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\begin{document}

\newpage

\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Gender Demographics in Engineering}


\markboth{Gender Demographics in Engineering}{Population Growth}

\author{Brody Johnson, Elodie Pozzi \\
Saint Louis University\\
Saint Louis MO 63103 USA\\
\teacher{brody.johnson@slu.edu, elodie.pozzi@slu.edu}
}

\makeStitlePDFLaTex


\begin{abstract}
The goal of this activity is to show students how population models can be used to examine social issues.  The students will examine three different population models and will use numerical methods to apply each model to demographic data for the percentage of engineering degrees awarded to women in the United States.  Finally, the students will discuss possible implications and weaknesses of the three models in predicting the percentage of engineering degrees awarded to women over the next ten years.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

In 1966, fewer than 1\% of the 35,826 bachelor's degrees in engineering awarded in the United States were earned by women \cite{APS2021}.  This percentage showed rapid growth over the following ten years, but undergraduate women in the United States still remain underrepresented in engineering some fifty years later, despite comprising the majority of the undergraduate student body.  In fact, women comprised 57\% of the undergraduate student body in 2017 while earning just 22\% of the bachelor's degrees in engineering awarded that year \cite{NCES2021, APS2021}.  This project makes no attempt to explain why women remain underrepresented in undergraduate engineering programs; however, students are encouraged to learn more about this issue by visiting the Society of Women Engineers website (\url{https://swe.org/}) or reading about the history of this organization \cite{SWE2021}.  Rather, the goal of this project is to use historical data to examine the following question:
\begin{quote}
    \emph{Does the growth in the percentage of women in undergraduate engineering programs over the last fifty years  appear to be on track to achieve a gender balance mirroring that of the entire undergraduate population?}
\end{quote}

\noindent
Towards this end, a mathematical model will be developed for the percentage of bachelor's degrees in engineering earned by women in the United States as a function of time, based on real data.  The data comes from the American Physical Society (\url{https://www.aps.org/}) and includes the percentage of bachelor's degrees in engineering earned by women in the United States for each year between 1966 and 2018 \cite{APS2021}.  The time history is shown in Figure \ref{fig1}.  

\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=4.50in]{stemgraph.png}    
\end{center}
\caption{Percentage of Bachelor's degrees in engineering earned by women in the United States \cite{APS2021}.} \label{fig1}
\end{figure}

Let $P(t)$ represent the percentage of bachelor's degrees in engineering that are awarded to women in year $t$.  It will be assumed that $P(t)$ obeys a differential equation of the form
\begin{equation} \label{genericDE}
\frac{dP}{dt} = f(P),
\end{equation}

\noindent
where $f$ is a continuous function.  As a percentage, the function $P(t)$ is confined to the range $0\le P(t) \le 100$ and thus it is reasonable to employ a population model that accounts for limited resources.  The \emph{growth function} $f(P)$ for such models usually satisfies the following general properties:
\begin{itemize}
\item $f(0)$ should be zero;
\item $f(P)$ should be positive for small positive values of $P$;
\item $f(P)$ should be negative for sufficiently large values of $P$.
\end{itemize}

\noindent
These assumptions reflect the fact that a small population grows steadily due to an abundance of resources, but growth will slow and possibly recede as the resources are exhausted.  Although the above criteria could be met by countless varieties of growth functions, two choices tend to receive the most attention.  The first of these is the so called \emph{logistic} growth function,
\begin{equation} \label{logisticf}
f(P) = kP\left ( 1 - \frac{P}{N} \right ),  \qquad k,\, N>0.
\end{equation}

\noindent
The second is called the \emph{Gompertz} growth function and is given by
\begin{equation} \label{Gompertzf}
f(P) = kP \ln{\left ( \frac{N}{P} \right )}, k,\, N>0.
\end{equation}

\noindent
One further growth function will be introduced for consideration. This growth function will be referred to as the \emph{cubic} growth function and is described by
\begin{equation} \label{cubicf}
f(P) = kP \left ( 1 - \frac{P}{N} \right)^{3}, k,\, N>0.
\end{equation}

\noindent
Figure \ref{fig2} offers a graphical comparison of the three different types of growth functions, each with $N=50$.  (The choice $N=50$ is used to reflect a population that is divided evenly between men and women.)  Do these functions satisfy the three general properties described above?  How will the differences in the shapes of curves influence the corresponding models?

