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TRANSCRIPT
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Running head: Final Project
Final Project
Elizabeth Enfield, Heather Davis, & Jodi Pitts
Seattle Pacific University
EDU 6976 –Interpreting & Applying Educational Research
Winter 2011
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Enrollment data was collected from public elementary and secondary schools in
fifty states and the District of Columbia in order to determine how the pupil to teacher
ratio, current expenditure per pupil, pupil to teacher ratio, and teacher salary would
affect students’ verbal SAT scores.
Part 1: Histograms, Box Plots, and Frequency Distribution
In this study, four continuous variables will be examined including pupil to
teacher ratio for fall 2005, expenditure per pupil, teacher salary, and student SAT verbal
scores.
The histogram represents the frequency of occurrence of scores. The box plot
divides the data into quartiles, and provides us with data to see if it is positively or
negatively skewed. Each of the four histograms and box plots represents the distribution
of scores for the four variables focused on in the study: pupil to teacher ratio for fall
2005 (ratio 1), expenditure per pupil, teacher salary, and student SAT verbal scores.
We can visually examine the distributions by looking at the data represented in the
histogram and box plots.
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<=10 (10, 12] (12, 14] (14, 16] (16, 18] (18, 20] >200
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4
6
8
10
12
14
16
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20 Frequency Distribution of
Pupil to Teacher Ratio (fall 2005)
Figure 1. Histogram for Pupil to Teacher Ratio
Lower
Whisker
Lower
Hinge Median
Upper
Hinge
Upper
Whisker
10.8 13.55 14.8 16.5 20.8
0 5 10 15 20 25 30
Figure 2. Box Plot for Pupil to Teacher Ratio
The distribution of average pupil to teacher ratio in 2005 seems to be close to a
normal distribution. The data is positively skewed with outliers to the right of the mean.
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<=5000 (5000, 7000]
(7000, 9000]
(9000, 11000]
(11000, 13000]
(13000, 15000]
(15000, 17000]
>170000
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4
6
8
10
12
14
16
18
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Frequency Distribution of Expendi-ture Per Pupil
Figure 3 Histogram for Expenditure per Pupil
Lower
Whisker
Lower
Hinge Median
Upper
Hinge
Upper
Whisker
5960 8639 9805 11426 14277
0 5000 10000 15000 20000 25000
Figure 4 Box Plot for Expenditure per Pupil
The distribution of expenditure per pupil seems to be close to a normal
distribution. The data is positively skewed with outliers to the right of the mean. The
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mean is higher than the median. Outliers have elevated the value of the mean. The
largest cluster of expenditure per pupil values is between 7,000 and 11,000.
In looking at the box plot, the line drawn from the minimum value to quartile one
and the line drawn from the third quartile to the maximum value are close to the same
length, which supports that it is close to normal distribution. The box itself contains the
middle 50% of the data. The median line within the box is not halfway from the hinges,
so the data is skewed.
<=35000
(35000, 41000]
(41000, 47000]
(47000, 53000]
(53000, 59000]
(59000, 65000]
>650000
5
10
15
20
25
30
Frequency Distribution ofTeacher Salary
Figure 5. Histogram for Teacher Salary
Lower
Whisker
Lower
Hinge Median
Upper
Hinge
Upper
Whisker
35607 42179.5 45575 53276.5 61372
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Figure 6. Box Plot for Teacher Salary
The distribution of average annual teacher salary is skewed to the right,
indicating that it is positively skewed. There are no outliers. This box plot also
signifies the mean is higher than the median. The largest cluster of salaries is
between 40,000 and 50,000.
