STAT 2141A - PROBABILITY AND
STATISTICS FOR ENGINEERS (FALL
2025) PROBLEMS & CERTIFIED
SOLUTIONS
Study Ace Smart
,STAT 2141A - Probability and Statistics for
Engineers (Fall 2025)
Tutorial 1 - Solutions
Chapter 1
Problem 1-5
Q: A group of 100 dental patients is randomly divided into two groups. One group (the
“treatment” group) receives supplemental fluoride treatments monthly for two years, while the
other group (the “control” group) receives standard semi-annual dental care. At the end of the
two-year study period, each patient’s tooth decay is evaluated and the two groups are
compared.
a) Does this study involve a conceptual or a physical population?
In this case, because the study results will be applied more generally to a larger population of
dental patients this is a conceptual population.
b) What is the purpose of randomly dividing the patients into two groups?
The random division is used to avoid a systematic difference between the patients in the
two groups. For example, one would not want all younger patients in one groups and older
patients in the other because any difference in groups might be due the fluoride, but might also
be due to the age difference. Consequently, the age effect would be confused (the term
confounded is used) with the fluoride effect. Similarly any other pattern between the patients in
the two groups would be a cause for concern of confounding. Therefore, the random
assignment is used to avoid such systematic differences.
c) Do you think that the study results would be valid if the patients elected which group
theybelonged to?
No the study would not be valid. Patients with more concern for tooth care, who more
consistently clean their teeth, may select the fluoride group. As in the previous part, the tooth
cleaning effect would be confounded with the fluoride treatment.
d) Why are two groups needed? Couldn’t valid results be obtained just by putting all
100patients into the “treatment” group?
, It is possible that a single group generates little or no tooth decay. One would not want to
attribute this result to the use of fluoride without a comparison. For example, if the control
group also generated little or no tooth decay then the effects of fluoride are not demonstrated.
One would need the control group to generate more tooth decay than the fluoride group to
demonstrate the positive effects of fluoride on tooth decay.
Problem 1-13
Q: A student in a laboratory course on quality control methods measures the length of a bolt
several times with a micrometer. When would it be reasonable to consider these measurements
a random sample? What is the population?
A: It would be reasonable to consider these measurements a random sample if the bolt and the
micrometer do not change between measurements. The population is conceptual—it consists
of all of the measurements that could be made on this single bolt with this micrometer. If the
goal of your quality control methods is to learn about the population of all bolts manufactured,
you need to measure all bolts or a sample of all bolts. See Example 1-1 for a similar problem.
It is not reasonable to consider these measurements a random sample of bolts because only
one bolt is measured multiple times. Variations in the production of bolts are not accounted for
in these measurements. If the analysis is to focus on only the single bolt then one might
consider these measurements a random sample from the populations of all measurements of
this single bolt.
Chapter 2
Problem 2-2
Q: In Applied Life Data Analysis (Wiley, 1982), Wayne Nelson presents the breakdown time of
an insulating fluid between electrodes at 34 kV. The times, in minutes, are as follows: 0.19,
0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91,
36.71, and 72.89. Calculate the sample average and sample standard deviation. Construct a
dot diagram of the data.
STATISTICS FOR ENGINEERS (FALL
2025) PROBLEMS & CERTIFIED
SOLUTIONS
Study Ace Smart
,STAT 2141A - Probability and Statistics for
Engineers (Fall 2025)
Tutorial 1 - Solutions
Chapter 1
Problem 1-5
Q: A group of 100 dental patients is randomly divided into two groups. One group (the
“treatment” group) receives supplemental fluoride treatments monthly for two years, while the
other group (the “control” group) receives standard semi-annual dental care. At the end of the
two-year study period, each patient’s tooth decay is evaluated and the two groups are
compared.
a) Does this study involve a conceptual or a physical population?
In this case, because the study results will be applied more generally to a larger population of
dental patients this is a conceptual population.
b) What is the purpose of randomly dividing the patients into two groups?
The random division is used to avoid a systematic difference between the patients in the
two groups. For example, one would not want all younger patients in one groups and older
patients in the other because any difference in groups might be due the fluoride, but might also
be due to the age difference. Consequently, the age effect would be confused (the term
confounded is used) with the fluoride effect. Similarly any other pattern between the patients in
the two groups would be a cause for concern of confounding. Therefore, the random
assignment is used to avoid such systematic differences.
c) Do you think that the study results would be valid if the patients elected which group
theybelonged to?
No the study would not be valid. Patients with more concern for tooth care, who more
consistently clean their teeth, may select the fluoride group. As in the previous part, the tooth
cleaning effect would be confounded with the fluoride treatment.
d) Why are two groups needed? Couldn’t valid results be obtained just by putting all
100patients into the “treatment” group?
, It is possible that a single group generates little or no tooth decay. One would not want to
attribute this result to the use of fluoride without a comparison. For example, if the control
group also generated little or no tooth decay then the effects of fluoride are not demonstrated.
One would need the control group to generate more tooth decay than the fluoride group to
demonstrate the positive effects of fluoride on tooth decay.
Problem 1-13
Q: A student in a laboratory course on quality control methods measures the length of a bolt
several times with a micrometer. When would it be reasonable to consider these measurements
a random sample? What is the population?
A: It would be reasonable to consider these measurements a random sample if the bolt and the
micrometer do not change between measurements. The population is conceptual—it consists
of all of the measurements that could be made on this single bolt with this micrometer. If the
goal of your quality control methods is to learn about the population of all bolts manufactured,
you need to measure all bolts or a sample of all bolts. See Example 1-1 for a similar problem.
It is not reasonable to consider these measurements a random sample of bolts because only
one bolt is measured multiple times. Variations in the production of bolts are not accounted for
in these measurements. If the analysis is to focus on only the single bolt then one might
consider these measurements a random sample from the populations of all measurements of
this single bolt.
Chapter 2
Problem 2-2
Q: In Applied Life Data Analysis (Wiley, 1982), Wayne Nelson presents the breakdown time of
an insulating fluid between electrodes at 34 kV. The times, in minutes, are as follows: 0.19,
0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91,
36.71, and 72.89. Calculate the sample average and sample standard deviation. Construct a
dot diagram of the data.
