, Test Bank For Probability And Statistics Forr1 r1 r1 r1 r1 r1
Engineering And The Sciences 8th Ed by Jay L.
r1 r1 r1 r1 r1 r1 r1 r1 r1
Devore.
r1
Chapter 1 – Overview and Descriptive Statistics
r1 r1 r1 r1 r1 r1
SHORT ANSWERr1
1. Give one possible sample of size 4 from each of the following populations:
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
a. All daily newspapers published in the United States
r1 r1 r1 r1 r1 r1 r1
b. All companies listed on the New York Stock Exchange
r1 r1 r1 r1 r1 r1 r1 r1
c. All students at your college or university
r1 r1 r1 r1 r1 r1
d. All grade point averages of students at your college or university
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
a. Houston Chronicle, Des Moines Register, Chicago Tribune, Washington Post
r1 r1 r1 r1 r1 r1 r1 r1
b. Capital One, Campbell Soup, Merrill Lynch, Pulitzer
r1 r1 r1 r1 r1 r1
c. John Anderson, Emily Black, Bill Carter, Kay Davis
r1 r1 r1 r1 r 1 r1 r1
d. 2.58. 2.96, 3.51, 3.69 r1 r1 r1
PTS: r 1 r 1 1
2. A Southern State University system consists of 23 campuses. An administrator wishes to make an inference
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
about the average distance between the hometowns of students and their campuses. Describe and discuss
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
several different sampling methods that might be employed. Would this be an enumerative or an analytic
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
study? Explain your reasoning.
r1 r1 r1 r1
ANS:
One could take a simple random sample of students from all students in the California State University
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
system and ask each student in the sample to report the distance from their hometown to campus.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
Alternatively, the sample could be generated by taking a stratified random sample by taking a simple
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
random sample from each of the 23 campuses and again asking each student in the sample to report the
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
distance from their hometown to campus.
r1 r1 r1 r1 r1 r1
Certain problems might arise with self reporting of distances, such as recording error or poor recall. This study is
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
enumerative because there exists a finite, identifiable population of objects from which to sample.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
3. A Michigan city divides naturally into ten district neighborhoods. How might a real estate appraiser select a
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 sample of single-family homes that could be used as a basis for developing an equation to predict
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 appraised value from characteristics such as age, size, number of bathrooms, and distance to the nearest
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 school, and so on? Is the study enumerative or analytic?
r1 r1 r1 r 1 r1 r1 r1 r1 r1
ANS:
One could generate a simple random sample of all single family homes in the city or a stratified random
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
sample by taking a simple random sample from each of the 10 district neighborhoods. From each of the
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
homes in the sample the necessary variables would be collected. This would be an enumerative study
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
because there exists a finite, identifiable population of objects from which to sample.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
, PTS: r 1 r 1 1
4. An experiment was carried out to study how flow rate through a solenoid valve in an automobile‘s pollution-
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
control system depended on three factors: armature lengths, spring load, and bobbin depth. Two different
r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1 r1 r 1 r1
levels (low and high) of each factor were chosen, and a single observation on flow was made for each
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
combination of levels.
r1 r1 r1
a. The resulting data set consisted of how many observations?
r1 r1 r1 r1 r1 r1 r1 r1
b. Is this an enumerative or analytic study? Explain your reasoning.
r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
a. Number observations equal 2 2 2=8 r1 r1 r1 r1 r 1 r1 r 1
b. This could be called an analytic study because the data would be collected on an existing
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
process. There is no sampling frame.
r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
5. The accompanying data specific gravity values for various wood types used in construction .
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
.41 .41 .42 .42. .42 .42 .42 .43 .44
.54 .55 .58 .62 .66 .66 .67 .68 .75
.31 .35 .36 .36 .37 .38 .40 .40 .40
.45 .46 .46 .47 .48 .48 .48 .51 .54
Construct a stem-and-leaf display using repeated stems and comment on any interesting features of the display.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
One method of denoting the pairs of stems having equal values is to denote the stem by L, for ‗low‘ and the
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
second stem by H, for ‗high‘. Using this notation, the stem-and-leaf display would appear as follows:
r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1 r1 r1 r1 r1
3L 1 stem: tenths r1
3H 56678 leaf: r 1 hundredths
4L 000112222234
5L 144
5H 58
6L 2
6H 6678
7L
7H 5
The stem-and-leaf display on the previous page shows that .45 is a good representative value for the data.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
In addition, the display is not symmetric and appears to be positively skewed. The spread of the data is
r 1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1
.75 - .31 = .44, which is .44/.45 = .978 or about 98% of the typical value of .45. This constitutes a
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1
reasonably large amount of variation in the data. The data value .75 is a possible outlier.
r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
6. Temperature transducers of a certain type are shipped in batches of 50. A sample of 60 batches was
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1
selected, and the number of transducers in each batch not conforming to design specifications was
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
determined, resulting in the following data:
r1 r1 r1 r1 r1 r1
0 4 r 1 r 1 2 r 1 1 1 3 4 1 2 3 2 2 8 4 5 1 3 1
r 1 1 r 1 r 1 3
2 1 r 1 r 1 2 r 1 1 3 2 0 5 3 3 1 3 2 4 7 0 2 3
r 1 4 r 1 r 1 0
5 0 r 1 r 1 2 r 1 1 0 6 4 2 1 6 0 3 3 3 6 1 2 3
r 1 3 r 1 r 1 2
, a. Determine frequencies and relative frequencies for the observed values of x = number of nonconforming
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
transducers in a batch.
r1 r1 r1 r1
b. What proportion of batches in the sample has at most four nonconforming transducers? What
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
proportion has fewer than four? What proportion has at least four nonconforming units?
