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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

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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

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10

15

20

25

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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

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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.