Chapter 4
RESULTS
Summary
The
researcher analyzed the results of the 2006 eighth grade Illinois Standards
Achievement Test (ISAT) math passing percentages available on a
schoolbyschool basis at http://www.isbe.net.
The schools used in this study were public schools that offered eighth
grade in Cook (excluding Chicago Public Schools), DuPage, Kane, Lake, McHenry and Will counties. The appendix contains one table for each
county that lists the schools in that county used in this preliminary
study. The total sample size of eighth
graders in this study exceeds 55,000.
This number is an approximation based on the total number of students in
each school divided by grade levels.
The data
gathering procedure required the researcher to record numerous characteristics
of controllable and noncontrollable factors specific to each of the 258
schools in this study. Controllable
factors are defined as factors that the school administration exhibits control
over. Noncontrollable factors are
defined as factors that the school administration has no control over. Figure 4 shows the controllable and
noncontrollable factors that were gathered for all of the schools combined.
Controllable
Factors

Noncontrollable
Factors

PupiltoTeacher
Ratio

Percentage of White
Students

Average Class Size
(8^{th} grade specific)

Percentage of Black
Students

Minutes of Math per
Day (8^{th} grade specific)

Percentage of
Hispanic Students

Percentage of Male
Teachers

Percentage of Asian
Students

Teachers’ Average
Years of Experience

Percentage of Native
Students

Percent of Teachers
with M.A.s or Higher

Percentage of
Multiracial Students

Average Teachers’
Salary

Percentage of Low
Income Students

Average
Administrators’ Salary

Mobility Rate
(percentage)

2005 Expenditure
per Student on Instruction

Attendance Rate
(percentage)

Figure 4:
Controllable and Noncontrollable Factors

The controllable and
noncontrollable factors, together with the eighth grade math ISAT passing
percentage, the county, the city, and the school, constituted more than 5,500
individual pieces of information that were entered into a Microsoft Excel
spreadsheet in preparation for this analyses.
This
preliminary study served to explore the connections between the eighth grade
math ISAT passing percentage and the controllable and noncontrollable
factors. Multiple linear regression
analysis was used to determine if relationships exist between the eighth grade
math ISAT passing percentage and the controllable and noncontrollable factors.
Results from the Analysis of Controllable
Education Factors
The purpose
of the first analysis performed by the researcher was to identify the
controllable factors that influence eighth grade math ISAT passing
percentages. The researcher employed the
multiple linear regression method described in Chapter III. The data used in the multiple linear
regression analysis included 258 schools with an eighth grade population of
more than 55,000 students.
The controllable factors that have
an influence on the eighth grade math ISAT passing percentage (y^) are:
1. pupiltoteacher
ratio, PtTR
2. average
class size, ACS
3. minutes
of math instruction per day, MMI
4. percentage
of teachers with M.A.s or higher, PerMA
5. average
administrator salary (based on a per $1,000 amount), AdminSal.
The results of the multiple regression analysis for the
controllable factors are show in Figure 5:
Controllable
Factor

Pvalue from
Multiple Regression Analysis

Pupiltoteacher ratio

0.014

Average class size

0.008

Minutes of math per day

Less than 0.001

Percentage of teachers with M.A.s or higher

Less than 0.001

Average administrator salary

0.003

Figure 5:
Controllable Factor Pvalues

The rsquare value is 0.368.
If persons were interested in predicting the eighth grade
math ISAT passing percentage based only on the controllable factors, they could
do so by employing the fixed value and the coefficients of the influential
independent variables that resulted from the multiple linear regression
analysis. From this analysis, the fixed
portion of the results, referred to as the yintercept, is equal to 63.8
percent. That is, the percentage of
eighth graders who meet or exceed the minimum ISAT math score is 63.8% before
the controllable factors are applied.
With the application of the controllable factors, the predictive model
equation is:
y^ = 63.8  0.82 PtTR + 0.48 ACS  0.28 MMI + 0.32 PerMA
+ 0.18 AdminSal
As an example of how
the controllable factors predictive model equation works, one needs to
calculate the expected percentage of eighth grade students attending Westchester Middle School
(Cook County) who meet or exceed the minimum
ISAT math score. The following are Westchester Middle School’s controllable factor
values:
a) pupiltoteacher
ratio is 16.6 students
b) average
class is 20.2 students
c) minutes
of math per day is 43
d) percent
of teachers with M.A.s or higher is 50.2
e) average
administrator salary is $95,128 and
The percent of students who meet or
exceed the minimum ISAT math score =
63.8  (0.82 x 16.6) + (0.48
x 20.2) 
(0.28 x 43) + (0.32 x 50.2) + (0.18 x 95.128) = 81.2%
According to this calculation, the
expected percentage of eighth grade students who meet or exceed the minimum
ISAT math score for this data would be about 81.2%. The actual percentage of eighth grade
students who met or exceeded the minimum ISAT math score was 85.0% or a
difference of 3.8%.
The predictive model equation is not considered a
reliable predictive tool unless there is a way of checking how accurately it
predicts. One of the most
straightforward ways of checking the predictive equation’s accuracy is to see
how well it can predict the passing percentages of the schools used in this
analysis. The process entails using the
predictive model equation on each school in the study to calculate the school’s
predictive passing percentage. Then the
predicted passing percentage is compared to the school’s actual passing
percentage. The difference between the
predicted and actual passing percentage is called a residual. The smaller the residuals, the better the
predictive model equation is for predicting a school’s expected passing
percentage.
The
first graphical analysis of the residuals from the controllable factors
predictive model equation is shown in Figure 6.