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
a.
Number Nonconforming r1 Relative Frequency Frequency r1
0 0.117 7
1 0.200 12
2 0.217 13
3 0.233 14
4 0.100 6
5 0.050 3
6 0.050 3
7 0.017 1
8 0.017 1
1.001
The relative r1 r 1 frequencies don’t add up exactly to 1because they have been rounded
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
b. The number of batches with at most 4 nonconforming items is 7+12+13+14+6=52, which is a proportion of
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
52/60=.867. The proportion of batches with (strictly) fewer than 4 nonconforming items is 46/60=.767.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
7. The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 for each wafer in a sample size 100, resulting in the following frequencies:
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
Number of particles r1 r 1 Frequency Number of particles r1 r 1 Frequency
0 1 8 12
1 2 9 4
2 3 10 5
3 12 11 3
4 11 12 1
5 15 13 2
6 18 14 1
7 10
a. What proportion of the sampled wafers had at least two particles? At least six particles?
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
b. What proportion of the sampled wafers had between four and nine particles, inclusive? Strictly between
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
four and nine particles?
r1 r1 r1 r1
ANS:
a. From this frequency distribution, the proportion of wafers that contained at least two particles is (100-1-2)/100 =
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
.97, or 97%. In a similar fashion, the proportion containing at least 6 particles is (100 – 1-2-3-12-11-
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
15)/100 = 56/100 = .56, or 56%. r1 r1 r1 r1 r1 r1
b. The proportion containing between 4 and 9 particles inclusive is (11+15+18+10+12+4)/100 = 70/100 =
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
.70, or 70%. The proportion that contain strictly between 4 and 9 (meaning strictly more than 4 and
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
strictly less than 9) is (15+ 18+10+12)/100= 55/100 = .55, or 55%.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
Engineering And The Sciences 8th Ed by Jay L.
r1 r1 r1 r1 r1 r1 r1 r1 r1
Devore.
r1
Chapter 1 – Overview and Descriptive Statistics
r1 r1 r1 r1 r1 r1
SHORT ANSWERr1
1. Give one possible sample of size 4 from each of the following populations:
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
a. All daily newspapers published in the United States
r1 r1 r1 r1 r1 r1 r1
b. All companies listed on the New York Stock Exchange
r1 r1 r1 r1 r1 r1 r1 r1
c. All students at your college or university
r1 r1 r1 r1 r1 r1
d. All grade point averages of students at your college or university
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
a. Houston Chronicle, Des Moines Register, Chicago Tribune, Washington Post
r1 r1 r1 r1 r1 r1 r1 r1
b. Capital One, Campbell Soup, Merrill Lynch, Pulitzer
r1 r1 r1 r1 r1 r1
c. John Anderson, Emily Black, Bill Carter, Kay Davis
r1 r1 r1 r1 r 1 r1 r1
d. 2.58. 2.96, 3.51, 3.69 r1 r1 r1
PTS: r 1 r 1 1
2. A Southern State University system consists of 23 campuses. An administrator wishes to make an inference
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
about the average distance between the hometowns of students and their campuses. Describe and discuss
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
several different sampling methods that might be employed. Would this be an enumerative or an analytic
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
study? Explain your reasoning.
r1 r1 r1 r1
ANS:
One could take a simple random sample of students from all students in the California State University
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
system and ask each student in the sample to report the distance from their hometown to campus.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
Alternatively, the sample could be generated by taking a stratified random sample by taking a simple
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
random sample from each of the 23 campuses and again asking each student in the sample to report the
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
distance from their hometown to campus.
r1 r1 r1 r1 r1 r1
Certain problems might arise with self reporting of distances, such as recording error or poor recall. This study is
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
enumerative because there exists a finite, identifiable population of objects from which to sample.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
3. A Michigan city divides naturally into ten district neighborhoods. How might a real estate appraiser select a
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 sample of single-family homes that could be used as a basis for developing an equation to predict
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 appraised value from characteristics such as age, size, number of bathrooms, and distance to the nearest
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 school, and so on? Is the study enumerative or analytic?
