Math and Verbal SAT Scores (Christine & Julie)
Generated Oct 10, 2011 by julierosario
We were wondering if we can find any relationship between Math and Verbal SAT Scores. The following table shows the Mathematics and Verbal SAT scores of a graduating class of high school seniors. We will investigate and prove the linear relationship between Math and verbal score in SAT.
<data1>
Below is the scatter diagram of student scores on the Mathematics and Verbal SAT scores of a graduating class of high school seniors. Verbal score is the explanatory (x) variable and Math score is the response (y) variable. The scatter diagram indicates that Math and Verbal scores are positively associated. The linear correlation between the two variables is 0.6846. Using Excel formula =sqrt(((tinv(0.05,160))^2)/(((tinv(0.05,160))^2)+160)) we find critical r is 0.154. Since 0.6846 > 0.154, we conclude that the two variables are linearly related.
<result1>
The leastsquares regression line is yhat = 209.55417 + 0.6750748x. If the verbal score in the SAT was 570 then the predicted score in Math would be 594.3468. One of the student's score in Verbal exam was 570 and the Math score was 530. Given that the predicted value when x = 570 is 594.3468, we would say that the Math score 530 was below the mean (expected score). That student underperformed in Math exam.
<result2>
The coefficient of determination is 0.4686855. So, 46.87% of the variation in Math score is explained by the Verbal score. We can say that Verbal scores contribute more information about the Math score.
The scatter plot of the residuals against the Verbal score shows no discernable pattern, so we are comfortable using a linear model.
<result3>
From the boxplot we can see that there is only one outlier. This correspond to the student who got a low score of 490 in the Verbal exam and got a high score of 800 in the Math exam.
<result4>
The slope and intercept Interpretation:
The slope of 0.6750748 means that for every 1 point increase in the Verbal score there is a corresponding increase of 0.6750748 points in the Math score on average.
The intercept is 209.55417. We cannot interpret the yintercept because the Verbal score cannot be zero. The lowest possible score in any SAT section is 200.
By: Christine G & Julie R.
Source: <A href="http://www.statcrunch.com/5.0/index.php?dataid=376245">SAT Scores</A>

Simple linear regression results:
Dependent Variable: Math Independent Variable: Verbal Math = 209.55417 + 0.6750748 Verbal Sample size: 162 R (correlation coefficient) = 0.6846 Rsq = 0.4686855 Estimate of error standard deviation: 71.754616 Parameter estimates:
Analysis of variance table for regression model:
Predicted values:
Residuals stored in new column, Residuals. 

