Part III
In part III, through scatterplots and regression analysis, we are trying to determine if there are relationships between Expenditure and SAT scores; Pupil/Teacher ratio and SAT scores; Teacher Salary and SAT scores; Revenues and SAT scores; Free and Reduced Lunch and SAT scores; and Students with Disabilities and SAT scores.
Descriptive StatisticsMean / Std. Deviation / N
Expenditure/ pupil (1000$) / 10,327.78 / 2,502.20 / 51
SAT score 2005-06 (verbal) / 534.94 / 37.80 / 51
SAT score 2005-06 (math) / 540.59 / 37.46 / 51
SAT score 2005-06 (writing) / 525.37 / 37.63 / 51
verbal SAT score 2005-06 / math SAT score 2005-06 / writing SAT score 2005-06
Expenditure/ pupil 2005-06 / Pearson r / -0.42** / -0.39** / -0.40**
p / 0.00 / 0.00 / 0.00
n / 51 / 51 / 51
In Figure 1, we are looking at the relationship between the current expenditure per pupil and the average math SAT score. Based on the Figure, I would say that the more money a district spends on a student, the more likely it is that they will have a lower math SAT score. There is a negative correlation between the math SAT scores and the expenditure per pupil with a slope of -.01 and a y-intercept at 601.42. With a slope of -.01 it means that for every dollarthat the expenditure increases, their math SAT score will decrease by .01. The coefficient correlation of -.39 tells us that there is a low correlation. There is definitely a relationship between the two variables, but it is a small relationship. Our p value tells us there is a significant difference. When we look at the coefficient of determination (r2), we see that it is low at .16, meaning that 16 percent of the variance in students’math SAT scores can be explained by the expenditure on that pupil. So 84 percent of their score is explained by other factors. There is a slight, almost negligible relationship. If the expenditure per pupil only explains 16 percent of the math SAT scores, it would be difficult to conclude that those scores are truly affected by the expenditure. Based on these findings, I don’t believe that spending more money on a student will cause their SAT scores to decrease. That doesn’t seem probable, but is nonetheless something we still need to take into consideration.
Ypred = -.01(X) + 601.42 So when X = 0, Y = 601.42
Figure 1
Model / R / R Square / Std. Error of the Estimate1 / .39a / .16 / 34.79
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 601.42 / 20.88 / 28.80 / .00
Expenditure/ pupil 2005-06 / -.01 / .00 / -.39 / -3.00 / .00
a. Dependent Variable: average math SAT score 2005-06
In Figure 2, we are looking at the relationship between the current expenditure per pupil and the average verbal SAT score. Based on the Figure, I would say that the more money a district spends on a student, the more likely it is that they will have a lower verbal SAT score. There is a negative correlation between the verbal SAT scores and the expenditure per pupil with a slope of -.01 and a y-intercept at 599.76. With a slope of -.01 it means that for every dollarthat the expenditure increases, their verbal SAT score will decrease by .01. The coefficient correlation of -.42 tells us that there is a moderate correlation. There is a substantial relationship between the two variables, and as the expenditure goes up, the verbal SAT score tends to decrease. Our p value tells us that there is a significant difference. When we look at the coefficient of determination (r2), we see that it is low at .17, meaning that 17 percent of the variance in students’ verbal SAT scores can be explained by the expenditure on that pupil. So 83 percent of their score is explained by other factors. Based on this figure, there is a slight, almost negligible relationship. If the expenditure per pupil only explains 17 percent of the verbal SAT scores, it would be difficult to conclude that those scores are truly affected by the expenditure. Based on these findings, I don’t believe that spending more money on a student will cause their SAT scores to decrease. That doesn’t seem probable, but is nonetheless something we still need to take into consideration. These findings are similar to the comparison of expenditure per pupil and math SAT scores.
