, SOLUTION MANUAL FOR Probability and Statistics 10th Edition Ellingson
Important Notes
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, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
Solution and Answer Guide
ELLINGSON AND PANORSKA, PROBABILITY AND STATISTICS FOR ENGINEERING AND THE SCIENCES,
10E, 2027, 9798214023823; CHAPTER 1: OVERVIEW AND DESCRIPTIVE STATISTICS
TABLE OF CONTENTS
Section 1.1 ................................................................................................................................... 1
Section 1.2 ................................................................................................................................... 3
Section 1.3 ................................................................................................................................. 18
Section 1.4 ................................................................................................................................. 24
SECTION 1.1
1.
Solutions:
a. Los Angeles Times, Chicago Tribune, Gainesville Sun, Washington Post
b. France, Sweden, Denmark, Malta
c. Alice Johnson, Catherine Miller, Emma Davis, Ken Lee
d. 2.97, 3.56, 2.20, 2.97
2.
Solutions:
a. 45.3 m, 52.7 m, 60.1 m, 48.9 m
b. 432 pp, 196 pp, 184 pp, 321 pp
c. 4.5, 4.9, 5.6, 6.4
d. 0.07 g, 1.58 g, 7.1 g, 27.2 g
3.
Solutions:
a. How likely is it that more than half of the sampled computers will need or have needed
warranty service? What is the expected number among the 100 that need warranty service?
How likely is it that the number needing warranty service will exceed the expected number by
more than 10?
b. Suppose that 15 of the 100 sampled needed warranty service. How confident can we be that
the proportion of all such computers needing warranty service is between .08 and .22? Does
the sample provide compelling evidence for concluding that more than 10% of all such
computers need warranty service?
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 1
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
4.
Solutions:
a. Concrete populations: all living citizens of U.S., all the mutual funds currently marketed in
U.S., all the books that were published in 1980. Hypothetical populations: All GPAs of the
University of California undergraduates in the next academic year, page lengths of all books
that will be published next year, the batting averages of all major league players in the next
baseball season.
b. (Concrete) Probability: In a sample of 5 mutual funds, what is the chance that all
5 have rates of return which exceeded 10% last year? Statistics: If previous year rates-of-
return for 5 mutual funds were 9.6, 14.5, 8.3, 9.9 and 10.2, can we conclude that the average
rate for all funds was below 10%? (Hypothetical) Probability: In a sample of 10 books to be
published next year, how likely is it that the average number of pages for the 10 is between
200 and 250? Statistics: If the sample average number of pages for 10 books is 227, can we
be highly confident that the average for all books is between 200 and 245?
5.
Solutions:
a. No. All students taking a large statistics course who participate in an SI program of this sort.
b. The advantage to randomly allocating students to the two groups is that the two groups
should then be fairly comparable before the study. If the two groups perform differently in the
class, we might attribute this to the treatments (SI and control). If students were allowed to
choose, stronger or more motivated students might prefer SI, which could bias the results.
c. If all students were put in the treatment group, there would be no firm basis for assessing the
effectiveness of SI (nothing to which the SI scores could reasonably be compared).
6.
Solution:
One could take a simple random sample of students from all students in the California State
University system and ask each student in the sample to report the distance from their hometown
to campus. Alternatively, the sample could be generated by taking a stratified random sample by
taking a simple random sample from each of the 23 campuses and again asking each student in
the sample to report the distance from their hometown to campus. Certain problems might arise
with self-reporting of distances, such as recording error or poor recall. This study is enumerative
because there exists a finite, identifiable population of objects from which to sample.
7.
Solution:
One could generate a simple random sample of all single-family homes in the city, or a stratified
random sample by taking a simple random sample from each of the 10 district neighborhoods. From
each of the selected homes, values of all desired variables would be determined. This is an
enumerative study as there is a finite, well-defined population of objects that can be used as a sample.
8.
Solutions:
a. Number observations equal
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 2
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
b. This could be called an analytic study because the data would be collected on an existing
process. There is no sampling frame.
9.
Solutions:
a. The measurements might vary for different reasons. Some of these include measurement
error resulting from mechanical or technical changes across measurements, recording errors,
differences in weather conditions at the time of measurements, etc.
b. No, because there is no sampling frame.
SECTION 1.2
10.
Solutions:
a.
5 9
6 33588
7 00234677889
8 127
9 077 stem: ones
10 7 leaf: tenths
11 368
A representative strength for these beams is around 7.8 MPa, but there is a reasonably large
amount of variation around that representative value.