\begin{figure}[thp]
\begin{center}
\begin{tikzpicture}
    \begin{axis}[width=4.0in,height=2.0in,
            xlabel=$P$,ylabel=$\frac{dP}{dt}$,xmin=0,xmax=105,ymin=-2,ymax=2.5,
            axis y line=left,axis x line=middle,
            xtick={0,50,100},
            xticklabels={0,50,100},
             every axis x label/.style={at={(current axis.right of origin)},anchor=west},
             every axis y label/.style={at={(current axis.north west)},anchor=south}]
    % logistic         
    \addplot[black,domain=0:100,samples=101]{0.15*x*(1-x/50)};
    \addlegendentry{Logistic}
    \addplot[blue,domain=0:100,samples=101]{0.1*x*ln(50/x)};
    \addlegendentry{Gompertz}
    \addplot[red,domain=0:100,samples=101]{0.4*x*(1-x/50)^3};
    \addlegendentry{Cubic}
    \end{axis}
\end{tikzpicture}
\end{center}
\caption{Comparison of the Logistic, Gompertz, and Cubic Population Models.} \label{fig2}
\end{figure}

Ultimately, these models will be used in an attempt to forecast the growth of the actual percentage of undergraduate engineering degrees awarded to women in the United States over the next five to ten years.  To accomplish this, it will be important to understand how the constants $k$ and $N$ affect the shape of the graph for $P(t)$.  It will also be important to recognize differences between the three models and determine which is best suited to the data.  Once this has been done, it will be possible to approach the main question by generating a forecast for the percentage of engineering degrees awarded to women in the future.  The remainder of this modeling scenario will break this large task into a series of steps, many of which rely on knowledge developed earlier in the course.


\section{Qualitative Analysis}

Qualitative analysis leads to the understanding of certain characteristics of the solutions of a differential equation (what is going to happen) without explicitly solving the differential equation (no solution formula is used).  This is especially important when the differential equation cannot be solved using available techniques.  Notice that the differential equation \eqref{genericDE} is \emph{autonomous} for any growth function $f(P)$, which means the slope $\frac{dP}{dt}$ at any point $(t,P)$ is dependent only on $P$.  Recall that the \emph{phase line} of an autonomous differential equation $\frac{dP}{dt} = f(P)$ is essentially an annotated graph of $f(P)$ versus $P$.  Positive values of $f(P)$ correspond to positive values of $\frac{dP}{dt}$, which means $P$ will increase (move to the right) as $t$ increases.  Similarly, negative values of $f(P)$ correspond to negative values of $\frac{dP}{dt}$, which means $P$ will decrease (move to the left) as $t$ increases.  These directions are noted on the $P$-axis using arrows to the right or left.  If $f(P)=0$, however, then $\frac{dP}{dt}=0$ and $P(t)$ will remain constant.  Such a point $P$ is called an \emph{equilibrium point} of the differential equation.  If nearby solutions tend towards the equilibrium point it is said to be \emph{stable}.  Otherwise, the equilibrium point is said to be \emph{unstable}.

\bigskip \noindent
{\bf Student Task:} Construct a phase line for the cubic model,
 $$ \frac{dP}{dt} = k P \left (1-\frac{P}{N} \right )^{3}, \qquad (0< N < 100) $$

\noindent
in the space provided below.  Include arrows on the $P$-axis that illustrate the direction of solutions.  Be sure to identify any equilibrium points and label each as stable or unstable.

\begin{center}
\begin{tikzpicture}
    \begin{axis}[width=4.0in,height=1.75in,
            xlabel=$P$,ylabel=$\frac{dP}{dt}$,xmin=0,xmax=105,ymin=-3,ymax=3,
            axis y line=left,axis x line=middle,
            xtick={57,100},ytick=\empty,
            xticklabels={$N$,100},
             every axis x label/.style={at={(current axis.right of origin)},anchor=west},
             every axis y label/.style={at={(current axis.north west)},anchor=south}]
\end{axis}
\node[scale=1] at (-0.2,1.25) {$0$};
\end{tikzpicture}
\end{center}

\begin{samepage}\noindent
Questions:

\begin{enumerate}[label=\thesection.\arabic*,start=1]
\item How many equilibrium points does this model have?  Describe the stability of each equilibrium point.
\item If $0<P(0)<100$, what can you say about $P(t)$ for large values of $t$?  Try to interpret your answer in the context of the percentage of women earning undergraduate degrees in engineering.
\item How do the parameters $k$ and $N$ affect the phase line?  Does one parameter have a more noticeable effect?
\item Would the phase line or any of your subsequent answers change if you replace the cubic model with the logistic model or the Gompertz model?  Explain.
\item How do the logistic, Gompertz, and cubic growth functions differ?  How do you think these differences will affect the solution curves for these models?  (You may want to graph each of the growth functions on a common plot.)
\end{enumerate}
\end{samepage}


