Warm-up
The city of Pittsburgh, PA was a major steel
producer in the first half of the twentieth
century. However, like many major industrial
American cities, the population of Pittsburgh
has decreased since 1950.
The table shows the population of Pittsburgh
in various years.
1. Graph the data, using years since 1950.
2. Find the equation of the regression line for
the data.
3. Explain what the slope of the
….regression line means in this setting.
4. In the year 2000, the census gave the
population of Pittsburgh as 334,563.
Was the regression model valid through
the year 2000? Explain.
Year
Population
1950
676,806
1960
1970
1980
1990
604,332
520,117
423,938
369,879
Source: http://en.wikipedia.org/wiki/Pittsburgh
Warm-up
The city of Pittsburgh, PA was a major steel
producer in the first half of the twentieth
century. However, like many major industrial
American cities, the population of Pittsburgh
has decreased since 1950.
The table shows the population of Pittsburgh
in various years.
1. Graph the data, using years since 1950.
Year
Population
1950
0
676,806
1960
1970
1980
1990
10
604,332
520,117
423,938
369,879
20
30
40
Source: http://en.wikipedia.org/wiki/Pittsburgh
2. Find the equation of the regression line for
the data. y = -7942.48x + 677864 where x = years since 1950 and y = population
3. Explain what the slope of the ….
It represents the average yearly decrease
….regression line means in this setting. in population, approximately 7942 people.
4. In the year 2000, the census gave the
No, the model is not valid through
population of Pittsburgh as 334,563.
the year 2000. The model’s estimate
Was the regression model valid through for the population of Pittsburgh in the
the year 2000? Explain.
year 2000 is 280,740, well below the
actual 2000 Pittsburgh population.
MATH 1101
FINAL GROUP PROJECT
Due – Last class before final exam (see
syllabus)
December
4
1. Work in groups of 3 or 4 people. More than four people in a group is not
acceptable. Working alone is not acceptable
2. Work should not be done on the question paper.
3. Include a cover page with the names of group members.
4. Show all work neatly, clearly, and completely. Whenever appropriate,
answer in full sentences. It is strongly recommended that group
members carefully proofread each other’s solutions.
5. Several questions ask for graphs. All graphs should be imported from a
computer and/or a graphing calculator. Include the viewing window or
show the scale on the axes.
6. Put thought into your explanations. Do not hesitate to research them on
the Web. Don’t take the easy way out!
7. Begin this group project as soon as possible and complete the problems
gradually. For example, questions 1 – 8 deal with subject matter we have
already covered in class and can be completed immediately.
8. Each group member must individually complete and turn in the form on
the following page: “Assessment of Group Effectiveness.” These will be
kept confidential.
9. This project is worth 60 points.
Assessment of Group Effectiveness
Your Name:
Names of the other group members:
___________________________
__________________________________
__________________________________
__________________________________
Please rate your group on each of the following statements by circling the rating
that applies. ( 1 = Strongly Disagree, 5 = Strongly Agree )
1. The group functioned smoothly, with all members contributing equally.
1
2
3
4
Strongly disagree
5
Strongly agree
2. Members of the group proofread each other’s work and gave appropriate input.
1
2
3
4
5
Strongly disagree
Strongly agree
3. Group members met deadlines established by the group for completing work.
1
2
3
4
5
Strongly disagree
4. Group members were present for all group meetings.
1
2
3
Strongly agree
4
Strongly disagree
If any group member made little or no contribution to the group project,
please indicate the name of that group member. This will be kept confidential.
5
Strongly agree
Answers to even-numbered HW problems
Section 3.4
Ex 2a) P = .181t – 349.775 where t = years, P = enrollment in millions
P =.181t + 12.406
using t = years since 2001
b) The slope means the enrollment increased, on average,
by .181 million (181,000) students per year.
c) P(2007) = 13.492 million (13,492,000)
d) No. If the trend had been valid in 1993, the expected
enrollment that year would have been 10.958 million,
not 11.2 million.
y
3x – 2y = 10
3
y= x–5
2
8
7
6
5
x + 2y = –2
1
y   x 1
2
4
3
2
1
1
(2, -2)
is is
the
solution
the
point to
this
system
of two
where
the two
linear
linesequations.
cross.
This method of solution is called
the “crossing graphs” method.
2
•
3
4
5
6
7
8
9
x
3x – 2y = 10
3
y= x–5
2
x + 2y = –2
1
y   x 1
2
(2, -2)
is is
the
solution
the
point to
this
system
of two
where
the two
linear
linesequations.
cross.
3x – 2y = 10
3
y= x–5
2
x + 2y = –2
1
y   x 1
2
(2, -2)
is is
the
solution
the
point to
this
system
of two
where
the two
linear
linesequations.
cross.
Substitution Method
3x – 2y = 10
x + 2y = –2
x = – 2y – 2
3x(–2y
– 2 y–=2)10 –2y = 10
–6y – 6 – 2y = 10
–8y = 16
x + 2y = –2
x + 2(–2) = –2
x=2
y = –2
Solution to the system: (2, -2)
Substitution Method
3x – 2y = 10
1
y   x 1
2
x + 2y = –2
 1