(What constitutes large or small variation usually depends on context, but variation is usually
considered large when the range of the data – the difference between the largest and
smallest value – is comparable to a representative value. Here, the range is
which is similar in size to the representative value of 7.8 MPa. So, most
researchers would call this a large amount of variation.)
b. The data display is not perfectly symmetric around some middle/representative value. There
is some positive skewness in this data.
c. Outliers are data points that appear to be very different from the pack. Looking at the stem-
and-leaf display in part (a), there appear to be no outliers in this data.
(A later section gives a more precise definition of what constitutes an outlier.)
d. From the stem-and-leaf display in part (a), there are 4 values greater than 10. Therefore, the
proportion of data values that exceed 10 is or, about 15%.
11.
Solution:
3L 1
3H 56678
4L 000112222234
4H 5667888 stem: tenths
5L 144 leaf: hundredths
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 3
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
5H 58
6L 2
6H 6678
7L
7H 5
The stem-and-leaf plot shows that .45 is a good representative value for the data. It also shows
that the plot is not symmetric and it is positively skewed. The range of the data is
which is comparable to the typical value of .45. This constitutes a reasonably large amount of
variation in the data. The data value .75 is a possible outlier.
12.
Solutions:
The sample size for this data set is .
a. The first four intervals correspond to observations less than 5, so the proportion of values
less than 5 is
b. The last four intervals correspond to observations at least 6, so the proportion of values at
least 6 is
c. The relative frequency and density histograms are shown below. The distribution of
particle sizes is not symmetric, but are positively skewed. Notice that the relative frequency
and density histograms are essentially identical, other than the vertical axis labeling, because
the bin widths are all the same.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 4
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
13.
Solution:
The histogram of ultimate strengths is symmetric and unimodal, with the point of symmetry at
approximately 135 ksi. There is a moderate amount of variation, and there are no gaps or outliers
in the distribution.
14.
Solutions:
a.
2 23 stem: 1.0
3 2344567789 leaf: .10
4 01356889
5 00001114455666789
6 0000122223344456667789999
7 00012233455555668
8 02233448
9 012233335666788
10 2344455688
11 2335999
12 37
13 8
14 36
15 0035
16
17
18 9
b. A representative is around 7.0.
c. The data exhibit a moderate amount of variation (this is subjective).
d. No, the data is skewed to the right, or positively skewed.
e. The value 18.9 appears to be an outlier, being more than two stem units from the previous
value.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 5
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
15.
Solution:
American French
8 1
755543211000 9 00234566
9432 10 2356
6630 11 1369
850 12 223558
8 13 7
14
15 8
2 16
American movie times are unimodal strongly positively skewed, while French movie times are
bimodal. A typical American movie runs about 95 minutes, while French movies are typically
either around 95 minutes or around 125 minutes. American movies are generally shorter than
French movies and are less variable in length. Finally, both American and French movies
occasionally run very long (outliers at 162 minutes and 158 minutes, respectively, in the
samples).
16.
Solutions:
a.
Beams Cylinders
9 5 8
88533 6 16
98877643200 7 012488
721 8 13359 stem: ones
770 9 278 leaf: tenths
7 10
863 11 2
12 6
13
14 1
The data are slightly skewed to the right, or positively skewed. The value of 14.1 MPa is an
outlier. About 15% of the measurements, that is, three out of twenty are 10 MPa.
b. The majority of observations are between 5 and 9 MPa for both beams and cylinders, with
the modal class being 7.0-7.9 MPa. The observations for cylinders arespread out, and the
maximum value of the cylinder observations is higher.
c.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 6
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
17.
Solutions:
The sample size for this data set is
a. “At most five bidders” means 2, 3, 4, or 5 bidders. The proportion of contracts that involved at
most 5 bidders is
Similarly, the proportion of contracts that involved at least 5 bidders (5 through 11) is equal to
b. The number of contracts with between 5 and 10 bidders, inclusive, is
so the proportion is “Strictly” between 5 and 10 means , or 9
bidders, for a proportion equal to
c. The distribution of number of bidders is positively skewed, ranging from 2 to
11 bidders, with a typical value of around 4-5 bidders.
18.
