Final answer:
To build tables for linear, quadratic, and cubic functions, one must create a scatter plot, calculate and overlay a least-squares regression line, find the correlation coefficient, and use the line to calculate specific values, like the average CPI for a year. For cubic functions, the Lagrangian Form can be used.
Step-by-step explanation:
To approximate equations and build tables for the given functions (1. y=a x+b, 2. y=a x², 3. y=a x³), we must perform the following steps:
- Draw a scatter plot of the data to visualize the relationship between the variables.
- Calculate the least-squares regression line, which minimizes the sum of the squares of the residuals. The equation for this line is written as ý = a + bx.
- Overlay the least-squares line on the scatter plot to assess how well it fits the data.
- Find the correlation coefficient to determine the strength and direction of the relationship between the variables. Check if the coefficient indicates a significant relationship.
- Use the fitted line equation to calculate the average Consumer Price Index (CPI) for the year 1990, or any other value specified.
To find a cubic polynomial that fits a set of points, one can use the Lagrangian Form of the Polynomial or other numerical methods.
Discussion Questions:
- Discuss the implications of the correlation coefficient on the data set.
- Examine the fit of the regression line and consider any outliers or anomalies.