<=450 (450, 475]
(475, 500]
(500, 525]
(525, 550]
(550, 575]
(575, 600]
(600, 625]
(625, 650]
>6500
2
4
6
8
10
12
14
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Frequency Distribution of SAT scores (verbal)
Figure 7. Histogram for SAT scores (verbal)
Lower
Whisker
Lower
Hinge Median
Upper
Hinge
Upper
Whisker
482 498 523 569 610
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200 300 400 500 600 700 800 900
Figure 8. Box Plot for SAT scores (verbal)
There is not a normal distribution in the distribution of SAT verbal scores. The
data is positively skewed, and there is less data occurring around numbers close to the
median and mean. While there is some variation in the placement of the data, there are
no distinct outliers.
Categorical variable: Distribution of Regions
A categorical variable in this study would be the geographical regions (West,
Mid-West, South, and Northeast). The regions are of unequal size. The south region
has the most states while the northeast region has the least number of states.
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Region 1 (West) Region 2 (Midwest)
Region 3 (South)
Region 4 (Northeast)
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2
4
6
8
10
12
14
16
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Figure 9. Bar Graph for region distribution
Part 2: Analysis of Variance
After analyzing the frequency distribution of the continuous variables, the
categorical variable, Region, will be examined. The four regions that will be compared
are Region 1 (West), Region 2 (Midwest), Region 3 (South), and Region 4 (Northeast).
The regions will be analyzed for differences in terms of four variables. ANOVA allows
comparison of differences among many sample groups. ANOVA tests will be used here
to determine whether there are differences among means of the four regions for these
four variables.
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Expenditure per Pupil
Table 1: ANOVA table for expenditure per pupil
ANOVA Table 5%
Source SS df MS F Fcritical
p-value
Between1.2E+0
8 34E+0
7 9.75122.802
40.000
0Rejec
t
Within1.9E+0
8 474E+0
6
Total3.1E+0
8 50
Table 2: Estimate of group means for expenditure per pupil
Estimates of Group MeansGroup Confidence Interval
R19244.9
2 ±1130.5 95%
R29905.4
2 ±1176.7 95%
R39720.8
8 ±988.61 95%
R413601.
4 ±1358.7 95%
Table 3:Tukey testforexpenditureper pupil
Tukey test for pairwise comparison of group meansR1
r 4R2 R2
n - r 47R3 R3
q0 3.79R4 Sig Sig Sig R4
T2559.7
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The first selected variable on which the four regions were compared was
expenditure per pupil. The mean descriptive statistics indicate the magnitude of the
differences between means (Table 2). This can also be seen in the 95% confidence
intervals for the means for the four regions. The Northeast had the largest expenditure
per pupil of all four regions; the West had the lowest.
An ANOVA test was conducted for expenditure per pupil. The critical value of F
is 2.80 for = .05. The obtained value of F is 9.75 (Table 1). When the obtained F
ratio is large, the variability between groups is greater than the variability within groups.
Here, the obtained value of F is larger than the critical, or table value of F, so the null
hypothesis is rejected. Therefore, there is a significant difference between the regions
in terms of expenditure per pupil.
The Levene test is a test of the homogeneity of variance among the groups. If
the significance level is greater than .05, homogeneity can be assumed. Here the
significance of Levene’s test is greater, .24. Therefore the variances are equal across
groups, which is the underlying assumption of the ANOVA.
Post-hoc tests are used as the next step in the analysis. When the F ratio has
been found to be significant, it should also be determined where the sample differences
come from. The effects may not be spread out evenly. To determine where significant
differences might be, Tukey’s HSD is used. By looking at Tukey HSD and comparing
group means, it can be seen that there are significant differences between Region 4
(Northeast) and each of the other three regions at the .05 level (Table 3). The mean
expenditure per pupil in the Northeast is significantly higher than in the other regions in
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the United States. There are no differences between the means of Region 1 (West) and
Region 2 (Midwest), Region 1 and Region 3 (South), or Region 2 and Region 3.