\section{Numerical Solution}

Regardless of the choice of the growth function, notice that \eqref{genericDE} is always separable, so one can attempt to find an explicit formula for the solutions via integration.  This works well for the logistic and Gompertz models, which is one of the reasons these models are often used.  The cubic model, on the other hand, is not easily solved in this manner.  In this case it makes sense to implement a numerical method to obtain an approximate solution of the differential equation.  The improved Euler method, summarized in Figure \ref{IEMethod}, will be adopted for this project.

\begin{figure}[thp]
\begin{center}
\fbox{\begin{minipage}{0.625\textwidth}
\begin{small}
\noindent
{\bf Improved Euler Method:}

\noindent
Choose $\Delta t>0$ and consider the initial value problem
$$ \frac{dP}{dt} = f(P) \quad \text{subject to} \quad P(t_{0})=P_{0}.$$

\noindent
Define $t_{j}=t_{0}+j\Delta t$.  The approximate value of $P(t_{j})$, denoted by $P_{j}$, can be computed from $P_{j-1}$ as follows:

\begin{enumerate}
\item Compute the initial slope estimate: $m_{1} = f(P_{j-1})$.

\medskip
\item Compute a temporary estimate for $P$: $\tilde{P} = P_{j-1} + m_{1} \Delta t$.

\medskip
\item Compute the second slope estimate: $m_{2} = f(\tilde{P})$.

\medskip
\item Compute $P_{j}$ using the average slope: 
$$P_{j} = P_{j-1} + \left ( \frac{m_{1}+m_{2}}{2} \right ) \Delta t.$$
\end{enumerate}
\end{small}
\end{minipage}}
\end{center}
\caption{The improved-Euler method.} \label{IEMethod}
\end{figure}

\begin{samepage}
\bigskip \noindent
{\bf Student Task:}  Create a spreadsheet to implement the improved Euler method for the cubic model to solve the initial value problem
$$ \frac{dP}{dt} = kP \left ( 1- \frac{P}{N} \right )^{3} \quad \text{subject to} \quad P(0)=1$$

\noindent
over the range $0\le t \le 52$. Use $\Delta t=1$, $k=0.2$, and $N=57$.
\end{samepage}

\begin{samepage}
\noindent
Tips for the Spreadsheet: (see Figure \ref{spreadsheet})
\begin{itemize}
\item Create six columns: (A) Iteration $j$, (B) Time $t_{j}$, (C) Population $P_{j}$, (D) Slope $m_{1}=f(P_{j})$, (E) Temporary Estimate $\tilde{P}$, (F) Slope $m_{2}=f(\tilde{P})$.

\item Create cells for the parameters $k$ and $N$ and reference these cells in the two slope formulas.

\item Add a scatter plot to graph $P_{j}$ as a function of $t_{j}$.
\end{itemize}
\end{samepage}

\begin{figure}[thp]
\centering
\includegraphics[width=5.0in]{spreadsheet-eg.png}
\caption{Screenshot of a sample spreadsheet.} \label{spreadsheet}
\vspace*{-.4 true cm} 
\end{figure}

\begin{samepage}
\noindent

Questions:
\begin{enumerate}[label=\thesection.\arabic*,start=1]
\item How long does it take before $P_{j}$ exceeds 25?  How sensitive is this time to the changes in te parameters $k$ and $N$?

\item How does the graph of the approximate solution change when $k$ is increased?  What if $k$ is decreased?

\item Create new sheets to implement the logistic and Gompertz models by copying and modifying the current sheet.  What changes are required? How do the solution curves differ?

