I) Analyzing and interpreting data
Dr. Fuller wanted to study whether Latino children's English ability affects their learning outcomes inkindergarten.
For this study Dr. Fuller analyzed data from the Early Childhood Longitudinal Study (ECLS-K) Kindergarten Class of 1998-99, sponsored by the National Center of Education Statistics..Using a multistage probabilistic sampling design, the ECLS-K is based on a nationally representative sample of approximately 20,000 kindergarteners. Dr. Fuller analyzed a randomly selected subsample of Latino children.
The variables analyzed are the following:
1) Math test score in kindergarten: Range from 6 to 84 points. Math tests are curriculum-based assessments based on the National Assessment of Educational Progress framework of content areas and domains.
2) English ability (OLDS), whichwas measured by students' test scores on the English Oral Language Development Scale taken in the fall of kindergarten. The ECLS-K study used the OLDS to determine whether students from non-English speaking homes had the necessary minimum English skills to take cognitive assessments in English. The OLDS scores range from 0-60 points.
Given that socioeconomic status is such a strong predictor of learning outcomes, he decided to include this variable in his analyses.
3) Students' socioeconomic status (SES) isa standardized measure with a mean of 0 and astandard deviation of 1. It represents student families'socioeconomic status based on mothers' and fathers' education, mothers' and fathers'occupational status, and householdincome, measured in the fall of kindergarten and in thespring of first grade.
Dr Fuller gave you the following tables/analyses and went traveling with his son to Turkey. You are responsible for finishing the required computations and reporting the mainresults of his study. He ran several analyses, including descriptive statistics, bivariate correlations, and several regression models (shown in the next page).
Table 1:Descriptive Information
|
Variable
|
Obs
|
Mean
|
Std. Dev
|
Min
|
Max
|
|
Math score
|
2108
|
17.57009
|
7.313663
|
6.24
|
71
|
|
SES
|
2108
|
-.4145897
|
.7070371
|
-3.515
|
2.785
|
|
OLDS
|
2108
|
36.42884
|
18.71745
|
0
|
60
|
Table 2: Bivariate correlation(obs=2108)
|
|
Math score
|
SES
|
OLDS
|
|
Math score
|
1.0000
|
|
|
|
SES
|
0.4711
|
1.0000
|
|
|
OLDS
|
0.3639
|
0.4807
|
1.0000
|
Model 1: English ability effects on Math Achievement
|
Source
|
Sum of Squares
|
Df
|
Mean Square
|
|
Number of obs = 2108
F(1, 2106) = 321.37
|
|
Model
|
14921.361
|
1
|
14921.361
|
|
Prob> F = 0.0000
|
|
Residual
|
97781.3687
|
2106
|
46.4298997
|
|
R-squared = 0.1324
|
|
Total
|
112702.73
|
2107
|
53.4896676
|
|
|
|
Math Score
|
Coefficient
|
Std Err.
|
T
|
P>|t|
|
|
OLDS
|
.1421757
|
.0079309
|
17.93
|
0.000
|
|
Constant
|
12.39079
|
.3248009
|
38.15
|
0.000
|
|
|
|
|
|
|
Model 2: SES effects on Math Achievement
|
Source
|
Sum of Squares
|
Df
|
Mean Square
|
|
|
|
|
Model
|
25008.6541
|
1
|
25008.6541
|
|
|
|
|
Residual
|
87694.0755
|
2106
|
41.6401118
|
|
|
|
|
Total
|
112702.73
|
2107
|
53.4896676
|
|
|
|
|
Math Score
|
Coefficient
|
Std Err.
|
T
|
P>|t|
|
|
SES
|
4.872712
|
.1988298
|
24.51
|
0.000
|
|
Constant
|
19.59027
|
.1629372
|
120.23
|
0.000
|
Model 3: Socioeconomic Status and English ability effects on Math Achievement
|
Source
|
Sum of Squares
|
Df
|
Mean Square
|
|
Number of obs = 2108
F(2, 2105) = 344.23
|
|
|
Model
|
27776.0744
|
2
|
13888.0372
|
|
Prob> F = 0.0000
|
|
|
Residual
|
84926.6553
|
2105
|
40.3452044
|
|
R-squared = 0.2465
|
|
Total
|
112702.73
|
2107
|
53.4896676
|
|
|
|
|
|
|
|
|
|
|
|
Math Score
|
Coefficient
|
Std Err.
|
T
|
P>|t|
|
|
|
OLDS
|
.0698275
|
.0084311
|
17.85
|
0.000
|
|
|
SES
|
3.984045
|
.2231973
|
8.28
|
0.000
|
|
|
Constant
|
16.6781
|
.3864713
|
43.15
|
0.000
|
|
|
|
|
|
|
|
|
Please respond to the following questions
3) What proportion of variance is explained by Model 2?
4) Write the regression equation of Model 3 and explain each of its parts. No need to describe results yet.
5) Based on Model 3, compute the standardized slopesfor SES and English ability; interpret them, and specify what additional information is learned from the standardized slopes that cannot be learned from the unstandardized slopes.
6) Describe in one or two paragraphs the importance of using multiple regression analysis(instead of only using bivariate regression) for studying the relationship between children's language ability and math score. What additional information is learned from multiple regression that cannot be learned from the simple bivariate regression?
7) Explain in two paragraphs the main results of Model 3.