he 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.

The cubic polynomial that fits the data is:

\[
P(x) = 0.00464x^3 - 0.3827x^2 + 9.0528x + 87.6524
\]

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

Would you like further details on how the regression was performed or any clarification on the results?

Here are 5 related questions for further exploration:
1. What does each coefficient in the cubic polynomial represent in this context?
2. How well does this cubic model fit the data?
3. Can we use this model to predict coal consumption beyond 1990?
4. What other types of regression models could be applied to this dataset?
5. How would the polynomial change if we added more recent data points?

**Tip:** When fitting a polynomial to data, always check the residuals to evaluate how well the model fits.

The amount of coal used by a small town over several decades is given in the table below. Perform regression to find a cubic polynomial that fits the data. Use years since 1930 as your inputs.

This problem involves performing regression analysis to find a cubic polynomial that fits coal consumption data from 1930 to 1990. The resulting cubic equation models coal usage trends based on years since 1930. Learn how to apply polynomial regression techniques and explore the relationship between time and coal consumption. Suitable for advanced high school or college students.

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