Pupil to Teacher Ratio:
Table 4: ANOVA table for pupil to teacher ratio
ANOVA Table 5%
Source SS df MS F Fcritical
p-value
Between 146.88 3 48.96 13.081
2.8024
0.0000
Reject
Within175.91
7 473.742
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Total322.79
7 50
Table 5: Estimates of group means for pupil to teacher ratio
Estimates of Group MeansGroup Confidence Interval
R117.807
7 ±1.0795 95%
R214.808
3 ±1.1235 95%
R314.864
7 ± 0.944 95%
R412.733
3 ±1.2973 95%
Table 6: Tukey test for pupil to teacher ratio
Tukey test for pairwise comparison of group meansR1
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r 4R2 Sig R2
n - r 47R3 Sig R3
q0 3.79R4 Sig R4
T2.4441
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The four regions were compared with respect to pupil to teacher ratio. The pupil
to teacher ratio is highest in the West and lowest in the Northeast. From the ANOVA,
the obtained value of F is 13.08 (Table 4). This F ratio is much larger than the critical
value of 2.80 so the null hypothesis is rejected. There is a significant difference
between the regions in terms of pupil to teacher ratio.
Levene’s test shows that p is .01, which is less than the critical value of .05. The
null hypothesis of equal variances is rejected and it is concluded that there is a
difference between the variances in the population. This does not meet the underlying
assumption of an ANOVA, which is that the variances are equal across the groups. A
modified procedure should be used.
Tukey’s HSD indicates significant differences between Region 1(West) and all
three of the other regions (Table 6). The pupil to teacher ratio in the West is
significantly higher than in the rest of the country.
The Dunnet C test shows the above difference between the West and the other
three regions, as well as a significant difference between Region 3 (South) and Region
4 (Northeast). The Northeast has a significantly smaller pupil to teacher ratio than the
South.
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Average Teacher Salary:
Table 7: ANOVA data for average teacher salary
ANOVA Table 5%
Source SS df MS F Fcritical
p-value
Between
4.3E+08 3
1E+08 3.4505
2.8024
0.0238
Reject
Within 2E+09 474E+0
7
Total2.4E+0
9 50
Table 8: Estimates of group means for average teacher salary
Estimates of Group MeansGroup Confidence Interval
R147223.
4 ±3616.6 95%
R246312.
5 ±3764.3 95%
R345717.
4 ±3162.6 95%
R453864.
9 ±4346.6 95%
Table 9: Tukey table for average teacher salary
Tukey test for pairwise comparison of group meansR1
r 4R2 R2
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n - r 47R3 R3
q0 3.79R4 Sig R4
T8188.7
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The four regions were next compared in terms of average teacher salary. The
average teacher salary is highest in Region 4 (Northeast) and lowest in Region 3
(South). From the ANOVA, the obtained value of F is 3.45 and the critical value of F at
= .05 is 2.80 (Table 7). The obtained value of F and the critical value of F are close,
but the null hypothesis is rejected. There is a significant difference between the regions
in terms of average teacher salary.
Levene’s test shows that p is .66, which is greater than the critical value of .05.
Because the p value is greater, we can assume homogeneity. The null hypothesis is
accepted. There is no difference between variances across groups. This meets the
underlying assumption of an ANOVA.
Tukey’s HSD indicates significant differences between Region 3 (South) and
Region 4 (Northeast) (Table 9). The South had the lowest salaries of all four regions
while the Northeast had the highest salaries.
Verbal SAT Scores:
Table 10: ANOVA data for verbal SAT scores
ANOVA Table 5%
Source SS df MS F Fcritical
p-value
Between 30986 3
10329 12.001
2.8024
0.0000
Reject
Within40450.
8 47860.6
6
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Total71436.
8 50
Table 11: Estimates of group means for verbal SAT scores
Estimates of Group MeansGroup Confidence Interval
R1528.69
2 ±16.369 95%
R2 576.5 ±17.037 95%
R3526.76
5 ±14.314 95%
R4 504 ±19.673 95%
Table 12: Tukey table for verbal SAT scores
Tukey test for pairwise comparison of group meansR1
r 4R2 Sig R2
n - r 47R3 Sig R3
q0 3.79R4 Sig R4
T 37.062
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Finally, the four regions were compared with respect to verbal SAT scores.
