Curve-Fitting Project: Linear Model of School Enrollment Data

This project investigates the changes in public school enrollment over time, specifically focusing on 11th and 12th-grade student enrollment from 1995 to 2004. The analysis uses data obtained from the National Center for Education Statistics (NCES) to determine if a linear relationship exists between the two variables. The project includes a scatter plot visualizing the relationship and uses a linear equation derived from Excel to model the data. The slope of the line is interpreted as the rate of change in enrollment. The project calculates and analyzes the correlation coefficient and the coefficient of determination to assess the goodness of fit for the linear model, predicting future enrollment values. The results show a strong linear relationship, with a high correlation coefficient indicating that as 11th-grade enrollment increases, so does 12th-grade enrollment.

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Curve-fitting Project - Linear Model
Student’s Name
Institutional Affiliation
Date

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LR1, LR2, LR3
The aim of this project is to investigate how the enrolment in public schools changes over
time (years). The period of investigation is from 1995 to 2004. Specifically, the project
investigates how the enrolment of 11th grade students and 12th grade students changes and if
there is a linear relationship between the two variables. Only public schools are considered in
this case. The data for this assignment was obtained from the National Center for Education
Statistics (NCES). The link for the data is
https://nces.ed.gov/programs/digest/d16/tables/dt16_201.20.asp
Table 1: 11th and 12th grade student enrolment in public schools that are part of state and
local systems from the year 1995 to 2004.
11th grade 12th grade
2826.023 2487.135
2930.297 2586.448
2971.923 2672.932
3020.899 2721.709
3033.941 2781.621
3082.842 2802.793
3173.939 2862.861
3228.867 2989.509
3277.218 3046.491
3369.339 3094.349

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2700 2800 2900 3000 3100 3200 3300 3400 3500
0
500
1000
1500
2000
2500
3000
3500
f(x) = 1.16189760619897 x − 787.455112215159
R² = 0.980447636361984
The relationship between 11th and 12th grade
enrolment
11th grade `
12th grade
Figure 1: The relationship between student enrolment in public schools from 11th and 12th
grades from 1995 to 2004
The equation for the straight line model obtained from excel is,
y=1.1619 x−787.46
This equation is then compared with the general equation for a straight line which is,
y=mx+c
Where,
m=slope=1.1619
c= y−intercept=787.46
LR-4
The slope of the graph can be interpreted as a measure of the rate of increase of the enrolment
of students with time (in this case years). The steepness of the slope is an indication of the

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magnitude of the rate of change. A steeper slope represents a higher rate of change and vice
versa.
LR-5
From figure 1, the value of the coefficient of determination was determined to be,
R2=0.9804
The table below shows the correlation between the two variables,
Table 2: The correlation coefficient obtained from excel
11th
grade
12th
grade
11th grade 1 0.990176
12th grade 0.990176 1
The correlation coefficient was obtained from excel and has a value of,
r =0.99
The correlation coefficient can be positive or negative. A positive correlation coefficient
indicates that there is a positive relationship between the two variables. That is, as one of the
variables increases, the other one increases too. On the other hand, if the correlation
coefficient is negative, one of the variables decreases as the other increases. Therefore, our
variables have a positive relationship. The goodness of the straight line to fit the given data is

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indicated by the correlation coefficient. The high correlation coefficient indicates that this is a
good model. The coefficient of determination has a value of 0.9804 which is very strong.
LR-6
Predicting future values
A prediction of any of the two variables at some time in the future can be made by
selecting a data value for the x variable (In this case a certain number of 11th grade students),
and substituting in the straight line equation to obtain the corresponding value for the y
variable.
y=1.1619 x−787.46
let x=3400
y= (1.1619 ×3400 )−787.46
y=3163
The coordinates can be written as,
3400,3163
LR-7
The analysis of the given data shows that a linear relationship exists between the
number of students enrolled in 11th grade and those enrolled in 12th grade from 1995 to 2004
in public schools. The coefficient of determination was very strong at 0.9804. the coefficient
of determination is a good indicator of the accuracy of the model connecting the two
variables. The higher the value, the better the model. This then means that predicting future

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values gives better results close to the actual values. The very high correlation coefficient of
0.99 indicates a very strong relationship between the two variables. Therefore, as the number
of 11th grade students enrolled in public schools increases, the 12th grade enrolment also
increases.
References

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Enrolment in grades 9 through 12 in public and private schools compared with population
14 to 17 years of age: Selected years, 1889-90 through fall 2016. (n.d.). Retrieved
from https://nces.ed.gov/programs/digest/d16/tables/dt16_201.20.asp