Question

a data set is shown in the table. the line of best fit modeling the data is y = 2.69x - 7.95.

x	y
2	-3.2
3	1.0
4	2.3
5	5.6

what is the residual value when x = 3?

• -0.88
• -0.12
• 0.12
• 0.88

Explanation:

Step1: Recall residual formula

Residual = Observed \( y \) - Predicted \( y \).

Step2: Find predicted \( y \) at \( x = 3 \)

Use the line of best fit \( y = 2.69x - 7.95 \). Substitute \( x = 3 \):
\( y_{\text{predicted}} = 2.69(3) - 7.95 \)
\( y_{\text{predicted}} = 8.07 - 7.95 = 0.12 \).

Step3: Find observed \( y \) at \( x = 3 \)

From the table, when \( x = 3 \), observed \( y = 1.0 \).

Step4: Calculate residual

Residual = \( 1.0 - 0.12 = 0.88 \)? Wait, no—wait, residual is observed - predicted. Wait, observed \( y = 1.0 \), predicted \( y = 0.12 \). So residual = \( 1.0 - 0.12 = 0.88 \)? Wait, no, wait the options: wait maybe I mixed up. Wait, no, let's recheck. Wait, line of best fit: \( y = 2.69x - 7.95 \). At \( x = 3 \), \( 2.69*3 = 8.07 \), 8.07 - 7.95 = 0.12. Observed \( y \) is 1.0. So residual = observed - predicted = \( 1.0 - 0.12 = 0.88 \)? But wait the options have 0.88 as an option. Wait, but let me check again. Wait, maybe I made a mistake. Wait, the formula for residual is \( e_i = y_i - \hat{y}_i \), where \( y_i \) is observed, \( \hat{y}_i \) is predicted. So \( 1.0 - 0.12 = 0.88 \). So the residual is 0.88.

Answer:

0.88