Friday, July 18, 2014

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Summary
Science Score had a mean of 51.85 (SD = 9.90). Math Score had a mean of 52.65 (SD = 9.37). Read Score had a mean of 52.23 (SD = 10.25). Further, shape of histograms together box-plots suggested an approximate normal distribution for all these variables.

Among 200, 84% of schools were of Public type and 16% of schools were of Private type. In Types of Programs, 22.5% had general program, 52.50% had academic programs and 25% had vocational programs. There was a significant association prevailing between Types of programs and Types of schools.
Math and reading scores had highest coefficient of correlation in the tune of 0.662.
The prediction equation thrown by regression analyis was , Science score = 16.76 + 0.667*Math score. Overall the model was found fit as the F-statistics was significant with a p-value as 0.000. Math score was able to explain approximately 40% of total variance in science score.  
Social Studies scores had mean 52.405 and standard deviation as 10.736. There were 109 females and 91 males out of total of 200 cases. 147 cases had shown writing score less than 60, whereas 53 cases had scored equal to more than 60. Low, middle and high categories of ses were 47, 95 and 58 respectively. 
The prediction equation for predicting Science scores thrown by multiple linear regression was: Science score = 12.325 + 0.389*math score + 0.05*social studies score + 0.335*reading score -2.010*female. Predictors were able to explain approximately 49% of total variance in science score.  There was no linear relationship between female and science score found through t-test. Confidence Interval for regression coefficient of female were found between -4.0202 and 0.0002 at 95% level.
Logistic regression Model was found fit as Hosmer Lameshow Chi-square statistics was found non-significant. The Logistic regression equation was found as: Honcomp = -10.201 + 0.098*read + 0.066*science + 0.110*ses. Overall classification accuracy was found as 78.51% with 91.80% for group-0 and 41.50% for group-1.  
Other than attsc4, first 5 items measuring attitude towards schools were found in second factor. Factor analysis had thrown twelve factors. However, based on scree plot, it is recommended that we should consider only three factors. Factor loading tells us the relationship between individual items with their corresponding factors. When only two factors were extracted, all ten items measuring attitude towards school belonged to second factor. Internal consistency of variables measuring attitude towards school was found ok.  
Conclusions:
1.      Science, Math, Read, Write, Social studies were found approximately normal.
2.      Significant association between Types of programs and Types of schools was found through Chi-square test.
3.      The coefficients of correlation between science, math and reading score were found between 0.630 and 0.661.
4.      Scatter plot showed a positive relationship between science and math scores.
5.      The simple linear regression for predicting Science score with math score as independent variable had R-square as 39.80%. The model was found fit.
6.      The multiple linear regression for predicting Science score with math score, social studies score, reading score and female as independent variables had R-square as 48.90%. the model was found fit. Linear relationship between female and science score was not confirmed by T-test.    
7.      Overall classification accuracy was found as 78.51% with 91.80% for group-0 and 41.50% for group-1 through Logistic regression model for predicting honcomp. 
8.      Based on scree plot, we should consider only three factors. The internal consistency of variables measuring attitude towards school was ok.
Recommendations
1.      Science score was better predicted by Multiple regression model hence should be used for prediction.
2.      Classification accuracies were comparatively very low in group-0 against group-1, hence, alternate models should be explored.
3.      Analyst/researcher should use the factor analysis results along with his past experience and judgment for data reduction purpose.
3.1.1 Summary of science, math and read scores.
Descriptive statistics has been shown in Table 3.1.1-1: Descriptives and Histograms and Box plots are shown in Figure 3.1.1-1: Histograms and Box Plots.

Science Score had a mean of 51.85 (SD = 9.90). The median value was found as 53.23 which was almost same as that of mean. Hence, mean can be considered as a reasonable estimator of central tendency. Minimum and maximum values were found as 26 and 74. Skewness and kurtosis values were found within +,- 2 and shape of histogram together suggested an approximate normal distribution. Further, box-plots had not shown any outliers. 

Math Score had a mean of 52.65 (SD = 9.37). The median value was found as 52.00 which was almost same as that of mean. Hence, mean can be considered as a reasonable estimator of central tendency. Minimum and maximum values were found as 33 and 75. Skewness and kurtosis values were found within +,- 2 and shape of histogram together suggested an approximate normal distribution. Further, box-plots had not shown any outliers.