Solutions:
a. The most interesting feature of the histogram is the three very large outliers (21, 24, and 32
directors). Without these, the distribution of number of directors would be roughly symmetric
with a typical value of around 9.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 7
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
Note: One way to have Minitab automatically construct a histogram from grouped data such
as this is to use Minitab’s ability to enter multiple copies of the same number by typing, for
example, 42(9) to enter 42 copies of the number 9. The frequency data in this exercise was
entered using the following Minitab commands:
MTB > set c1
DATA> 3(4) 12(5) 13(6) 25(7) 24(8) 42(9) 23(10) 19(11) 16(12)
11(13) 5(14) 4(15) 1(16) 3(17) 1(21) 1(24) 1(32)
DATA> end
b. The frequency distribution shown below is almost the same as the one in the textbook,
except the three largest values are grouped into the “ ” category. If the data is presented
this way, we could not create a histogram, because we wouldn’t know the upper boundary for
the rectangle corresponding to the “ ” category.
No. dir. 4 5 6 7 8 9 10 11
Freq. 3 12 13 25 24 42 23 19
No dir. 12 13 14 15 16 17
Freq. 16 11 5 4 1 3 3
c. The sample size is So, the proportion of these corporations
that have at most 10 directors is
d. Similarly, the proportion of these corporations with more than 15 directors is
19.
Solutions:
a. From this frequency distribution, the proportion of wafers that contained at least one particle
is or . Note that it is much easier to subtract 1 (which is the number
of wafers that contain 0 particles) from 100 than it would be to add all the frequencies for
particles. In a similar fashion, the proportion containing at least 5 particles is
or,
b. The proportion containing between 5 and 10 particles is
or The proportion that contain strictly between 5 and 10 (meaning
strictly more than 5 and strictly less than 10) is or
44%.
c. The following histogram was constructed using Minitab. It is almost symmetric and unimodal.
It has a few smaller modes and has a very slight positive skew.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 8
accessible website, in whole or in part.
Important Notes
The file includes the complete test bank, organized chapter by chapter.
A sample of selected pages has been provided for preview.
All available appendices and Excel files (if included in the original resources) are
provided.
We continuously update our files to ensure you receive the latest and most accurate
editions.
New editions are added regularly – stay connected for updates!
✅ Why Buy From Us?
📚 Complete & organized chapter-by-chapter – no missing content, no guessing.
⚡ Instant digital delivery – get your file the moment you pay, no waiting.
📅 Always up to date – we track new editions so you always get the latest version.
💬 Friendly support – real humans ready to help, anytime you need us.
🔒 Safe & secure – thousands of satisfied students trust us every semester.
🛡️Our Guarantees
💰 Money-Back Guarantee: Not satisfied? We offer a full refund – no questions asked.
🔄 Wrong File? No Problem: Contact us and we will replace it immediately with the
correct version, free of charge.
⏰ 24/7 Support: We are always here – reach out anytime and expect a fast response.
Contact Email:
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
Solution and Answer Guide
ELLINGSON AND PANORSKA, PROBABILITY AND STATISTICS FOR ENGINEERING AND THE SCIENCES,
10E, 2027, 9798214023823; CHAPTER 1: OVERVIEW AND DESCRIPTIVE STATISTICS
TABLE OF CONTENTS
Section 1.1 ................................................................................................................................... 1
Section 1.2 ................................................................................................................................... 3
Section 1.3 ................................................................................................................................. 18
Section 1.4 ................................................................................................................................. 24
SECTION 1.1
1.
Solutions:
a. Los Angeles Times, Chicago Tribune, Gainesville Sun, Washington Post
b. France, Sweden, Denmark, Malta
c. Alice Johnson, Catherine Miller, Emma Davis, Ken Lee
d. 2.97, 3.56, 2.20, 2.97
2.
Solutions:
a. 45.3 m, 52.7 m, 60.1 m, 48.9 m
b. 432 pp, 196 pp, 184 pp, 321 pp
c. 4.5, 4.9, 5.6, 6.4
d. 0.07 g, 1.58 g, 7.1 g, 27.2 g
3.
Solutions:
a. How likely is it that more than half of the sampled computers will need or have needed
warranty service? What is the expected number among the 100 that need warranty service?
How likely is it that the number needing warranty service will exceed the expected number by
more than 10?
b. Suppose that 15 of the 100 sampled needed warranty service. How confident can we be that
the proportion of all such computers needing warranty service is between .08 and .22? Does
the sample provide compelling evidence for concluding that more than 10% of all such
computers need warranty service?
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 1
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
4.
