A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line, and the computer output is shown below. Along with the paired sample data, the program was also given an [tex]$x$[/tex] value of 2 (years of study) to be used for predicting test score.

Equation:
[tex]\[ y = mx + b \][/tex]

Statistics:
- [tex]$r^2 = 0.83$[/tex]
- [tex]$r = 0.91$[/tex]

Parameters:
- [tex][tex]$m = 10.90$[/tex][/tex]
- [tex]$b = 31.55$[/tex]

Answer the following question based on the information given above:

What percentage of the total variation in test scores can be explained by the linear relationship between years of study and test scores? (1 point)



Answer :

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.