Given R code with fitted regression line conclude if researchers should use this formula to estimate

Question

> test = lm(GPA~SAT)
> summary(test)

Call:
lm(GPA~SAT)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.73842 -0.26146  0.01016  0.24413  1.00182 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.9293955  0.4235782  -2.194   0.0305 *  
math_SAT     0.0065119  0.0006773   9.614 5.34e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3713 on 103 degrees of freedom
Multiple R-squared:  0.473,     Adjusted R-squared:  0.4678 
F-statistic: 92.43 on 1 and 103 DF,  p-value: 5.34e-16

Consider this R code.

Basically $\hat Y = -0.9293955 + 0.0065119X$

The coefficient of determination is $0.473$.

So based on this data, the regression line and coefficient of determination, can we say that researchers should use SAT (X) scores to estimate GPA (Y). I think no but not sure how to explain it

Answer 1

You can interpret the $R^2$ value as the proportion of variance in $y$ explained by the model. Since there is only one explanatory variable, you could say that SAT scores explain about $47.3\%$ of the variance in GPA scores. Equivalently, $52.7\%$ of the variance in GPA remains, even after accounting for SAT scores in this model.

Whether this is enough to consider the model to be useful is entirely up to you. However, most fields have guidelines for minimal $R^2$ values to consider.

Suppose these GPA values were to be used in some sort of assessment, and students for which only the SAT is available would be imputed with this model. Do you think this would be a fair way to determine their GPA?

If the researchers were merely trying to determine how well SAT predicts GPA, then of course they could use this model.