Solutions:
a. Concrete populations: all living citizens of U.S., all the mutual funds currently marketed in
U.S., all the books that were published in 1980. Hypothetical populations: All GPAs of the
University of California undergraduates in the next academic year, page lengths of all books
that will be published next year, the batting averages of all major league players in the next
baseball season.
b. (Concrete) Probability: In a sample of 5 mutual funds, what is the chance that all
5 have rates of return which exceeded 10% last year? Statistics: If previous year rates-of-
return for 5 mutual funds were 9.6, 14.5, 8.3, 9.9 and 10.2, can we conclude that the average
rate for all funds was below 10%? (Hypothetical) Probability: In a sample of 10 books to be
published next year, how likely is it that the average number of pages for the 10 is between
200 and 250? Statistics: If the sample average number of pages for 10 books is 227, can we
be highly confident that the average for all books is between 200 and 245?
5.
Solutions:
a. No. All students taking a large statistics course who participate in an SI program of this sort.
b. The advantage to randomly allocating students to the two groups is that the two groups
should then be fairly comparable before the study. If the two groups perform differently in the
class, we might attribute this to the treatments (SI and control). If students were allowed to
choose, stronger or more motivated students might prefer SI, which could bias the results.
c. If all students were put in the treatment group, there would be no firm basis for assessing the
effectiveness of SI (nothing to which the SI scores could reasonably be compared).
6.
Solution:
One could take a simple random sample of students from all students in the California State
University system and ask each student in the sample to report the distance from their hometown
to campus. Alternatively, the sample could be generated by taking a stratified random sample by
taking a simple random sample from each of the 23 campuses and again asking each student in
the sample to report the distance from their hometown to campus. Certain problems might arise
with self-reporting of distances, such as recording error or poor recall. This study is enumerative
because there exists a finite, identifiable population of objects from which to sample.
7.
Solution:
One could generate a simple random sample of all single-family homes in the city, or a stratified
random sample by taking a simple random sample from each of the 10 district neighborhoods. From
each of the selected homes, values of all desired variables would be determined. This is an
enumerative study as there is a finite, well-defined population of objects that can be used as a sample.
8.
Solutions:
a. Number observations equal
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 2
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
b. This could be called an analytic study because the data would be collected on an existing
process. There is no sampling frame.
9.
Solutions:
a. The measurements might vary for different reasons. Some of these include measurement
error resulting from mechanical or technical changes across measurements, recording errors,
differences in weather conditions at the time of measurements, etc.
b. No, because there is no sampling frame.
SECTION 1.2
10.
Solutions:
a.
5 9
6 33588
7 00234677889
8 127
9 077 stem: ones
10 7 leaf: tenths
11 368
A representative strength for these beams is around 7.8 MPa, but there is a reasonably large
amount of variation around that representative value.
(What constitutes large or small variation usually depends on context, but variation is usually
considered large when the range of the data – the difference between the largest and
smallest value – is comparable to a representative value. Here, the range is
which is similar in size to the representative value of 7.8 MPa. So, most
researchers would call this a large amount of variation.)
b. The data display is not perfectly symmetric around some middle/representative value. There
is some positive skewness in this data.
c. Outliers are data points that appear to be very different from the pack. Looking at the stem-
and-leaf display in part (a), there appear to be no outliers in this data.
(A later section gives a more precise definition of what constitutes an outlier.)
d. From the stem-and-leaf display in part (a), there are 4 values greater than 10. Therefore, the
proportion of data values that exceed 10 is or, about 15%.
11.
Solution:
3L 1
3H 56678
4L 000112222234
4H 5667888 stem: tenths
5L 144 leaf: hundredths
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 3
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
5H 58
6L 2
6H 6678
7L
7H 5
The stem-and-leaf plot shows that .45 is a good representative value for the data. It also shows
that the plot is not symmetric and it is positively skewed. The range of the data is
which is comparable to the typical value of .45. This constitutes a reasonably large amount of
variation in the data. The data value .75 is a possible outlier.
12.
Solutions:
The sample size for this data set is .
a. The first four intervals correspond to observations less than 5, so the proportion of values
less than 5 is
b. The last four intervals correspond to observations at least 6, so the proportion of values at
least 6 is
c. The relative frequency and density histograms are shown below. The distribution of
particle sizes is not symmetric, but are positively skewed. Notice that the relative frequency
and density histograms are essentially identical, other than the vertical axis labeling, because
the bin widths are all the same.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 4
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
13.
Solution:
The histogram of ultimate strengths is symmetric and unimodal, with the point of symmetry at
approximately 135 ksi. There is a moderate amount of variation, and there are no gaps or outliers
in the distribution.
