The percentage of the total variation in test scores that can be explained by the linear relationship between years of study and test scores is found using the coefficient of determination, denoted as [tex]\( r^2 \)[/tex].
The coefficient of determination ([tex]\( r^2 \)[/tex]) represents the proportion of the variance in the dependent variable (test scores) that is predictable from the independent variable (years of study). In this case, [tex]\( r^2 = 0.83 \)[/tex].
To convert this proportion into a percentage, you simply multiply [tex]\( r^2 \)[/tex] by 100:
[tex]\[ \text{Percentage explained} = r^2 \times 100 \][/tex]
Substituting the given value of [tex]\( r^2 \)[/tex]:
[tex]\[ \text{Percentage explained} = 0.83 \times 100 = 83.0 \][/tex]
Therefore, 83.0% of the total variation in test scores can be explained by the linear relationship between years of study and test scores.