Ypred = -.01(X) + 599.76 So when X = 0, Y = 599.76
Figure 2
Model / R / R Square / Std. Error of the Estimate1 / .42a / .17 / 34.73
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 599.76 / 20.85 / 28.77 / .00
Expenditure/pupil 2005-06 / -.01 / .00 / -.42 / -3.20 / .00
a. Dependent Variable: average verbal SAT score 2005-06
Descriptive Statistics
Mean / Std. Deviation / N
pupil/teacher ratio Fall 06 / 15.16 / 2.52 / 51
writing SAT score 2005-06 / 525.37 / 37.63 / 51
verbal SAT score 2005-06 / 534.94 / 37.80 / 51
math SAT score 2005-06 / 540.59 / 37.46 / 51
average writing SAT score 2005-06 / average verbal SAT score 2005-06 / average math SAT score 2005-06
average pupil/teacher ratio Fall 2006 / Pearson r / -.07 / -.03 / -.03
p / .64 / .82 / .85
n / 51 / 51 / 51
In Figure 3, we are looking at the relationship between the average pupil per teacher and the average verbal SAT score. Based on the Figure, it is difficult to say what the correlation is because our points are widely distributed with a slope of -.49 and a y-intercept of 542.32. With a slope of -.49 it means that for every student added to a teacher’s class, the verbal SAT average will decrease by .49. The coefficient correlation of -.03 tells us that there is a slight correlation. There is an insignificant relationship between the two variables. Our p value tells us there is no significant difference. When we look at the coefficient of determination (r2), we see that it is low at .00, saying that there is no correlation between the two variables or that 0 percent of a student’s verbal SAT score can be explained by the number of pupils per teacher. So approximately 100 percent of the scores are explained by other factors. This is unexpected. I would be under the assumption that if you have more pupils per teacher, it would be more difficult for an educator to teach well to all students. However, when we look at the slope, it appears as though, with an increase in pupil per teacher, the average SAT score decreases by nearly half a point. From a practical standpoint, I would say there is a relationship between the two variables.
Ypred = -.49(X) + 542.32 So when X = 0, Y = 542.32
Figure 3
Model / R / R Square / Std. Error of the Estimate1 / .03a / .00 / 38.16
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 542.32 / 32.86 / 16.50 / .00
pupil/teacher ratio Fall 2006 / -.49 / 2.14 / -.033 / -.23 / .82
a. Dependent Variable: average verbal SAT score 2005-06
In Figure 4, we are looking at the relationship between the average pupil per teacher and the average math SAT score. Based on the Figure, it is difficult to say what the correlation is because our points are widely distributed with a slope of -.41 and a y-intercept of 546.76. With a slope of -.41 it means that for every student added to a teacher’s class, the math SAT average will decrease by .41. The coefficient correlation of -.03 tells us that there is a slight correlation. There is an insignificant relationship between the two variables. Our p value tells us there is no significant difference. When we look at the coefficient of determination (r2), we see that it is low at .00, saying that there is no correlation between the two variables or that 0 percent of a student’s math SAT score can be explained by the number of pupils per teacher. So approximately 100 percent of the scores are explained by other factors. This is unexpected. I would be under the assumption that if you have more pupils per teacher, it would be more difficult for an educator to teach well to all students. However, when we look at the slope, it appears as though, with an increase in pupil per teacher, the average SAT score decreases by just under half a point. From a practical standpoint, I would say there is a relationship between the two variables.
Ypred = -.41(X) + 546.76 So when X = 0, Y = 546.76
Figure 4
Model / R / R Square / Std. Error of the Estimate1 / .03a / .00 / 37.82
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 546.76 / 32.57 / 16.79 / .00
average pupil/teacher ratio Fall 2006 / -.41 / 2.12 / -.03 / -.19 / .85
a. Dependent Variable: average math SAT score 2005-06
Descriptive Statistics
Mean / Std. Deviation / N
ave salary 2005-2006 / 47,679.08 / 6,942.01 / 51
writing SAT score 2005-06 / 525.37 / 37.63 / 51
verbal SAT score 2005-06 / 534.94 / 37.80 / 51
math SAT score 2005-06 / 540.59 / 37.46 / 51
average writing SAT score 2005-06 / average verbal SAT score 2005-06 / average math SAT score 2005-06
estimated ave salary 2005-2006 / Pearson r / -.45** / -.48** / -.41**
p / .00 / .00 / .00
n / 51 / 51 / 51
In Figure 5, we are looking at the relationship between the estimated average salary and the average math SAT score. Based on a first glimpse of the Figure, I would say that there is a negative correlation, but there is a slope of .00 and a y-intercept of 645.24. With a slope of .00 it tells us that there is no correlation between the estimated salary and the average math SAT. However, the coefficient correlation of -.41 tells us that there is a moderate correlation; there is a substantial relationship between the two variables. Our p value tells us there is a significant difference. When we look at the coefficient of determination (r2), we see that it is low at .17, meaning that only 17 percent of the variance in students’ math SAT scores can be explained by the estimated average salary. So 83 percent of their score is explained by other factors. The relationship is not visible on the scatterFigure, but the coefficient correlation tells us that there is still a relationship. Some would assume that as the salary for an educator increases, scores for the SAT would also increase, but this is not necessarily true. It says that regardless of how much an educator is paid, whether they are paid a lot of money on the East coast and work in a suburb, or whether they are paid substantially less in a West coast city, students are still scoring the same on SAT’s across the nation.
