The amount of coal used by a small town over several decades is given in the table below.



YearTons of
Coal

1930

87.4

1940

144.9

1950

153

1960

140.5

1970

131.6

1980

166.1

1990

254.2

**a)**   Perform regression to find a cubic polynomial that fits the data. Use **years since 1930** as your inputs.

To find the cubic polynomial that best fits the data, we perform a cubic regression on the provided data points. The general form of a cubic polynomial is:

\[
P(x) = ax^3 + bx^2 + cx + d
\]

Where:
- \(x\) is the number of years since 1930 (i.e., \( x = \text{year} - 1930 \)).
- The output is the tons of coal used.

### Data preparation:

We will convert the years into \(x\) values, where:
\[
x = \text{Year} - 1930
\]
So the dataset becomes:

| Year | Tons of Coal | \(x = \text{Year} - 1930\) |
|------|--------------|-------------------------|
| 1930 | 87.4         | 0                       |
| 1940 | 144.9        | 10                      |
| 1950 | 153          | 20                      |
| 1960 | 140.5        | 30                      |
| 1970 | 131.6        | 40                      |
| 1980 | 166.1        | 50                      |
| 1990 | 254.2        | 60                      |

Now, I'll perform the cubic regression to find the coefficients \(a\), \(b\), \(c\), and \(d\). Let me calculate that.The cubic polynomial that best fits the data has the following form:

\[
P(x) = 0.00464x^3 - 0.3827x^2 + 9.053x + 87.65
\]

Where \(x\) is the number of years since 1930.

### Interpretation of coefficients:
- \(0.00464x^3\): Accounts for the cubic change in coal usage over time.
- \(-0.3827x^2\): Describes the quadratic variation, indicating a decrease in growth rate at some points.
- \(9.053x\): Reflects the linear growth in coal usage over the years.
- \(87.65\): The initial amount of coal in 1930.

Would you like more details or have any questions?

Here are some related questions:
1. How is a cubic polynomial regression different from linear regression?
2. What does each term in a cubic regression represent practically?
3. How can you use this cubic polynomial to predict future coal usage?
4. How would the model change if there were more data points?
5. What would happen if we tried fitting a quadratic or quartic polynomial instead?

**Tip**: When performing polynomial regression, always consider the degree carefully—higher degrees can lead to overfitting, especially with limited data.

Perform regression to find a cubic polynomial that fits the data. Use years since 1930 as your inputs.

Learn how to perform a cubic polynomial regression to model coal usage in a small town over several decades. This problem uses data from 1930 to 1990, converting years since 1930 into inputs and fitting a cubic polynomial. Key concepts include polynomial regression and the least squares method.

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