14.
Solutions:
a.
2 23 stem: 1.0
3 2344567789 leaf: .10
4 01356889
5 00001114455666789
6 0000122223344456667789999
7 00012233455555668
8 02233448
9 012233335666788
10 2344455688
11 2335999
12 37
13 8
14 36
15 0035
16
17
18 9
b. A representative is around 7.0.
c. The data exhibit a moderate amount of variation (this is subjective).
d. No, the data is skewed to the right, or positively skewed.
e. The value 18.9 appears to be an outlier, being more than two stem units from the previous
value.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 5
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
15.
Solution:
American French
8 1
755543211000 9 00234566
9432 10 2356
6630 11 1369
850 12 223558
8 13 7
14
15 8
2 16
American movie times are unimodal strongly positively skewed, while French movie times are
bimodal. A typical American movie runs about 95 minutes, while French movies are typically
either around 95 minutes or around 125 minutes. American movies are generally shorter than
French movies and are less variable in length. Finally, both American and French movies
occasionally run very long (outliers at 162 minutes and 158 minutes, respectively, in the
samples).
16.
Solutions:
a.
Beams Cylinders
9 5 8
88533 6 16
98877643200 7 012488
721 8 13359 stem: ones
770 9 278 leaf: tenths
7 10
863 11 2
12 6
13
14 1
The data are slightly skewed to the right, or positively skewed. The value of 14.1 MPa is an
outlier. About 15% of the measurements, that is, three out of twenty are 10 MPa.
b. The majority of observations are between 5 and 9 MPa for both beams and cylinders, with
the modal class being 7.0-7.9 MPa. The observations for cylinders arespread out, and the
maximum value of the cylinder observations is higher.
c.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 6
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
17.
Solutions:
The sample size for this data set is
a. “At most five bidders” means 2, 3, 4, or 5 bidders. The proportion of contracts that involved at
most 5 bidders is
Similarly, the proportion of contracts that involved at least 5 bidders (5 through 11) is equal to
b. The number of contracts with between 5 and 10 bidders, inclusive, is
so the proportion is “Strictly” between 5 and 10 means , or 9
bidders, for a proportion equal to
c. The distribution of number of bidders is positively skewed, ranging from 2 to
11 bidders, with a typical value of around 4-5 bidders.
18.
Solutions:
a. The most interesting feature of the histogram is the three very large outliers (21, 24, and 32
directors). Without these, the distribution of number of directors would be roughly symmetric
with a typical value of around 9.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 7
accessible website, in whole or in part.
, Solution and Answer Guide: Ellingson and Panorska, Probability and Statistics for Engineering and the Sciences,
10e, 2027, 9798214023823; Chapter 1: Overview and Descriptive Statistics
Note: One way to have Minitab automatically construct a histogram from grouped data such
as this is to use Minitab’s ability to enter multiple copies of the same number by typing, for
example, 42(9) to enter 42 copies of the number 9. The frequency data in this exercise was
entered using the following Minitab commands:
MTB > set c1
DATA> 3(4) 12(5) 13(6) 25(7) 24(8) 42(9) 23(10) 19(11) 16(12)
11(13) 5(14) 4(15) 1(16) 3(17) 1(21) 1(24) 1(32)
DATA> end
b. The frequency distribution shown below is almost the same as the one in the textbook,
except the three largest values are grouped into the “ ” category. If the data is presented
this way, we could not create a histogram, because we wouldn’t know the upper boundary for
the rectangle corresponding to the “ ” category.
No. dir. 4 5 6 7 8 9 10 11
Freq. 3 12 13 25 24 42 23 19
No dir. 12 13 14 15 16 17
Freq. 16 11 5 4 1 3 3
c. The sample size is So, the proportion of these corporations
that have at most 10 directors is
d. Similarly, the proportion of these corporations with more than 15 directors is
19.
Solutions:
a. From this frequency distribution, the proportion of wafers that contained at least one particle
is or . Note that it is much easier to subtract 1 (which is the number
of wafers that contain 0 particles) from 100 than it would be to add all the frequencies for
particles. In a similar fashion, the proportion containing at least 5 particles is
or,
b. The proportion containing between 5 and 10 particles is
or The proportion that contain strictly between 5 and 10 (meaning
strictly more than 5 and strictly less than 10) is or
44%.
c. The following histogram was constructed using Minitab. It is almost symmetric and unimodal.
It has a few smaller modes and has a very slight positive skew.
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