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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.

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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.

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The least-squares regression line is y-hat = 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.

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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.

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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.

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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 y-intercept 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>

Result 1: Scatter Plot Verbal and Math score   [Info]

Result 2: Simple Linear Regression SAT   [Info]
Simple linear regression results:
Dependent Variable: Math
Independent Variable: Verbal
Math = 209.55417 + 0.6750748 Verbal
Sample size: 162
R (correlation coefficient) = 0.6846
R-sq = 0.4686855
Estimate of error standard deviation: 71.754616

Parameter estimates:
 Parameter Estimate Std. Err. Alternative DF T-Stat P-Value Intercept 209.55417 34.349354 ≠ 0 160 6.1006727 <0.0001 Slope 0.6750748 0.05682337 ≠ 0 160 11.880232 <0.0001

Analysis of variance table for regression model:
 Source DF SS MS F-stat P-value Model 1 726690.5 726690.5 141.13991 <0.0001 Error 160 823795.94 5148.7246 Total 161 1550486.4

Predicted values:
 X value Pred. Y s.e.(Pred. y) 95% C.I. for mean 95% P.I. for new 570 594.3468 5.832239 (582.8287, 605.86487) (452.17117, 736.5224)

Residuals stored in new column, Residuals.

Result 3: Scatter Plot Residual vs Verbal   [Info]

Result 4: Boxplot Residual   [Info]