Question

1. which regression model would best fit the following scatterplots below?

Explanation:

Step1: Analyze Left Scatterplot

The left scatterplot has a U - shape, which is characteristic of a quadratic (polynomial of degree 2) relationship. A quadratic regression model ($y = ax^{2}+bx + c$, $a\gt0$) would fit this as the data points decrease to a minimum and then increase.

Step2: Analyze Middle Scatterplot

The middle scatterplot has a curve that looks like a quadratic with $a\lt0$ (a downward - opening parabola - like shape) or could also be modeled by a logarithmic or reciprocal function, but the most common model for a curve that first decreases and then levels off (or has a concave - down shape) can be a quadratic regression (or sometimes a logarithmic, but quadratic is more likely for the shape here) or a power - law regression. However, the U - like or inverted U - like shapes are well - modeled by quadratic.

Step3: Analyze Right Scatterplot

The right scatterplot shows a positive linear trend? No, wait, the points seem to have a positive association and a bit of a curve, but more likely a linear regression? Wait, no, looking again, the right scatterplot has points that seem to have a positive correlation with a bit of scatter, but actually, the left is quadratic (U - shaped), middle is maybe quadratic (inverted U - shaped) or a logarithmic, and right is linear? Wait, no, the problem is about which regression model (linear, quadratic, exponential, etc.) best fits each.

Wait, let's re - examine:

1. Left Scatterplot: The points form a U - shape, so a quadratic regression model ($y = ax^{2}+bx + c$, $a\gt0$) is appropriate.
2. Middle Scatterplot: The points form a curve that decreases and then levels off, which can be modeled by a quadratic regression model with $a\lt0$ (inverted U - shape) or a logarithmic regression, but quadratic is a common choice for such symmetric or near - symmetric curves.
3. Right Scatterplot: The points seem to have a positive linear relationship (or a slightly curved positive relationship, but linear regression is a candidate, or maybe a quadratic with a small $a$). But the key is to identify the model based on the shape.

But since the question is about which regression model (we assume we have to state for each, but the problem is a bit unclear, but likely:

Left: Quadratic (Polynomial) Regression

Middle: Quadratic (or Logarithmic, but Quadratic is more for the curve) or maybe a reciprocal regression, but Quadratic is better for the U - like/inverted U - like.

Right: Linear Regression (if the points are roughly linear) or maybe a quadratic with a small curvature.

But since the problem is about "which regression model would best fit the following scatterplots", we can say:

• Left: Quadratic Regression (because of the U - shape)
• Middle: Quadratic Regression (inverted U - shape, or a logarithmic, but quadratic is standard for such curves)
• Right: Linear Regression (if the points have a linear trend) or Quadratic (if there is a slight curve)

But maybe the intended answer is:

Left: Quadratic Regression

Middle: Quadratic Regression (or Logarithmic, but Quadratic)

Right: Linear Regression

But the problem is a bit unclear, but assuming we have to state the models:

For the left scatterplot: Quadratic Regression (due to U - shaped distribution of points)

For the middle scatterplot: Quadratic Regression (inverted U - shaped, so a quadratic model with a negative leading coefficient)

For the right scatterplot: Linear Regression (positive linear association)

Answer:

• Left Scatterplot: Quadratic Regression Model
• Middle Scatterplot: Quadratic (or Logarithmic) Regression Model (Quadratic is more likely)
• Right Scatterplot: Linear Regression Model

(Note: The problem statement is a bit unclear, but based on the shapes of the scatterplots, the left is U - shaped (quadratic), middle is curved (quadratic or logarithmic), right is roughly linear.)