r1 r1 r1 r 1 r1 r1 r1 r1 r1
ANS:
One could generate a simple random sample of all single family homes in the city or a stratified random
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
sample by taking a simple random sample from each of the 10 district neighborhoods. From each of the
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
homes in the sample the necessary variables would be collected. This would be an enumerative study
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
because there exists a finite, identifiable population of objects from which to sample.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
, PTS: r 1 r 1 1
4. An experiment was carried out to study how flow rate through a solenoid valve in an automobile‘s pollution-
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
control system depended on three factors: armature lengths, spring load, and bobbin depth. Two different
r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1 r1 r 1 r1
levels (low and high) of each factor were chosen, and a single observation on flow was made for each
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
combination of levels.
r1 r1 r1
a. The resulting data set consisted of how many observations?
r1 r1 r1 r1 r1 r1 r1 r1
b. Is this an enumerative or analytic study? Explain your reasoning.
r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
a. Number observations equal 2 2 2=8 r1 r1 r1 r1 r 1 r1 r 1
b. This could be called an analytic study because the data would be collected on an existing
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
process. There is no sampling frame.
r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
5. The accompanying data specific gravity values for various wood types used in construction .
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
.41 .41 .42 .42. .42 .42 .42 .43 .44
.54 .55 .58 .62 .66 .66 .67 .68 .75
.31 .35 .36 .36 .37 .38 .40 .40 .40
.45 .46 .46 .47 .48 .48 .48 .51 .54
Construct a stem-and-leaf display using repeated stems and comment on any interesting features of the display.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
One method of denoting the pairs of stems having equal values is to denote the stem by L, for ‗low‘ and the
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
second stem by H, for ‗high‘. Using this notation, the stem-and-leaf display would appear as follows:
r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1 r1 r1 r1 r1
3L 1 stem: tenths r1
3H 56678 leaf: r 1 hundredths
4L 000112222234
5L 144
5H 58
6L 2
6H 6678
7L
7H 5
The stem-and-leaf display on the previous page shows that .45 is a good representative value for the data.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
In addition, the display is not symmetric and appears to be positively skewed. The spread of the data is
r 1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1
.75 - .31 = .44, which is .44/.45 = .978 or about 98% of the typical value of .45. This constitutes a
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1
reasonably large amount of variation in the data. The data value .75 is a possible outlier.
r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
6. Temperature transducers of a certain type are shipped in batches of 50. A sample of 60 batches was
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r 1 r1 r1 r1 r1 r1
selected, and the number of transducers in each batch not conforming to design specifications was
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
determined, resulting in the following data:
r1 r1 r1 r1 r1 r1
0 4 r 1 r 1 2 r 1 1 1 3 4 1 2 3 2 2 8 4 5 1 3 1
r 1 1 r 1 r 1 3
2 1 r 1 r 1 2 r 1 1 3 2 0 5 3 3 1 3 2 4 7 0 2 3
r 1 4 r 1 r 1 0
5 0 r 1 r 1 2 r 1 1 0 6 4 2 1 6 0 3 3 3 6 1 2 3
r 1 3 r 1 r 1 2
, a. Determine frequencies and relative frequencies for the observed values of x = number of nonconforming
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
transducers in a batch.
r1 r1 r1 r1
b. What proportion of batches in the sample has at most four nonconforming transducers? What
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
proportion has fewer than four? What proportion has at least four nonconforming units?
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
ANS:
a.
Number Nonconforming r1 Relative Frequency Frequency r1
0 0.117 7
1 0.200 12
2 0.217 13
3 0.233 14
4 0.100 6
5 0.050 3
6 0.050 3
7 0.017 1
8 0.017 1
1.001
The relative r1 r 1 frequencies don’t add up exactly to 1because they have been rounded
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
b. The number of batches with at most 4 nonconforming items is 7+12+13+14+6=52, which is a proportion of
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
52/60=.867. The proportion of batches with (strictly) fewer than 4 nonconforming items is 46/60=.767.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
PTS: r 1 r 1 1
7. The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
r1 for each wafer in a sample size 100, resulting in the following frequencies:
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
Number of particles r1 r 1 Frequency Number of particles r1 r 1 Frequency
0 1 8 12
1 2 9 4
2 3 10 5
3 12 11 3
4 11 12 1
5 15 13 2
6 18 14 1
7 10
a. What proportion of the sampled wafers had at least two particles? At least six particles?
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
b. What proportion of the sampled wafers had between four and nine particles, inclusive? Strictly between
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
four and nine particles?
r1 r1 r1 r1
ANS:
a. From this frequency distribution, the proportion of wafers that contained at least two particles is (100-1-2)/100 =
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
.97, or 97%. In a similar fashion, the proportion containing at least 6 particles is (100 – 1-2-3-12-11-
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
15)/100 = 56/100 = .56, or 56%. r1 r1 r1 r1 r1 r1
b. The proportion containing between 4 and 9 particles inclusive is (11+15+18+10+12+4)/100 = 70/100 =
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
.70, or 70%. The proportion that contain strictly between 4 and 9 (meaning strictly more than 4 and
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
strictly less than 9) is (15+ 18+10+12)/100= 55/100 = .55, or 55%.
r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1
