Question

Portray the following hypothetical data on a two-variable diagram: $$ \begin{array}{ccc} \text { Academic Year } & \begin{array}{c} \text { Total } \\ \text { Enrollment } \end{array} & \begin{array}{c} \text { Enrollment in } \\ \text { Economics Courses } \end{array} \\ \hline 2012-2013 & 3,000 & 300 \\ \hline 2013-2014 & 3,100 & 325 \\ \hline 2014-2015 & 3,200 & 350 \\ \hline 2015-2016 & 3,300 & 375 \\ 2016-2017 & 3,400 & 400 \\ \hline \end{array} $$ Measure the slope of the resulting line, and explain what this number means.

Short answer

Answer: The slope of the line in a two-variable diagram represents the rate of change of enrollment in economics courses (Y) in relation to total enrollment (X). It indicates how enrollment in economics courses increases or decreases for every unit increase in total enrollment. A positive slope implies that economics enrollment increases as total enrollment increases, while a negative slope implies a decrease in economics enrollment as total enrollment increases.

Step-by-step solution

1. Understand the variables and their representation

The given data has two variables: Total Enrollment (variable X) and Enrollment in Economics Courses (variable Y). Each row of the table represents a data point (X, Y) for a particular academic year.

2. Plot the data points

Using graph paper or any graphing tool, create a two-variable diagram (also known as a scatterplot) with Total Enrollment on the horizontal (X) axis and Enrollment in Economics Courses on the vertical (Y) axis. For each academic year, plot a point using the corresponding X and Y values on the graph. For example, for the 2012-2013 academic year, plot point (3000, 300).

3. Draw the best-fit line

In the scatterplot, draw a line that follows the trend of the plotted points and passes through as many of the points as possible. Ideally, the line should have an equal number of points above and below it. This is called a best-fit line or a line of best fit.

4. Calculate the slope of the line

To calculate the slope of the line, follow these steps: a) Choose two points on the line (preferably endpoints), call them A (x1, y1) and B (x2, y2). b) Find the difference in Y values (change in Y): ΔY = y2 - y1 c) Find the difference in X values (change in X): ΔX = x2 - x1 d) Calculate the slope by dividing ΔY by ΔX: slope = ΔY / ΔX

5. Interpret the slope

The slope of the line represents the rate of change of Y in relation to X. In this context, it represents the rate at which enrollment in economics courses (Y) increases for every unit increase in total enrollment (X). If the slope is positive, it implies that enrollment in economics courses increases as total enrollment increases, and if the slope is negative, it implies a decrease in economics enrollment with an increase in total enrollment.

Key concepts

Enrollment Trends

Enrollment trends refer to how the number of students signing up for courses changes over time. In the context of our data, we have observed the total enrollment as well as the enrollment specifically in economics courses across different academic years. Noticing trends in these enrollments can provide insights into the popularity of different subjects or shifts in student interest.
To effectively understand trends, visualize the data on a chart that tracks changes over successive years. This helps in clearly seeing upward or downward trends. An increase in total enrollment alongside a consistent rise in economics course enrollment, as given in our problem, indicates growing interest in economics, or possibly increased requirement offerings.
Understanding these trends aids educational institutions in decision-making processes such as course offerings, resource allocations, and departmental funding.

Data Analysis

Data Analysis involves examining datasets to make informed decisions. When analyzing enrollment data, the main task is to identify patterns or trends. These patterns assist in understanding how certain variables interact or change over time.
In our exercise, the dataset gives total enrollment figures and enrollment in economics courses across five academic years. By plotting these data points on a scatterplot, the relationship between these two variables becomes visually evident. This relationship is crucial as it reveals whether economics courses' popularity increases or if it's simply in proportion with total enrollment growth.
From there, we utilize the best-fit line to demonstrate the overall tendency in the data. This line, often used in statistics, shows how two variables might correlate. Calculating the slope of this line further helps quantify how these variables interact, reflecting how much one changes in response to the other.

Academic Year

An academic year is the period during which school operates, typically split into semesters or quarters. It’s important to note that data presented by academic years, such as our example of total enrollments, captures consistent changes and trends annually.
Each academic year mentioned in the dataset provides a snapshot of student enrollments during that time. Tracking individual academic years allows for a clear year-to-year comparison of how both total enrollment and economics course enrollments change.
Focusing on each academic year helps institutions plan better. They use these insights to adjust curricula, manage resources efficiently, and anticipate future educational demands.