Figure 6: Residuals Versus Observations for Controllable
Factors Model Equation

Figure 6
shows that the residuals for the controllable factors predictive model equation
are well distributed. The graph in Figure
6 shows a handful of schools that have residuals above 20 and below –20.
Figure 7
shows another graphical analysis of the residuals from the controllable factors
predictive model equation.

Figure
7: Residuals Versus Passing Percentage for
Controllable Factors Model Equation

Figure 7
shows a plot of the residuals compared to the passing percentage for the
controllable factors predictive model equation.
The graph in Figure 7 shows a handful of schools that have residuals
above 20 and below –20.
Results from the Analysis of
NonControllable Education Factors
The purpose of the second analysis
performed by the researcher was to identify the noncontrollable factors that
influence eighth grade math ISAT passing percentages. Again, the researcher employed the multiple
linear regression method described in Chapter III. The data used in this second multiple linear
regression analysis included all 258 schools with an eighth grade population of
more than 55,000 students.
The noncontrollable factors that
have an influence on the eighth grade math ISAT passing percentage (y) are:
1. percentage
of black students, PBS
2. percentage
of Hispanic students, PHS
3. percentage
of lowincome students, PLI
4. student
mobility rate, SMR
5. student
attendance rate, SAR.
The results of the multiple regression analysis for the
noncontrollable factors are show in Figure 8:
Noncontrollable
Factor

Pvalue from Multiple
Regression Analysis

Percentage of black students

Less than 0.001

Percentage of Hispanic students

0.001

Percentage of lowincome students

0.004

Student mobility rate

Less than 0.001

Student attendance rate

Less than 0.001

Figure 8:
Noncontrollable Factor Pvalues

The rsquare value is
0.786.
If persons were interested in predicting the eighth grade
math ISAT passing percentage based only on the noncontrollable factors, they
could do so by employing the fixed value and the coefficients of the
influential independent variables that resulted from the multiple linear
regression analysis. From this
preliminary analysis, the fixed portion of the results, referred to as the
yintercept, is equal to 121.3 percent.
With the application of the noncontrollable factors, the predictive
model equation is:
y^ = 2.25 SAR  0.21 PBS  0.12 PHS  0.11
PLI 
0.18 SMR 
121.3
As an
example of how the noncontrollable factors predictive model equation works,
let us calculate the expected percentage of eighth grade students attending Westchester Middle School
(Cook County) who meet or exceed the minimum
ISAT math score. The following are Westchester Middle School’s noncontrollable factor
values:
a) the
black student percentage is 24.9
b) the
Hispanic student percentage is 17.9
c) the
lowincome percentage is 6.7
d) the
mobility rate is 7.6 percent and
e) the
attendance rate is 96.5 percent
The percent of students who meet or
exceed the minimum ISAT math score =
(2.25 x 96.5)  (0.21 x 24.9) 
(0.12 x 17.9) 
(0.11 x 6.7) 
(0.18 x 7.6) 
121.3 = 86.7%
According to this calculation, the
expected percentage of eighth grade students who meet or exceed the minimum
ISAT math score for these factors would be about 86.7%. The actual percentage of eighth grade
students who met or exceeded the minimum ISAT math score was 85.0% or a
difference of 1.7%.
As described in the previous section, the reliability of
the predictive equation needs to be tested to determine if it is an accurate
means of predicting math ISAT passing percentages based on noncontrollable
factors. The same methodology will be
used to check the reliability of the noncontrollable factors predictive
equation as was employed with the controllable factors predictive equation. The process involves calculating the residuals
for each school used in the study and then plotting the residuals to determine
whether a pattern exists.
Figure
9 shows the first graphical analysis of the residuals from the noncontrollable
factors predictive model equation.