x

1
y


3x – 2 = 10
 2

= 10
3x + x + 2 = 10
2 + 2y = –2
x=2
y = –2
Solution to the system: (2, -2)
Elimination Method (Linear Combination)
3x – 2y = 10
Because the coefficients of y in the two equations
are opposites, adding the equations will eliminate
the y-variable.
x + 2y = -2
3x – 2y = 10
+
x + 2y = -2
4x
=8
x=2
2 + 2y = -2
y = -2
Solution to the system: (2, -2)
3x – 2y = 19
3(3x – 2y = 19)
7x + 3y = 6
2(7x + 3y = 6)
9x – 6y = 57
+
14x + 6y = 12
23x = 69
x=3
3x – 2y = 19
(3, -5)
3(3) – 2y = 19
–2y = 10
y = –5
Practice
Two systems of linear equations are given below. Solve
one using the substitution method, and one using the
elimination method.
3x – 5y = 47
2x + 3y = 7
8x + 3y = 11
5x – y = 9
(4, –7)
(2, 1)
Year
Kennesaw
State
Valdosta
State
1980
1981
1982
1983
3,903
4,195
4,779
5,383
4,901
4,909
5,548
5,835
http://www.usg.edu/research/documents/enrollment_reports/rpt79-88.pdf
Shown above are the enrollments at Kennesaw
State College and Valdosta State College at
the beginning of the school years listed.
1. Using years since 1980, Make scatter plots of the
enrollment data for both schools on the same
calculator screen.
2. Using linear regression models for both schools,
and assuming the trends continued, identify the
year in which the Kennesaw State’s enrollment
overtook Valdosta State’s enrollment.
Year
1980
1981
1982
1983
Kennesaw
State
Valdosta
State
3,903
4,195
4,779
5,383
4,901
4,909
5,548
5,835
Regression model for KSU
y = 502.4x +3811.4
Regression model for VSU
y = 344.1x + 4782.1
http://www.usg.edu/research/documents/enrollment_reports/rpt79-88.pdf
Shown above are the enrollments at Kennesaw
State College and Valdosta State College at
the beginning of the school years listed.
1. Using years since 1980, Make scatter plots of the
enrollment data for both schools on the same
calculator screen.
2. Using linear regression models for both schools,
and assuming the trends continued, identify the
year in which the Kennesaw State’s enrollment
overtook Valdosta State’s enrollment.
HOMEWORK
Section 3.5:
Do not read Section 3.5
Page 298 # S-11, S-12, S-16
Pages 299 – 300 # X
5, 6, 10
ALSO
DOWNLOAD, PRINT, AND COMPLETE
THE PRACTICE TEST.
DON’T FORGET TO DOWNLOAD AND
PRINT A COPY OF THE FINAL GROUP
PROJECT.