Read Score had a mean of 52.23 (SD = 10.25). The median value was found as 50.00 which was almost same as that of mean. Hence, mean can be considered as a reasonable estimator of central tendency. Minimum and maximum values were found as 28 and 76. Skewness and kurtosis values were found within +,- 2 and shape of histogram together suggested an approximate normal distribution. Further, box-plots had not shown any outliers.
Table 3.1.1-1
Descriptives
Descriptive Statistics
science
math
read
Mean
51.85
52.65
52.23
95% Confidence Interval for Mean
Lower Bound
50.47
51.34
50.80
Upper Bound
53.23
53.95
53.66
5% Trimmed Mean
51.96
52.39
52.14
Median
53.00
52.00
50.00
Variance
98.03
87.77
105.12
Std. Deviation
9.90
9.37
10.25
Minimum
26.00
33.00
28.00
Maximum
74.00
75.00
76.00
Range
48.00
42.00
48.00
Interquartile Range
14.00
14.00
16.00
Skewness
-0.19
0.29
0.20
Kurtosis
-0.56
-0.65
-0.62

Figure 3.1.1-1
Histograms and Box Plots


Variable↓
Histogram
Box plot


     
3.1.2 (a) Table of type of school and type of program.
Interpretation of Row and Column percentages:
Screen shot 3.1.2-1: Cross tabulation shows percentages within each cells. Overall Public schools were 84% and Private schools were 16%. Under types of program, general. Academic and vocation were 22.50%, 52,50% and 25% respectively.
Within Public type of schools, general were 19.50%, academic were 40.50% and vocation were 24%. Within Private type of schools, general were 3%, academic were 12% and vocation were only 1%.
Screen shot 3.1.2-1
Cross tabulation

3.1.2 (b) Association between type of school and type of program.
Hypotheses were designed as follows:
SPSS Output is shown in Screen shot 3.1.2-1 for Chi-square test. The chi-square statistics was found as 9.269 with 2 degrees of freedom and significance value was less than the level of significance, 5%. This indicated the rejection of null hypothesis and we can conclude that a significant association between both the types was existing. Further, the strength of association, as indicated by Phi was found as 0.215.
Screen shot 3.1.2-1:
Chi-square tests

3.1.3 Correlation between science, math and read.
Screen shot 3.1.3: Correlations shows the SPSS output for bi-variate correlation analysis. The maximum correlation was found between Math and Read in the tune of 0.662 followed by Science and Math as 0.631. Almost the same correlation was found between Science and Read as 0.630.

Screen shot 3.1.3:
Correlations
3.1.4 Scatter plot between science and math.
Screen shot 3.1.4: Scatter Plot shows the SPSS output for scatter plot between Science and Math scores. The trend line shows that there is a Positive relationship existing between two variables. Further, the coefficient of determination is 0.398 which shows that 39.80 percent of total variance in Science score was explained by Math scores.
Screen shot 3.1.4:
Scatter Plot
3.1.5 Simple Linear Regression between science and math.
Screen shot 3.1.5: Regression Output shows the SPSS output for Simple Linear Regression between Science (Response Variable) and Math scores (Explanatory Variable).
The relationship between Science and Math scores were found positive as indicated by the positive sign of regression coefficient 0.667. The linearity of relationship was further supported by the significance value of regression coefficient.  It was found less than 0.05. The coefficient of correlation was indicated a 0.631 which was same as found in previous analysis under section 3.1.3. R-square value was found as 0.395 which was reflected in scatter plot also.
The Durbin Watson Statistics was 1.878 which showed that there was no autocorrelation. Overall model was found good as F-statistics was significant (p-value was <.= 0.01). Standard Error of estimate was found as 7.702. 
Screen shot 3.1.5:
Regression Output
3.2
Descriptive Analysis
As I have already presented the Descriptive Analysis for science, math and read under 3.1.1, I am presenting for Descriptive Analysis for socst, female, honcomp and ses under the following section.
Variable: Social Studies Score (socst)
Social Studies Score had a mean of 52.405 (SD = 10.736). The median value was found as 52.41 which was almost same as that of mean. Hence, mean can be considered as a reasonable estimator of central tendency. Minimum and maximum values were found as 45 and 71. Descriptive statistics is shown in Screen shot 3.2-1.
Skewness and kurtosis values were found within +,- 2 and shape of histogram together suggested an approximate normal distribution. Further, box-plots had not shown any outliers as shown in Figure 3.2-1.
Screen shot 3.2-1
Descriptive Statistics of Social Studies Score (socst)

Figure 3.2-1
Histogram & Box Plot of Social Studies Score
Variable: Female, honcomp and ses
The frequencies of female, honcomp and ses were shown in Bar Diagrams in Figure 3.2-2. There were 91 males and 109 female respondents. 53 respondents have scored more than or equal to 60 in writing score. Rest 147 scored less than 60 in writing score. There were 47, 95 and 58 respondents under low, middle and high categories of ses variable.
Figure 3.2-2
Bar Diagrams of female, honcomp and ses

3.2.1 (a) Regression Equation and Output
The regression equation found is as follows:

Regression output is shown in Screen shots 3.2-2 and 3.2-3.

Screen shot 3.2-2
Regression Output-Coefficients
Screen shot 3.2-3
Regression Output-Model Summary and ANOVA results


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