Question

a data set is shown in the table. the line of best fit modeling the data is ( y = 2.69x - 7.95 ). what is the residual value when ( x = 3 )?

Explanation:

Step1: Find predicted y at x=3

Substitute \( x = 3 \) into \( y = 2.69x - 7.95 \).
\( y_{\text{predicted}} = 2.69(3) - 7.95 \)
\( y_{\text{predicted}} = 8.07 - 7.95 = 0.12 \)

Step2: Calculate residual

Residual = Observed \( y \) - Predicted \( y \).
From the table, when \( x = 3 \), observed \( y = 1.0 \).
Residual = \( 1.0 - 0.12 = 0.88 \)? Wait, no—wait, residual is observed - predicted. Wait, no: Wait, the formula is residual = actual (observed) - predicted. Wait, let's recalculate:

Wait, predicted at \( x=3 \) is \( 2.69*3 -7.95 = 8.07 -7.95 = 0.12 \). Observed \( y \) when \( x=3 \) is \( 1.0 \). So residual = \( 1.0 - 0.12 = 0.88 \)? Wait, but let's check again. Wait, maybe I mixed up? Wait, no: residual is \( y_{\text{actual}} - y_{\text{predicted}} \). So \( 1.0 - 0.12 = 0.88 \). Wait, but the options include 0.88. Wait, but let's recheck the calculation:

\( 2.69 * 3 = 8.07 \). \( 8.07 - 7.95 = 0.12 \). Then actual \( y \) is 1.0. So \( 1.0 - 0.12 = 0.88 \). So the residual is 0.88.

Wait, but let me confirm the formula for residual: Residual = Observed value - Predicted value. So yes, that's correct. So when \( x=3 \), observed \( y=1.0 \), predicted \( y=0.12 \), so residual is \( 1.0 - 0.12 = 0.88 \).

Answer:

0.88