\item Does the data in Figure \ref{fig1} exhibit any characteristics of the solution curves found here?  Explain.
\end{enumerate}
\end{samepage}

\section{Model Implementation}

The last step in this project involves the application of the three population models to actual data for the percentage of bachelor's degrees in engineering earned by women in the United States between 1966 and 2018.  The parameters $k$ and $N$ will be adjusted so that each model predicts the data as accurately as possible.  The previous steps should have provided some insight about how changes to the parameters $k$ and $N$ will influence the shape of the solution predicted by the model.  This insight will help determine whether a given value of $k$ or $N$ in the model is too small or too large based on the comparison with the historical data.  For convenience, the raw data depicted in Figure \ref{fig1} is presented as Table \ref{table1}.  This data is also available in a spreadsheet among the supporting documents.



\bigskip \noindent
{\bf Student Task:} Modify your spreadsheet to compare the observed data to predicted values of $P(t)$ for any values of the parameters $k$ and $N$.  Optimize the values of $k$ and $N$ for each model so that the sum of the squared error (SSE) is minimized.  The sum of the squared error is computed as follows:
$$ \text{SSE} = \sum_{j=1}^{J} \left ( P_{j}- P(t_{j}) \right )^{2},$$

\noindent
where $J$ is the number of data points.

\begin{table}[h!]
\centering
\begin{tabular}{|c|c|g|g|c|c|g|g|}
\hline
Year & $P(\text{Year})$ & \text{Year} & $P(\text{Year})$ & \text{Year} & $P(\text{Year})$ & \text{Year} & $P(\text{Year})$ \\
\hline
1966 & 0.4075 & 1980 & 10.1207 & 1994 & 16.5096 & 2008 & 18.4923 \\
1967 & 0.5083 & 1981 & 11.0850 & 1995 & 17.2792 & 2009 & 18.0604 \\
1968 & 0.5732 & 1982 & 12.2665 & 1996 & 17.9368 & 2010 & 18.4281 \\
1969 & 0.7503 & 1983 & 13.2820 & 1997 & 18.3960 & 2011 & 18.7971 \\
1970 & 0.7527 & 1984 & 14.0887 & 1998 & 18.6147 & 2012 & 19.1987 \\
1971 & 0.7978 & 1985 & 14.4975 & 1999 & 19.7700 & 2013 & 19.3157 \\
1972 & 1.0763 & 1986 & 14.4988 & 2000 & 20.5188 & 2014 & 19.8257 \\
1973 & 1.2313 & 1987 & 15.3228 & 2001 & 20.1118 & 2015 & 20.0755 \\
1974 & 1.6139 & 1988 & 15.3648 & 2002 & 20.9667 & 2016 & 20.9124 \\
1975 & 2.1218 & 1989 & 15.2180 & 2003 & 20.3679 & 2017 & 21.5032 \\
1976 & 3.3952 & 1990 & 15.4130 & 2004 & 20.5111 & 2018 & 22.2550 \\
1977 & 4.9423 & 1991 & 15.5418 & 2005 & 19.9740 & & \\
1978 & 7.3692 & 1992 & 15.5567 & 2006 & 19.5750 & & \\
1979 & 9.1287 & 1993 & 15.9174 & 2007 & 18.5692 & & \\
\hline
\end{tabular}
\caption{Percentage of Bachelor's degrees in engineering earned by women in the United States \cite{APS2021}.} \label{table1}
\end{table}


\bigskip \noindent
Tips for the Spreadsheet:
\begin{itemize}
\item Add two additional columns: (G) Actual Data $P(t_{j})$, (H) Squared Error.
\item Interpret 1966 as $t=0$ so that $2018$ corresponds to $t=52$.
\item Modify $P_{0}$ to match the starting value of the data.
\item Create a cell for the sum of the squared error column.
\item Add a line or scatter plot to compare $P(t_{j})$-observed with $P_{j}$-predicted.
\item Start with $k=0.1$ and $N=50$.  Make small adjustments to $k$ or $N$ while watching the graphs and the sum of the squared error.  Think about how the changes will affect the graph of $P(t)$-predicted before implementing them.
\item If the spreadsheet is equipped with Solver, use the Solver to optimize the values of $k$ and $N$ so that the sum of the squared errors is minimized.
\item Repeat these steps for each of the three models.
\end{itemize}


\begin{samepage}
\noindent
Questions: 

\begin{enumerate}[label=\thesection.\arabic*,start=1]
\item Do any of the models fail to capture important trend(s) in the data?  Explain.