Ypred = .00(X) + 645.24 So when X = 0, Y = 645.24
Figure 5
Model / R / R Square / Std. Error of the Estimate1 / .41a / .17 / 34.57
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 645.24 / 33.92 / 19.02 / .00
estimated ave salary 2005-2006 / -.00 / .00 / -.41 / -3.12 / .00
a. Dependent Variable: average math SAT score 2005-06
In Figure 6, we are looking at the relationship between the estimated average salary and the average verbal SAT score. Based on a first glimpse of the Figure, I would say that there is a negative correlation, but there is a slope of .00 and a y-intercept of 658.20. With a slope of .00 it tells us that there is no correlation between the estimated salary and the average verbal SAT. However, the coefficient correlation of -.48 tells us that there is a moderate correlation; there is a substantial relationship between the two variables. Our p value tells us there is a significant difference. When we look at the coefficient of determination (r2), we see that it is low at .23, meaning that only 23 percent of the variance in students’ verbal SAT scores can be explained by the estimated average salary. So 77 percent of their score is explained by other factors. The relationship is not visible on the scatterFigure, butthe coefficient correlation tells us that there is still a relationship. Some would assume that as the salary for an educator increases, scores for the SAT would also increase, but this is not necessarily true. This is very similar to the comparison of estimated average salary and average math SAT scores. It says that regardless of how much an educator is paid, whether they are paid a greater amount of money on the East coast and work in a suburb, or whether they are paid substantially less in a West coast city, students are still scoring the same on SAT’s across the nation.
Ypred = .00(X) + 658.20 So when X = 0, Y = 658.20
Figure 6
Model / R / R Square / Std. Error of the Estimate1 / .48a / .23 / 33.61
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 658.20 / 32.98 / 19.96 / .00
ave salary 2005-2006 / -.00 / .00 / -.48 / -3.78 / .00
a. Dependent Variable: average verbal SAT score 2005-06
Descriptive Statistics
Mean / Std. Deviation / N
Total revenues 2005-06 (1000s) / 10,208,705.02 / 12,216,880.36 / 51
writing SAT score 2005-06 / 525.37 / 37.63 / 51
verbal SAT score 2005-06 / 534.94 / 37.80 / 51
math SAT score 2005-06 / 540.59 / 37.46 / 51
writing SAT score 2005-06 / verbal SAT score 2005-06 / math SAT score 2005-06
Total revenues for the year 2005-06 (in thousands) / Pearson r / -.25 / -.29* / -.20
p / .07 / .04 / .16
N / 51 / 51 / 51
In Figure 7, we are looking at the relationship between the total revenues and the average verbal SAT score. Based on a first glimpse of the Figure, I would say that there is a negative correlation with a slope of -9.070E-7 and a y-intercept of 544.20. With a slope of -9.070E-7 it tells us that there is negative correlation between the total revenues and the average verbal SAT score. For every thousand dollars in revenue, the average SAT score will drop by only a fraction of a point. The coefficient correlation of -.29 tells us that there is a low correlation; there is a definite relationship between the two variables, but the relationship is small. Our p value tells us there is a significant difference. When we look at the coefficient of determination (r2), we see that it is low at .09, meaning that only 9 percent of the variance in students’verbal SAT scores can be explained by the estimated average salary. So 91 percent of their score is explained by other factors. I would assume that the higher the revenue, the higher the scores for the SAT would be, but this is not necessarily true. Based on the coefficient of determination, the relationship between the two variables is low. Based on the slope, the relationship is almost nonexistent. This says that regardless of how much a state earns in revenue, a student’s score on the SAT’s will not be greatly affected.
