas all have at least 0.475 pounds of cheese?

Q513

7.111 Suppose that fuel efficiency for a particular model car under specified conditions is normally distributed with a mean value of 30.0 mpg and a standard deviation of 1.2 mpg.

a.       What is the probability that the fuel efficiency for a randomly selected car of this type is between 29 and 31 mpg?

b.       Would it surprise you to find that the efficiency of a randomly selected car of this model is less than 25 mpg?

c.       If three cars of this model are randomly selected, what is the probability that all three have efficiencies exceeding 32 mpg?

d.       Find a number * such that 95% of all cars of this model have efficiencies exceeding * (i.e., P(*) = .95).

Q514

7.112 A coin is flipped 25 times. Let be the number of flips that result in heads (H). Consider the following rule for deciding whether or not the coin is fair:

Judge the coin fair if 8 17.

Judge the coin biased if either 7 or > 18.

a.      What is the probability of judging the coin biased when it is actually fair?

b.       What is the probability of judging the coin fair when P(H) = .9, so that there is a substantial bias? Repeat for P(H) = .1.

c.       What is the probability of judging the coin fair when P(H) = .6? when P(H) = .4? Why are these proba- bilities so large compared to the probabilities in Part (b)?

d.      What happens to the “error probabilities” of Parts (a) and (b) if the decision rule is changed so that the coin is judged fair if 7 18 and unfair otherwise? Is this a better rule than the one first proposed? Explain.

Q515

7.113 The probability distribution of x, the number of defective tires on a randomly selected automobile checked at a certain inspection station, is given in the following table:

x01234
p(x).54.16.06.04.20

The mean value of is m5 1.2. Calculate the values of

s2  

x  and sx.

7.108 Refer to the probability distribution of Exercise 7.107.

a.      Calculate the mean value and standard deviation of x.

b.       What is the probability that the number of lines in use is farther than 3 standard deviations from the mean value?

Q515

7.114 The amount of time spent by a statistical consul- tant with a client at their first meeting is a random vari- able having a normal distribution with a mean value of 60 minutes and a standard deviation of 10 minutes.

a.      What is the probability that more than 45 minutes is spent at the first meeting?

b.       What amount of time is exceeded by only 10% of all clients at a first meeting?

c.       If the consultant assesses a fixed charge of $10 (for overhead) and then charges $50 per hour, what is the mean revenue from a client’s first meeting?

Q516

7.115 The lifetime of a certain brand of battery is nor- mally distributed with a mean value of 6 hours and a standard deviation of 0.8 hours when it is used in a par- ticular DVD player. Suppose that two new batteries are

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independently selected and put into the player. The player ceases to function as soon as one of the batteries fails.

a.       What is the probability that the player functions for at least 4 hours?

b.       What is the probability that the DVD player works for at most 7 hours?

c.       Find a number * such that only 5% of all DVD players will function without battery replacement for more than c* hours.

Q517

7.117 The Wall Street Journal (February 15, 1972) re- ported that General Electric was sued in Texas for sex discrimination over a minimum height requirement of 5 ft. 7 in. The suit claimed that this restriction elimi- nated more than 94% of adult females from consider- ation. Let represent the height of a randomly selected adult woman. Suppose that is approximately normally distributed with mean 66 inches (5 ft. 6 in.) and stan- dard deviation 2 inches.

a.       Is the claim that 94% of all women are shorter than 5 ft. 7 in. correct?

b.       What proportion of adult women would be ex- cluded from employment as a result of the height restriction?

Q518

7.118 The longest “run” of ’s in the sequence SSFSSSSFFS has length 4, corresponding to the ’s on the fourth, fifth, sixth, and seventh trials. Consider a binomial experiment with = 4, and let be the length (number of trials) in the longest run of ’s.

a.      When = .5, the 16 possible outcomes are equally likely. Determine the probability distribution of in this case (first list all outcomes and the value for each one). Then calculate my.

b.       Repeat Part (a) for the case = .6.

c.       Let denote the longest run of either ’s or ’s. De- termine the probability distribution of when = .5.

Q519

7.119 Two sisters, Allison and Teri, have agreed to meet between 1 and 6 p.m. on a particular day. In fact, Allison is equally likely to arrive at exactly 1 p.m., 2 p.m., 3 p.m., 4 p.m., 5 p.m., or 6 p.m. Teri is also equally likely to ar-

rive at each of these six times, and Allison’s and Teri’s arrival times are independent of one another. Thus there are 36 equally likely (Allison, Teri) arrival-time pairs, for example, (2, 3) or (6, 1). Suppose that the first person to arrive waits until the second person arrives; let be the amount of time the first person has to wait.

a.       What is the probability distribution of w?

b.       On average, how much time do you expect to elapse between the two arrivals?

Q520

7.120 Four people—a, b, c, and d—are waiting to give blood. Of these four, a and b have type AB blood, whereas c and d do not. An emergency call has just come in for some type AB blood. If blood donations are taken one by one from the four people in random order and is the number of donations needed to obtain an AB in- dividual (so possible values are 1, 2, and 3), what is the probability distribution of x?

7.116 A machine producing vitamin E capsules operates so that the actual amount of vitamin E in each capsule is normally distributed with a mean of 5 mg and a standard deviation of 0.05 mg. What is the probability that a ran- domly selected capsule contains less than 4.9 mg of vita- min E? at least 5.2 mg?

Q521

7.121 Kyle and Lygia are going to play a series of Trivial Pursuit games. The first person to win four games will be declared the winner. Suppose that outcomes of succes- sive games are independent and that the probability of Lygia winning any particular game is .6. Define a ran- dom variable as the number of games played in the series.

a.       What is p(4)? (Hint: Either Kyle or Lygia could win four straight games.)

b.       What is p(5)? (Hint: For Lygia to win in exactly five games, what has to happen in the first four games and in Game 5?)

c.       Determine the probability distribution of x.

d.       On average, how many games will the series last?