\item What do the optimal values of $k$ and $N$ convey about the progress towards a gender balance in the engineering sciences through 2018?  Do the models agree in this regard?

\item Which model led to the smallest value for the sum of the squared error?

\item Which model do you think will predict $P(t)$ the most accurately over the next ten years?
\end{enumerate}
\end{samepage}


\begin{thebibliography}{98}
\bibitem{APS2021} American Physical Society. \url{https://www.aps.org/programs/education/statistics/womenmajors.cfm}. Accessed 9 June 2021.

\bibitem{NCES2021} National Council for Educational Statistics. \url{https://nces.ed.gov/fastfacts/display.asp?id=98}. Accessed 9 June 2021.

\bibitem{SWE2021} Wikipedia article: 'Society of Women Engineers'. \url{https://en.wikipedia.org/wiki/Society_of_Women_Engineers}. Accessed 16 June 2021.


\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

% How to use in a class? Background information needed.
\subsection*{Background Knowledge and Placement in a Course}

The goal of this modeling scenario is to provide a real world example of population modeling and to help students recognize subtle differences between models whose phase lines are almost identical in order to prepare them to choose (or possibly invent) an appropriate model for a given set of data.  This modeling scenario makes use of {\bf phase lines} for autonomous first-order differential equations and the {\bf improved Euler's method} for a first-order initial value problem.  This project could be undertaken either in class or out of class, in groups or individually, and could be instructor-led or student-driven.  It would be reasonable to expect students to complete this project in groups of 2--3 students over a period of at most two weeks.  The content would fit most naturally at a point when population models are being discussed; however, it may also make sense to introduce the project after both population modeling and numerical methods have been discussed.  In particular, it would be useful for the students to have been exposed to both Euler's method and the improved Euler method.

The project makes use of the well-known logistic and Gompertz models, while introducing a third model which is referred to here as the cubic model.  Each model uses a constant $k$ to represent the growth rate and a constant $N$ to represent the carrying capacity of the population.  For a fixed value of $N$, the phase lines for the three models are similar, so it will be important for students to examine the graphs of the growth functions in order to understand differences between the models.  Looking at Figure \ref{fig1}, it is apparent that the logistic model has a symmetric growth function, while the growth function of the Gompertz model is biased towards a faster initial growth and a slower final approach towards the carrying capacity.  The cubic model exhibits even greater asymmetry than the Gompertz model, which seems to work well with the provided data.  It is hoped that students will discover how the shape of each growth function relates to the shape of the corresponding solution curves; however, instructors are certainly encouraged to provide guidance as needed.

% Materials needed to implement the activity in class
\subsection*{Materials}

Students will need to use some form of spreadsheet software in order to implement the improved Euler method and compare the results with the provided data.  If the class meets in a computer classroom or if it is reasonable to expect students to bring a laptop to class, then it would make sense to give this project to students during class and to provide the data in spreadsheet form.  Otherwise, the scenario could be given as an out-of-class assignment or used for an instructor-led demonstration.

\subsection*{Customization}
% alternative implementations

As written, this modeling scenario focuses on the population of women in undergraduate engineering programs.  The original data set includes gender breakdowns for a variety of STEM disciplines and can be found on the website of the American Physical Society using the link in the references of the student version.  If students are working in groups, the instructor could give each group a different data set.  The students may find it interesting to see how the findings vary across disciplines.

Instructors may also choose to find and incorporate data from other contexts.  Within the realm of social issues, this might involve looking at trends for other underrepresented groups within education or may shift the focus to topics outside of academics.  Instructors may also consider introducing data for tumor growth  \cite{Vaidya1982}, population data in biology \cite{USFWS2021}, sales data for electric vehicles \cite{AFDC2021}, etc.

Some instructors may choose to eliminate the cubic model and focus on the logistic and Gompertz models.  This would have the advantage of removing the dependence on numerical methods so that students could be asked to find the general solutions for the logistic and Gompertz models and use these to estimate the parameters $k$ and $N$.  The one disadvantage of this approach is that the cubic model produces the best fit for the given data and illustrates how the symmetry or asymmetry of the growth function can be tailored to a specific problem.

