The table asks you to calculate residuals for the given values based on the model $y = 4x - 5$.

Here's the process:

1. Calculate the y-value from the model:
   The formula is $y = 4x - 5$.
   For each $x$, calculate $y_{\text{model}}$.

2. Find the residual:
   The residual is $y_{\text{observed}} - y_{\text{model}}$.
   For each value of $x$, subtract the calculated $y_{\text{model}}$ from the observed $y$.

Let’s go through it step by step:

1. First row (already filled out)

• $x = -4$:
  $y_{\text{model}} = 4(-4) - 5 = -16 - 5 = -21$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = -18 - (-21) = 3$.

2. Second row (partially filled out)

• $x = -3$:
  $y_{\text{model}} = 4(-3) - 5 = -12 - 5 = -17$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = -13 - (-17) = 4$.

3. Third row

• $x = -2$:
  $y_{\text{model}} = 4(-2) - 5 = -8 - 5 = -13$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = -10 - (-13) = 3$.

4. Fourth row

• $x = -1$:
  $y_{\text{model}} = 4(-1) - 5 = -4 - 5 = -9$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = -7 - (-9) = 2$.

5. Fifth row

• $x = 0$:
  $y_{\text{model}} = 4(0) - 5 = 0 - 5 = -5$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = -2 - (-5) = 3$.

6. Sixth row

• $x = 1$:
  $y_{\text{model}} = 4(1) - 5 = 4 - 5 = -1$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = 0 - (-1) = 1$.

7. Seventh row

• $x = 2$:
  $y_{\text{model}} = 4(2) - 5 = 8 - 5 = 3$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = 6 - 3 = 3$.

8. Eighth row

• $x = 4$:
  $y_{\text{model}} = 4(4) - 5 = 16 - 5 = 11$.
  Residual: $y_{\text{observed}} - y_{\text{model}} = 15 - 11 = 4$.

Completed Table:

Would you like more details on the residuals, or do you have any questions?

Here are some related questions:

1. How do residuals help measure the accuracy of a model?
2. Can residuals be negative, and what does that indicate?
3. What would the residual be if the observed $y$ matches the predicted $y_{\text{model}}$?
4. How can you tell if a model is a good fit based on residuals?
5. How do you calculate residuals for non-linear models?

Tip: When residuals are close to zero, it indicates the model predicts the data well.