can you write out linear cubic and quadratic equations and fill them in with numbers from the chart

Certainly! I'll write out the equations for the linear, quadratic, and cubic regression models using the coefficients derived from the data. Let's start with the general forms of each equation:

1. **Linear Regression Equation:**
   \[
   y = mx + c
   \]
   Where:
   - \( y \) is the predicted height.
   - \( x \) is the femur length.
   - \( m \) is the slope.
   - \( c \) is the y-intercept.

2. **Quadratic Regression Equation:**
   \[
   y = ax^2 + bx + c
   \]
   Where:
   - \( y \) is the predicted height.
   - \( x \) is the femur length.
   - \( a \), \( b \), and \( c \) are coefficients.

3. **Cubic Regression Equation:**
   \[
   y = ax^3 + bx^2 + cx + d
   \]
   Where:
   - \( y \) is the predicted height.
   - \( x \) is the femur length.
   - \( a \), \( b \), \( c \), and \( d \) are coefficients.

I'll now calculate the exact coefficients using the data provided and fill in the equations.Here are the regression equations filled in with the coefficients derived from the dataHere are the regression equations filled in with the coefficients derived from the 

1. **Linear Regression Equation:**
   \[
   y = 1.7544x + 90.2881
   \]

2. **Quadratic Regression Equation:**
   \[
   y = -0.0174x^2 + 3.2821x + 56.9604
   \]

3. **Cubic Regression Equation:**
   \[
   y = 0.0062x^3 - 0.8313x^2 + 39.0011x - 462.8151
   \]

These equations can now be used to predict height based on femur length.

Would you like to delve deeper into any of these equations or have any further questions? Here are some questions you might find interesting:

1. How can we use these equations to predict heights for other femur lengths?
2. What does the sign and magnitude of the coefficients tell us about the relationship between femur length and height?
3. Why might the cubic model have more terms than the quadratic or linear models?
4. How does the cubic model improve over the quadratic and linear models?
5. In what scenarios might a simpler linear model be preferred despite lower accuracy?

**Tip:** Always check the domain of your data when using higher-order polynomial models, as they can sometimes produce unrealistic predictions outside the range of your data.

Learn how to solve linear, quadratic, and cubic regression equations using specific data points. This problem covers key mathematical concepts in regression analysis, including the formulas for linear, quadratic, and cubic regression models. Suitable for advanced high school students.

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