\subsection*{Solutions}

\noindent 
{\bf Qualitative Analysis}

\noindent
The phase line below includes the graph of $f(P)$ for the cubic model with $k=0.4$ and $N=50$.

\begin{center}
\begin{tikzpicture}
    \begin{axis}[width=4.0in,height=1.75in,
            xlabel=$P$,ylabel=$\frac{dP}{dt}$,xmin=-5,xmax=105,ymin=-3,ymax=3,
            axis y line=middle,axis x line=middle,
            xtick={57,100},ytick=\empty,
            xticklabels={$N$,100},
             every axis x label/.style={at={(current axis.right of origin)},anchor=west},
             every axis y label/.style={at={(current axis.north west)},anchor=south}]
             
        \addplot[red,domain=-5:100,samples=106]{0.4*x*(1-x/50)^3};
     
        \node at (axis cs:57,0)[circle,fill,inner sep=2pt]{};     
        \node at (axis cs:0,0)[circle,fill,inner sep=2pt]{};     
        
        \node at (axis cs:-3,0) {$<$}; 
        \node at (axis cs:16.25,0) {$>$}; 
        \node at (axis cs:32.5,0) {$>$}; 
        \node at (axis cs:48.75,0) {$>$}; 
        \node at (axis cs:71,0) {$<$}; 
        \node at (axis cs:85,0) {$<$}; 

\end{axis}
\node[scale=1] at (-0.2,1.25) {$0$};
\end{tikzpicture}
\end{center}

\noindent
Questions:

\begin{enumerate}[label=1.\arabic*,start=1]
\item How many equilibrium points does this model have?  Describe the stability of each equilibrium point.

\begin{proof}[Answer:]
The equilibrium points occur at $P=0$ and $P=N$.  The equilibrium point at $P=0$ is an unstable source because nearby solutions tend away from this point, while the equilibrium point at $P=N$ is stable sink because nearby solutions tend towards this point.
\end{proof}

\item If $0<P(0)<100$, what can you say about $P(t)$ for large values of $t$?  Try to interpret your answer in the context of the percentage of women earning undergraduate degrees in engineering.

\begin{proof}[Answer:]
If $P(0)<N$, then $P(t)$ will increase towards $N$ as $t\rightarrow \infty$.  If $P(0)>N$, then $P(t)$ will decrease towards $N$ as $t\rightarrow \infty$.  Finally, if $P(0)=N$, then $P(t)$ will remain constant for all $t>0$.  In each case the slope of the graph will approach zero for large values of $t$.
\end{proof}

\item How do the parameters $k$ and $N$ affect the phase line?  Does one parameter have a more noticeable effect?

\begin{proof}[Answer:]
The stable equilibrium moves as $N$ is changed.  As long as $k>0$, the phase line is not affected by changes to $k$.  Thus changes to $N$ will be more noticeable when looking at the phase line.
\end{proof}

\item Would the phase line or any of your subsequent answers change if you replace the cubic model by the logistic model or the Gompertz model?  Explain.

\begin{proof}[Answer:]
No.  Each model has an unstable equilibrium point at $P=0$ and a stable equilibrium point at $P=N$.  The sign of $\frac{dP}{dt}$ is identical for all values of $P$ as long as the same values for $k$ and $N$ are used for each model.
\end{proof}

\item The graphs in Figure \ref{fig2} illustrate differences between the growth functions of the logistic, Gompertz, and cubic models.  How do you think these differences will affect the solution curves for these models?

\begin{proof}[Answer:]
The cubic model shows greater initial growth near $P=0$ than either of the other two models and the Gompertz model shows slightly greater initial growth in comparison with the logistic model.  As the solutions approach $N$, the growth for the cubic model will be the slowest, followed by the Gompertz model, and then the logistic model.  One would expect $P(t)$ to grow more quickly with the cubic model than the other models initially, but it may take longer to get close to the carrying capacity due to the slower growth as $P(t)$ approaches $N$.  As before, the Gompertz model exhibits similar characteristics, but to a lesser extent than the cubic model.
\end{proof}
\end{enumerate}


\noindent 
{\bf Numerical Solution}

\medskip \noindent
The instructor resources include a spreadsheet that can be used to verify student work for both this portion of the project and the model implementation.  Some instructors may choose to create a mostly blank version of the spreadsheet that includes the appropriate columns and parameter cells to make it easier for students to complete the tasks during a class meeting.