Ypred = -9.070(X) + 544.20 So when X = 0, Y = 544.20
Figure 7
Model / R / R Square / Std. Error of the Estimate1 / .29a / .09 / 36.50
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 544.20 / 6.69 / 81.36 / .00
Total revenues 2005-06 (in thousands) / -9.070E-7 / .00 / -.29 / -2.15 / .04
a. Dependent Variable: average verbal SAT score 2005-06
In Figure 8, we are looking at the relationship between the total revenues and the average math SAT score. Based on a first glimpse of the Figure, I would say that there is a negative correlation with a slope of -6.089E-7 and a y-intercept of 546.80. With a slope of -6.089E-7 it tells us that there is negative correlation between the total revenues and the average math SAT score. For every thousand dollars in revenue, the average SAT score will drop by a fraction of a point. The coefficient correlation of -.20 tells us that there is a low correlation; there is a definite relationship between the two variables, but the relationship is small. Our p value tells us there is no significant difference. When we look at the coefficient of determination (r2), we see that it is low at .04, meaning that only 4 percent of the variance in students’ math SAT scores can be explained by the estimated average salary. So 96 percent of their score is explained by other factors. I would assume that the higher the revenue, the higher the scores for the SAT would be, but this is not necessarily true. Based on the coefficient of determination, the relationship between the two variables is low. And based on the slope, the relationship is almost nonexistent. This says that regardless of how much a state earns in revenue, a student’s score on the SAT’s will not be greatly affected.
Ypred = -6.089(X) + 546.80 So when X = 0, Y = 546.80
Figure 8
Model / R / R Square / Std. Error of the Estimate1 / .20a / .04 / 37.09
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 546.80 / 6.80 / 80.47 / .00
Total revenues 2005-06 (in thousands) / -6.089E-7 / .00 / -.20 / -1.42 / .16
a. Dependent Variable: average math SAT score 2005-06
Descriptive Statistics
Mean / Std. Deviation / N
% of students eligible for free/reduced lunch 2006-07 / 39.82 / 10.64 / 50
writing SAT score 2005-06 / 525.37 / 37.63 / 51
verbal SAT score 2005-06 / 534.94 / 37.80 / 51
math SAT score 2005-06 / 540.59 / 37.46 / 51
writing SAT score 2005-06 / verbal SAT score 2005-06 / math SAT score 2005-06
% of students eligible for free/reduced lunch 2006-07 / Pearson r / .08 / .02 / -.08
p / .58 / .87 / .57
n / 50 / 50 / 50
In Figure 9, we are looking at the relationship between the percentage of students eligible for free and reduced lunch and the average math SAT score. Based on a first glimpse of the Figure, I would say that there is a negative correlation with a slope of -.292 and a y-intercept of 552.85. With a slope of -.292 it tells us that there is negative correlation between the percentage of students eligible for free and reduced lunch and the average math SAT score. For every one percent increase in eligible students, the average SAT score will drop by approximately .3 points. The coefficient correlation of -.08 tells us that there is only a slight correlation; there is almost no relationship between the two variables. Our p value tells us there is no significant difference. When we look at the coefficient of determination (r2), we see that it is low at .01, meaning that only 1 percent of the variance in students’ math SAT scores can be explained by the percentage of eligible students. So 99 percent of their score is explained by other factors. Regardless of the percentage of students in a state that are eligible for free and reduced lunch, the math SAT scores will not be affected. We can say that there is a relationship, but it is so low it is almost non existent.
Ypred = -.292 (X) + 552.85 So when X = 0, Y = 552.85
Figure 9
Model / R / R Square / Std. Error of the Estimate1 / .08a / .01 / 37.80
Coefficientsa
Model / Unstandardized Coefficients / Standardized Coefficients / t / Sig.
B / Std. Error / Beta
1 / (Constant) / 552.85 / 20.90 / 26.45 / .00
% of students eligible for free/reduced lunch 2006-07 / -.292 / .51 / -.08 / -.58 / .57
a. Dependent Variable: average math SAT score 2005-06
In Figure 10, we are looking at the relationship between the percentage of students eligible for free and reduced lunch and the average verbal SAT score. Based on a first glimpse of the Figure, it is difficult to say what the correlation might be. There is a slope of .09 and a y-intercept of 532.29. With a slope of .09 it tells us that there is positive correlation between the percentage of students eligible for free and reduced lunch and the average verbal SAT score. For every one percent increase in eligible students, the average SAT score will increase by approximately one tenth of a point. The coefficient correlation of .02 tells us that there is only a slight correlation; there is almost no relationship between the two variables. Our p value tells us there is no significant difference. When we look at the coefficient of determination (r2), we see that it is low at .00, meaning that only 0 percent of the variance in students’ math SAT scores can be explained by the percentage of eligible students. So 100 percent of their score is explained by other factors. Regardless of the percentage of students in a state that are eligible for free and reduced lunch, the verbal SAT scores will not be affected. Based on the slope, we can say that there is a relationship, but it is so low it is almost non existent.