To determine the residual and its interpretation, we need to follow a step-by-step process.
First, we will calculate the predicted science score ([tex]\(\hat{y}\)[/tex]) using the given line of best fit equation and the student's IQ score.
The given line of best fit is:
[tex]\[ \hat{y} = -20.3 + 0.7489s \][/tex]
Here, [tex]\( s \)[/tex] is the student's IQ score, which is 100.
Plugging [tex]\( s = 100 \)[/tex] into the equation, we get:
[tex]\[ \hat{y} = -20.3 + 0.7489 \times 100 \][/tex]
Calculating this gives:
[tex]\[ \hat{y} = -20.3 + 74.89 \][/tex]
[tex]\[ \hat{y} = 54.59 \][/tex]
So, the predicted science test score for the student is 54.59.
Next, we calculate the residual. The residual is defined as the difference between the actual score and the predicted score:
[tex]\[ \text{Residual} = \text{Actual score} - \text{Predicted score} \][/tex]
Given, the actual science test score for the student is 52.6.
Therefore, we calculate the residual as:
[tex]\[ \text{Residual} = 52.6 - 54.59 \][/tex]
[tex]\[ \text{Residual} = -1.99 \][/tex]
The residual is -1.99.
To interpret the residual, we examine its sign. Since the residual is negative ([tex]\(-1.99\)[/tex]), it means the actual score is less than the predicted score. Thus, the line of best fit overpredicts the student's science test score.
Given this information, the correct interpretation is:
-1.99; The line of best fit overpredicts the student's science test score.
