MAT 114 QUANTITATIVE REASONING
SUMMER 2014
PROJECT TWO
The objective of this project is to use a spreadsheet program to test correlation strength and to create and
utilize mathematical functions which model relationships between pairs of quantities, especially relationships
which demonstrate linear or exponential growth/decay.
Important definitions and helpful examples related to this project are provided in Chapter 3 of the NAU MAT
114 course website. Specifically:
3A – independent/dependent variables, positive/negative correlation, correlation coefficient
3B – comparing linear and exponential growth/decay
3E – implementing functions using Microsoft Excel (video example)
A spreadsheet file containing the data sets required for this project must be downloaded from the NAU MAT
114 course website.
All required charts/graphs must be created within a spreadsheet program. Handwritten or hand-drawn
charts/graphs will not be accepted.
Answers to the following questions must be compiled into a single word-processing document (Microsoft
Word, etc.), including necessary explanatory text and supporting charts/graphs. Explanations and discussions
of answers must be in complete, grammatically correct sentences. The final document must be neat and
organized, with answers to individual problems clearly labeled.
REEDS
1. Ecologists are studying a certain species of riparian reed, Arundo donax L., and they collected data by
finding examples of the reed along the bank of a river in Southern California and measuring the distance
each reed was from the bank and also its diameter. The ecologists believe there is a linear relationship
between these quantities. A portion of the data which was collected is provided in the accompanying
spreadsheet.
a) Which quantity is most likely the independent variable? Which quantity is most likely the dependent
variable? Please provide a brief explanation of your answers.
b) Does it appear these quantities are positively related or negatively related? What biological factors may
explain the “direction” of this relationship?
c) Create a scatterplot of the data.
(This is accomplished in Microsoft Excel by highlighting all of the data, clicking the “Insert” tab, and
selecting “Scatter”.)
d) Have your spreadsheet program create a linear regression trendline for the data, displaying the
equation and the coefficient of determination (R2) on the graph.
(This is accomplished in Microsoft Excel by right-clicking on a data point in your scatterplot, selecting
“Add Trendline…”, and selecting “Display Equation on chart” and “Display R-squared value on chart”.)
**Include a copy of this scatterplot in your report.
e) Find the correlation coefficient R for your linear regression trendline. Does that value indicate a strong
correlation or a weak correlation?
Note: You must calculate and discuss R, but the spreadsheet program displays R2.
f) State and interpret the slope of the linear regression trendline. Include units.
g) The ecologists claim in their paper that “reed diameter decreases one inch for every ten feet from the
bank.” Does this statement agree with the slope you just stated? Explain.
h) Find and interpret the y-intercept of the linear regression trendline.
i) Find and interpret the x-intercept of your trendline.
j) Based on your linear model, what diameter of reed would you expect to find 28 feet from the bank of the
river?
TOADS
2. There has been a well-documented worldwide decline of amphibian species since the late 1980s. Various
causes have been hypothesized, including both human and natural forces, but the root cause of this decline
remains a mystery and is currently the focus of intense research.
In the accompanying spreadsheet, you will find census data documenting the size of Sonoran Green Toad
(Bufo retiformis) populations from the year 1987 to 2011 at two distinct locations, labeled Site A and Site B.
Note that the population was tracked by counting the number of female toads present each year.
A scatterplot has been created for each Site. A linear regression trendline as well as an exponential
regression trendline have been generated for each scatterplot. The trendline equations and coefficients of
determination (R2) are also provided on the graphs.
Tables are provided for you in the spreadsheet for entering your results for the following problems.
a) Find the correlation coefficient R for each of the regression trendlines provided on the scatterplots.
b) Based on the correlation coefficients for the two trendlines provided (linear and exponential), which
type of function (linear or exponential) appears to be a better fit for the data for each Site?
c) Use the trendline equations provided to project the population at each Site in the year 2020. Perform
these calculations for both the linear and exponential trendlines for each Site.
Note: The trendline equations use the year 1987 as x = 1, 1988 as x = 2, etc. Thus the year 2011, for
example, corresponds to an x-value of 25.
d) Use the trendline equations provided to project the population at each Site in the year 2040. Perform
these calculations for both the linear and exponential trendlines for each Site.
e) Briefly discuss the population predictions you calculated in the previous two parts. Are all of the
predictions realistic? Are you optimistic that the toad population at either Site will survive long term?
3. In addition to studying the two Sites individually, biologists are interested in the overall combined toad
population represented at these Sites.
a) Create a column in the spreadsheet which displays the Total toad population by adding the populations
given at both Sites.
b) Create a scatterplot of the data.
c) Have your spreadsheet program create a linear regression trendline for the data, displaying the
equation and the coefficient of determination (R2) on the graph. Find the linear correlation coefficient (R).
d) Have your spreadsheet program create an exponential regression trendline for the data, displaying the
equation and the coefficient of determination (R2) on the graph. Find the exponential correlation
coefficient (R).
**Include a copy of this scatterplot in your report.
e) Based on the two correlation coefficients you have just calculated, which type of function (linear or
exponential) appears to be a better fit for the Total toad population?
WASTE MANAGEMENT
Growing cities face the challenge of efficiently disposing of or recycling the solid waste generated by their
increasing populations. Fortunately, thanks to advances in recycling technologies, the amount of waste which
can be reused or recycled increases every year.
The town of Florence, Arizona has experienced population growth of approximately 6.191% per year since 1991.
4. a) If the population of Florence was 7418 in 1991, create a table of values indicating the population’s
growth through 2011.
b) In recent years, the growth of Florence has slowed significantly. Town officials project growth of only
1.65% for the next several years. Assuming this lower growth rate began in 2011-2012, continue your table
of values to project the population of Florence through 2030.
c) It is estimated that each individual generates approximately 1424 pounds of solid waste each year.
Create a new column of values which indicates how many pounds of solid waste are generated by the
population of Florence from 1991 to 2030.
Advances in recycling technologies have reduced the amount of solid waste which goes to a landfill each year.
In a typical municipality, the amount of solid waste which could be recycled in the early 1990s was
approximately one million pounds per year. Depending on the municipality, this amount may have grown by
as much as 600,000 pounds per year since.
5. a) Assuming Florence could recycle 1,110,000 pounds of solid waste in 1991 and that this capacity has
increased by a fixed amount of 509,850 pounds per year since, create a column of values indicating how
many pounds of solid waste are recycled by Florence from 1991 to 2030.
b) Assuming the number of pounds of solid waste which end up in a landfill is the total solid waste
generated minus the number of pounds recycled, generate a column of values indicating how many pounds
of solid waste end up in a landfill due to the population of Florence from 1991 to 2030.
c) Create a multi-line graph displaying the total solid waste generated and the solid waste which ends up in
a landfill due to the population of Florence from 1991 to 2030.