Q522

7.124                   Suppose that the pH of soil samples taken from a certain geographic region is normally distributed with a mean pH of 6.00 and a standard deviation of 0.10. If the pH of a randomly selected soil sample from this region is determined, answer the following questions about it:

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a.      What is the probability that the resulting pH is be- tween 5.90 and 6.15?

b.       What is the probability that the resulting pH exceeds 6.10?

c.       What is the probability that the resulting pH is at most 5.95?

d.      What value will be exceeded by only 5% of all such pH values?

Q523

7.125 The lightbulbs used to provide exterior lighting for a large office building have an average lifetime of 700 hours. If length of life is approximately normally distributed with a standard deviation of 50 hours, how often should all the bulbs be replaced so that no more than 20% of the bulbs will have already burned out?

7.122 Refer to Exercise 7.121, and let be the number of games won by the series loser. Determine the proba- bility distribution of y.

Q524

7.126 Let denote the duration of a randomly selected pregnancy (the time elapsed between conception and birth). Accepted values for the mean value and standard deviation of are 266 days and 16 days, respectively. Suppose that the probability distribution of is (ap- proximately)  normal.

a.      What is the probability that the duration of preg- nancy is between 250 and 300 days?

b.       What is the probability that the duration of preg- nancy is at most 240 days?

c.      What is the probability that the duration of preg- nancy is within 16 days of the mean duration?

d.      A “Dear Abby” column dated January 20, 1973, contained a letter from a woman who stated that the duration of her pregnancy was exactly 310 days. (She wrote that the last visit with her husband, who was in the Navy, occurred 310 days before birth.) What is the probability that duration of pregnancy is at least 310 days? Does this probability make you a bit skeptical of the claim?

e.      Some insurance companies will pay the medical ex- penses associated with childbirth only if the insur- ance has been in effect for more than 9 months (275 days). This restriction is designed to ensure that the insurance company pays benefits for only those pregnancies for which conception occurred during coverage. Suppose that conception occurred 2 weeks after coverage began. What is the probability that the insurance company will refuse to pay benefits because of the 275-day insurance requirement?

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            Q525

CR7.1 The paper “The Psychology of Security” (Com- munications of the ACM [2008]: 36–41) states that people are more likely to gamble for a loss than to accept a guaranteed loss. For example, the authors indicate that when presented with a scenario that allows people to choose between a guaranteed gain of $5 and a gain   of

$10 if the flip of a coin results in a head, more people will choose the guaranteed gain. However, when pre- sented with a scenario that involves choosing between a guaranteed loss of $5 and a loss of $10 if the flip of a coin results in a head, more people will choose the gambling option. Describe how you would design an experiment that would allow you to test this theory.

Q526

49   52   53 59 59 61 69 70 70 73   79   80 85   87   89 90 92 93 93 95 96 97 114 149 208 318 334                  

CR7.2 The article “Water Consumption in the Bay Area” (San Jose Mercury News, January 16, 2010) re- ported the accompanying values for water consumption (gallons per person per day) for residential customers of 27 San Francisco Bay area water agencies:

a.      Use the given water consumption values to construct a boxplot. Describe any interesting features of the boxplot. Are there any outliers in the data set?

b.       Compute the mean and standard deviation for this data set. If the two largest values were deleted from the data set, would the standard deviation of this new data set be larger or smaller than the standard deviation you computed for the entire data set?

c.       How do the values of the mean and median of the entire data set compare? Is this consistent with the shape of the boxplot from Part (a)? Explain.

Q527

CR7.3 Red-light cameras are used in many places to deter drivers from running red lights. The following graphical display appeared in the article “Communities Put a Halt to Red-Light Cameras” (USA Today, January 18, 2010). Based on this graph, would it be correct to conclude that there were fewer red-light cameras in 2009 than in 2002? Explain.

7.123 Suppose that your statistics professor tells you that the scores on a midterm exam were approximately normally distributed with a mean of 78 and a standard deviation of 7. The top 15% of all scores have been des- ignated A’s. Your score is 89. Did you receive an A? Ex- plain.

Q528

CR7.4   The following quote is from USA Today (Janu- ary 21, 2010):

Most Americans think the Census is important, and the majority say they will participate in the count this spring—although Hispanics, younger people and the less educated are not as enthusiastic, a Pew Research Center survey found. The poll of 1504 adults found that perceptions differ: 74% of blacks and 72% of Hispanics rate the Census “very important” vs. 57% of whites. Believing it’s impor- tant doesn’t necessarily mean participating: Fewer than half of Hispanics, 57% of blacks and 61% of whites say they will fill out and mail the forms. Age is the greatest factor in participation: Only 36% of respondents who are 30 or younger say they will definitely participate.

Suppose that it is reasonable to regard survey participants as representative of adult Americans.

a. Suppose that an adult American is to be selected at random and consider the following events:

5 the event that the selected individual thinks that the Census is very important

5 the event that the selected individual is white

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Cumulative Review Exercises      495

Are the events and independent or dependent events? Explain.

Give an example of two census-related events differ- 1990 199 ent  than  the  ones  defined  in  Part  (a)  and  indicate 1991 166 whether they are independent or dependent events. 1992 156 Explain your reasoning. 1993 137   1994 177    

b.

    Year         Number of Manatee Deaths        

Q529

CR7.5 Beverly is building a fireplace. She is using spe- cial decorative 8-inch bricks for the bottom row of the fireplace. Although the bricks are advertised as 8-inch bricks, the actual length of a brick is variable, and the brick lengths are approximately

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