19

Running head: Final Project

Final Project

Elizabeth Enfield, Heather Davis, & Jodi Pitts

Seattle Pacific University

EDU 6976 –Interpreting & Applying Educational Research

Winter 2011

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.

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

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.

Figure 3 Histogram for Expenditure per Pupil

Lower Whisker

Lower Hinge

Median

Upper Hinge

Upper Whisker

5960

8639

9805

11426

14277

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 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.

Figure 5. Histogram for Teacher Salary

Lower Whisker

Lower Hinge

Median

Upper Hinge

Upper Whisker

35607

42179.5

45575

53276.5

61372

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.

Figure 7. Histogram for SAT scores (verbal)

Lower Whisker

Lower Hinge

Median

Upper Hinge

Upper Whisker

482

498

523

569

610

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.

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.

Expenditure per Pupil

Table 1: ANOVA table for expenditure per pupil

ANOVA Table

5%

Source

SS

df

MS

F

Fcritical

p-value

Between

1.2E+08

3

4E+07

9.7512

2.8024

0.0000

Reject

Within

1.9E+08

47

4E+06

Total

3.1E+08

50

Table 2: Estimate of group means for expenditure per pupil

Estimates of Group Means

Group

Confidence Interval

R1

9244.92

±

1130.5

95%

R2

9905.42

±

1176.7

95%

R3

9720.88

±

988.61

95%

R4

13601.4

±

1358.7

95%

Table 3:

Tukey test

for

expenditure

per pupil

Tukey test for pairwise comparison of group means

 

 

 

R1

 

 

 

r

4

R2

 

R2

 

n - r

47

R3

 

 

R3

 

q0

3.79

R4

Sig

Sig

 Sig

 R4

T

2559.73

 

 

 

 

 

 

 

 

 

 

 

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 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

Within

175.917

47

3.7429

Total

322.797

50

Table 5: Estimates of group means for pupil to teacher ratio

Estimates of Group Means

Group

Confidence Interval

R1

17.8077

±

1.0795

95%

R2

14.8083

±

1.1235

95%

R3

14.8647

±

0.944

95%

 R4

12.7333

±

1.2973

95%

 

 

 

 

 

Table 6: Tukey test for pupil to teacher ratio

Tukey test for pairwise comparison of group means

 

 

 

R1

 

 

 

r

4

R2

Sig

R2

 

n - r

47

R3

Sig

 

R3

 

q0

3.79

R4

Sig

 

 

 R4

T

2.44412

 

 

 

 

 

 

 

 

 

 

 

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.

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

47

4E+07

Total

2.4E+09

50

Table 8: Estimates of group means for average teacher salary

Estimates of Group Means

Group

Confidence Interval

R1

47223.4

±

3616.6

95%

R2

46312.5

±

3764.3

95%

R3

45717.4

±

3162.6

95%

 R4

53864.9

±

4346.6

95%

 

 

 

 

 

Table 9: Tukey table for average teacher salary

Tukey test for pairwise comparison of group means

 

 

 

R1

 

 

 

r

4

R2

 

R2

 

n - r

47

R3

 

 

R3

 

q0

3.79

R4

 

 

 Sig

 R4

T

8188.72

 

 

 

 

 

 

 

 

 

 

 

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

Within

40450.8

47

860.66

Total

71436.8

50

Table 11: Estimates of group means for verbal SAT scores

Estimates of Group Means

Group

Confidence Interval

R1

528.692

±

16.369

95%

R2

576.5

±

17.037

95%

R3

526.765

±

14.314

95%

R4

504

±

19.673

95%

 

 

 

 

 

Table 12: Tukey table for verbal SAT scores

Tukey test for pairwise comparison of group means

 

 

 

R1

 

 

 

r

4

R2

Sig

R2

 

n - r

47

R3

 

Sig

R3

 

q0

3.79

R4

 

Sig

 

 R4

T

37.0623

 

 

 

 

 

 

 

 

 

 

 

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 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. Figure 1

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

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

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.

Frequency Distribution of

Teacher Salary

<=35000(35000, 41000](41000, 47000](47000, 53000](53000, 59000](59000, 65000]>6500005249850

3560742179.542179.545575455754557553276.553276.56137253276.553276.542179.542179.522331332221123560761372228888.58888.51325534255341369922699221386567.586567.513356073560735607356073560761372613726137261372613722222222222356073560735607356073560761372613726137261372613722222222222

Frequency Distribution of

SAT scores (verbal)

<=450(450, 475](475, 500](500, 525](525, 550](550, 575](575, 600](600, 625](625, 650]>6500014125108200

48249849852352352356956961056956949849822331332221124826102228528513391.5391.513675.5675.5137827821348248248248248261061061061061022222222224824824824824826106106106106102222222222

Region 1 (West)Region 2 (Midwest)Region 3 (South)Region 4 (Northeast)1312179

Frequency Distribution of

Pupil to Teacher Ratio (fall 2005)

<=10(10, 12](12, 14](14, 16](16, 18](18, 20]>20041418933

10.813.5513.5514.814.814.816.516.520.816.516.513.5513.55223313322211210.820.8224.70000000000000284.7000000000000028139.12500000000000189.12500000000000181320.92499999999998620.9249999999999861325.34999999999998725.3499999999999871310.810.810.810.810.820.820.820.820.820.8222222222210.810.810.810.810.822.121.320.820.820.82222222222

Frequency Distribution of Expenditure Per Pupil

<=5000(5000, 7000](7000, 9000](9000, 11000](11000, 13000](13000, 15000](15000, 17000]>170000117197421

596086398639980598059805114261142614277114261142686398639223313322211259601427722278278134458.54458.51315606.515606.51319787197871359605960596059605960142771427714277142771427722222222225960596059605960596018339165111575914277142772222222222