Figure 9: Residuals Versus Observations for Noncontrollable Factors Model Equation

The
residuals for the noncontrollable factors predictive model equation are well
distributed. The graph in Figure 9 shows
a few schools that have residuals above 20 and below –20.
The second
graphical analysis of the residuals from the noncontrollable factors
predictive model equation is shown in Figure 10.

Figure
10: Residuals Versus Passing Percentage for Noncontrollable
Factors Model Equation

Figure 10 shows a
plot of the residuals compared to the passing percentage for the
noncontrollable factors predictive model equation. The graph in Figure 10 shows a few schools
that have residuals above 20 and below –20.
Results from the Analysis of the
Combination of Controllable and NonControllable Factors
The purpose of the third and last
analysis was to identify the combined controllable and noncontrollable factors
that influence eighth grade math ISAT passing percentages. Again, the researcher employed the multiple
linear regression method described in Chapter III. The data used in this third multiple linear
regression analysis included all 258 schools with an eighth grade population of
more than 55,000 students.
The combined controllable and
noncontrollable factors that have an influence on the eighth grade math ISAT
passing percentage are:
1. percentage
of black students, PBS
2. percentage
of Hispanic students, PHS
3. percentage
of lowincome students, PLI
4. student
mobility rate, SMR
5. student
attendance rate, SAR
6. pupiltoteacher
ratio, PtTR.
The results of the multiple regression analysis for the
combined controllable and noncontrollable factors are show in Figure 11:
Controllable
and Noncontrollable Factors

Pvalue from
Multiple Regression Analysis

Percentage of black students

Less than 0.001

Percentage of Hispanic students

0.001

Percentage of lowincome students

0.006

Student mobility rate

0.001

Student attendance rate

Less than 0.001

Pupiltoteacher ratio

0.001

Figure 11: Controllable
and Noncontrollable Factor Pvalues

The rsquare value is 0.795.
If persons were interested in predicting the eighth grade
math ISAT passing percentage based on the controllable and noncontrollable
factors, they could do so by employing the fixed value and the coefficients of
the influential independent variables that resulted from the multiple linear
regression analysis. From this
preliminary analysis, the fixed portion of the results, referred to as the
yintercept, is equal to 65.4 percent.
With the application of the controllable and noncontrollable factors,
the predictive model equation would be:
y^ = 1.78 SAR  0.23 PBS  0.12 PHS  0.11
PLI 
0.15 SMR 
0.57 PtTR 
65.4
As an example of how
the controllable and noncontrollable factors predictive model equation works,
one needs to calculate the expected percentage of eighth grade students
attending Westchester Middle School (Cook County)
who meet or exceed the minimum ISAT math score.
The following are Westchester
Middle School’s
controllable and noncontrollable factor values:
a) the
black student percentage is 24.9
b) the
Hispanic student percentage is 17.9
c) the
lowincome percentage is 6.7
d) the
mobility rate is 7.6 percent
e) the
attendance rate is 96.5 percent and
f) the
pupiltoteacher ratio is 16.6.
The percent of students who meet or
exceed the minimum ISAT math score =
(1.78x96.5)  (0.23x24.9) 
(0.12x17.9) 
(0.11x6.7) 
(0.15x7.6) 
(0.57x16.6) 
65.4 = 86.7%
According to this calculation, the
expected percentage of eighth grade students who meet or exceed the minimum
ISAT math score for these factors would be about 86.7%. The actual percentage of eighth grade
students who met or exceeded the minimum ISAT math score was 85.0% or a
difference of 1.7%.
Once, again the reliability of the predictive equation
needs to be tested to determine if it is an accurate means of predicting math
ISAT passing percentages based on controllable and noncontrollable
factors. The same methodology will be
used to check the reliability of the combined controllable and noncontrollable
factors predictive equation as was employed in the previous two analyses. The process involves calculating the
residuals for each school used in the study and then plotting the residuals to
determine whether a pattern exists.
Figure 12 shows the first graphical
analysis of the residuals from the controllable and noncontrollable factors
predictive model equation.

Figure 12: Residuals Versus Observations for Controllable and Noncontrollable Factors Model Equation

The
residuals for the combined controllable and noncontrollable factors predictive
model equation are well distributed. The
graph in Figure 12 shows a few schools that have residuals above 20 and below
–20.
The second
graphical analysis of the residuals from the combined controllable and
noncontrollable factors predictive equation is shown in Figure 13.

Figure
13: Residuals Versus Passing Percentage for Controllable
and Noncontrollable Factors Model Equation

Figure 13
shows a plot of the residuals compared to the passing percentage for the
combined controllable and noncontrollable factors predictive model
equation. The graph in Figure 13 shows a
few schools that have residuals above 20 and below –20.