Verbal SAT scores were highest in the Midwest and lowest in the Northeast. From the
ANOVA, the obtained value of F is 12.00 and the critical value at = .05 is 2.80 (Table
10). Here, the obtained value of F is much higher than the critical value so the
variability between groups is greater than the variability within groups. The null
hypothesis is rejected indicating that there is a significant difference between the
regions in terms of verbal SAT scores.
However Levene’s test shows that p is .00. This is less than the critical value
of .05, so the null hypothesis is rejected. This signifies that there is a difference
between variances in the population, which does not meet the underlying assumption of
an ANOVA.
Tukey’s HSD indicates significant differences between Region 1 (West) and
Region 2 (Midwest), Region 2 and Region 3 (South), and Region 2 and Region 4
(Northeast). The verbal SAT scores in Region 2 (Midwest) are significantly higher than
any region in the country (Table 12).
The Dunnet C test shows the above differences, as well as an additional
significant difference between Region 1 (West) and Region 4 (Northeast).
Part 3: Scatterplots
Scatterplots allow the consumer to understand the relationship between
variables, however it is important to understand that correlation does not mean
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causation. One can determine if the relationship is positive, negative, or if there is a
relationship at all. Depending on how close the points on the scatterplots are to the
regression line determines the strength of the correlation. By analyzing the scatterplots
the consumer can also determine if the study is significant and the estimated
percentage of accuracy.
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Figure 1
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Figure 1 shows a negative correlation because Verbal SAT scores get smaller as
expenditure per pupil increases. The slope is slightly downward and the slope of the
line is -0.0063 which further supports the negative correlation. Figure 1 shows a
moderate relationship between verbal SAT scores and expenditure per pupils because
the points are close to the regression line between 7,000 and 11,000, but are far away
from each other as the expenditure per pupil increases. The correlation is linear
because the scatterplot is oval shaped. The results are significant because r=.42
which is above the critical values of .279 (α=.05) and .361 (α=.01). Only 17% of the
variance in verbal SAT scores can be explained by expenditure per pupil (r²=0.1726).
The regression equation is y = -0.0063x + 599.76. There doesn’t seem to be enough
difference to take action based on the results.
Figure 2
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Figure 2 has a negative correlation, which can be seen with the downward slope and
the slope of the line -0.0026. Teacher salary and verbal SAT scores have a moderate
correlation because most of the points are close to the line of best fit. The correlation is
linear because of the oval shape of the scatterplot. The results are significant because
r=.47 which is above the critical values of .279 (α=.05) and .361 (α=.01). Only 23% of
the variance in verbal SAT scores can be explained by teacher salary (r²=0.2254). The
regression equation is y = -0.0026x + 658.2. There doesn’t seem to be enough
difference to take action based on the results.
Figure 3
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Figure 3 has a zero correlation, which can be seen by the line of regression. Pupil to
teacher ratio and verbal SAT scores has a moderate correlation because most of the
points are close to the line of best fit. The correlation is linear because of the oval
shape of the scatterplot. The results are not significant because r=.0141 which is below
the critical values of .279 (α=.05) and .361 (α=.01). 0% of the variance in the verbal
SAT scores can be explained by pupil to teacher ratio (r²=0.0002). The regression
equation is y = -0.2154x + 538.22.
Based on the percentages of r² there is the most correlation between teacher salary and
verbal SAT scores. According to the scatterplots one can infer that both expenditures
per pupil and teacher salary have the largest impact on verbal SAT scores. The greatest
impact to verbal SAT scores occurs when $7,000 to $11,000 are spent on students and
$37,000 to $50,000 spent on teacher salaries. We can infer from these results that
school funding and teacher salaries are important to achievement, but more money
doesn’t guarantee better scores. More research into pupil to teacher ratio and verbal
SAT scores should be conducted because of the strength of the correlation.