To determine whether the model $y = 4x - 5$ is a good fit for the data, we can use residuals. A residual is the difference between the observed value (from the table) and the predicted value (from the model).

Steps:

1. Calculate the predicted values using the model $y = 4x - 5$ for each $x$ value.
2. Find the residuals by subtracting the predicted value from the observed value.
3. Analyze the residuals to determine if they are randomly scattered (good fit) or show a pattern (poor fit).

Here are the calculations:

$x$	Observed $y$	Predicted $y = 4x - 5$	Residual (Observed $y$ - Predicted $y$)
-4	-18	$4(-4) - 5 = -16 - 5 = -21$	$-18 - (-21) = 3$
-3	-13	$4(-3) - 5 = -12 - 5 = -17$	$-13 - (-17) = 4$
-2	-10	$4(-2) - 5 = -8 - 5 = -13$	$-10 - (-13) = 3$
-1	-7	$4(-1) - 5 = -4 - 5 = -9$	$-7 - (-9) = 2$
0	-2	$4(0) - 5 = 0 - 5 = -5$	$-2 - (-5) = 3$
1	0	$4(1) - 5 = 4 - 5 = -1$	$0 - (-1) = 1$
2	6	$4(2) - 5 = 8 - 5 = 3$	$6 - 3 = 3$
3	10	$4(3) - 5 = 12 - 5 = 7$	$10 - 7 = 3$
4	15	$4(4) - 5 = 16 - 5 = 11$	$15 - 11 = 4$

Residual Analysis:

• The residuals are: 3, 4, 3, 2, 3, 1, 3, 3, 4.
• Since the residuals do not consistently increase or decrease and seem relatively small (mostly around 3), the model does a decent job of predicting the values. However, the residuals are not exactly zero, which means the model is not a perfect fit.
• The residuals show some fluctuation but no strong pattern like a curve or increasing trend, suggesting that the linear model is reasonable, though there might be slight room for improvement.

Conclusion:

The residuals suggest that the model $y = 4x - 5$ is a fairly good fit for the data, though there are some small deviations. The linearity of the model seems appropriate.

Do you want more details on any of the steps? Here are 5 related questions:

1. How do you interpret residuals to assess model fit?
2. Can we graph the residuals to visualize their pattern?
3. What are some signs that suggest a poor model fit based on residuals?
4. How do you handle larger residuals in a data set?
5. Would a different type of model (like quadratic) fit this data better?

Tip: Residual plots are a great way to visually check the fit of a model, especially when looking for non-random patterns.