To determine how well the data point (7, 90) fits the line of best fit given by the equation ŷ = 5.6x + 53.7, we need to calculate the predicted value of y (ŷ) when x = 7.
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Substitute x = 7 into the equation: \[ \hat{y} = 5.6(7) + 53.7 \]
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Calculate: \[ \hat{y} = 5.6 \times 7 = 39.2 \] \[ \hat{y} = 39.2 + 53.7 = 92.9 \]
The predicted value ŷ is 92.9.
Now, we can find the difference between the observed value and the predicted value:
- Observed value (y) = 90
- Predicted value (ŷ) = 92.9
To find the error or residual: \[ \text{Residual} = y - \hat{y} = 90 - 92.9 = -2.9 \]
Thus, the value of the residual is -2.9. This indicates that the observed value is 2.9 units below the predicted value.
So the answer is: -2.9