\noindent
Questions:
\begin{enumerate}[label=2.\arabic*,start=1]
\item How long does it take before $P_{j}$ exceeds 25?

\begin{proof}[Answer:]
According to the numerical results it will take 29 years for $P(t)$ to increase from 1 to 25 with $k=0.2$ and $N=57$.
\end{proof}

\item How does the graph of the approximate solution change when $k$ is increased?  What if $k$ is decreased?

\begin{proof}[Answer:]
As $k$ is increased the slope of the solution curve increases.  For example, with $k=0.3$, $P(t)$ reaches 25 after only 19 years.  With $k=0.1$ it will take more than 52 years for $P(t)$ to reach 25.
\end{proof}

\item Create new sheets to implement the logistic and Gompertz models by copying and modifying the current sheet.  What changes are required? How do the solution curves differ?

\begin{proof}[Answer:]
The columns for $m_{1}$ and $m_{2}$ must be modified to use the appropriate growth functions for the logistic and Gompertz models, respectively.  You should find that the value of $k$ is sensitive to the model chosen.  For example, in order that $P(t)$ increase from 1 to 25 within 29 years using $N=57$ (as above), the logistic model would need $k\approx 0.1625$, which is a little smaller than the value used above for the cubic model.
\end{proof}

\item Does the data in Figure \ref{fig1} exhibit any characteristics of the solution curves found here?  Explain.

\begin{proof}[Answer:]
Yes.  For each model, the graph of $P(t)$ is initially concave up and then changes concavity as $P(t)$ increases.  The solutions for the logistic model appear to approach $N$ the fastest and flatten off significantly for large $t$, while the solutions for the cubic model approach $N$ more slowly and do not flatten off as much.  The Gompertz model falls between these two extremes.
\end{proof}
\end{enumerate}

\noindent 
{\bf Model Implementation}

\noindent
The instructor resources include a spreadsheet that can be used to verify student work for both this portion of the project and the numerical solution.  The graphs of the data along with the best fit produced for each model are given below.

\begin{center}
\includegraphics[width=0.75\textwidth]{modelsols.png}
\end{center}

\noindent
Questions: 

\begin{enumerate}[label=3.\arabic*,start=1]
\item Do any of the models fail to capture important trend(s) in the data?  Explain.

\begin{proof}[Answer:]
The data shows some oscillatory trends between $t=20$ and $t=52$ with at least two up-and-down swings.  None of the models can predict this behavior and, in fact, no autonomous model could do so.  The logistic model does a poor job of fitting the most recent data, especially the generally upward trend in $P(t)$ over the final twenty years.  The cubic model does the best job for this portion of the data, while the Gompertz model is better than the logistic but not as good as the cubic model.
\end{proof}

\item What do the optimal values of $k$ and $N$ convey about the progress towards a gender balance in the engineering sciences through 2018?  Do the models agree in this regard?

\begin{proof}[Answer:]
The logistic and Gompertz models are somewhat pessimistic with parameter estimates of $N=19.20$ and $N=20.55$, respectively.  These values are actually below the most recent values of $P(t)$.  The cubic model was optimized with $N=27.34$, which suggests that $P(t)$ will continue to increase in the near future, although more slowly than one might hope.  The models agree to the extent that the data suggests that a gender balance in undergraduate engineering programs does not appear to be imminent.
\end{proof}

\item Which model produced the smallest value for the sum of the squared error?

\begin{proof}[Answer:]
The cubic model achieved the best fit ($\text{SSE}=72.25$), followed by the Gompertz model ($\text{SSE}=82.18$) and the logistic model ($\text{SSE}=87.41$).  This seems to be due to the fact that the asymmetry in the cubic model is better suited to the data.  The data exhibits rapid initial growth combined with very slow growth as $P(t)$ increases, which mirrors the shape of the growth function for the cubic model.
\end{proof}

\item Which model do you think will predict $P(t)$ the most accurately over the next ten years?

\begin{proof}[Answer:]
The cubic model best captures the growth of $P(t)$ over the last 10 years and, as such, it is expected to do the best job predicting $P(t)$ over the next ten years.  Looking at the graphs, however, it appears that the recent growth rate of $P(t)$ exceeds even that of the cubic model, which is hopefully an encouraging sign.
\end{proof}
\end{enumerate}




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}

\end{document}

\emph{Are current efforts to diversify the undergraduate student body in the engineering sciences on track to achieve a gender balance mirroring that of the entire undergraduate population?}