Question

Use exponential regression to find an exponential function that best fits this data.

f(x) =

Use linear regression to find an linear function that best fits this data.

g(x) =

Of these two, which equation best fits the data?

-Exponential

-Linear

image of table attached

Answer 1

`f(x)=610 \cdot 1.01^x`

`g(x) = 5.4x+611`

The linear function best fits the data.

Explanation:

Exponential Regression

Exponential regression is a statistical technique that models the relationship between a dependent variable (x) and an independent variable (y) using the exponential function y = a · b^x.

Entering the data from the table into a regression calculator gives:

• a = 610.0819384856
• b = 1.0090645306
• R² = 0.1310079944

Therefore, the exponential function that best fits the data is:

`f(x)=610 \cdot 1.01^x`

where each value is rounded to three significant figures.

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable (y) and an independent variable (x) using the linear function y = ax + b.

Entering the data from the table into a regression calculator gives:

• a = 5.4
• b = 611.2666666667
• R² = 0.1361223492

Therefore, the linear function that best fits the data is:

`g(x) = 5.4x+611`

where each value is rounded to three significant figures.

Best Fit

The coefficient of determination (R²) is a statistical measure used to assess the goodness of fit of a regression model. It is expressed as a value between 0 and 1, where a greater R² value indicates a better fit of the regression model to the data.

As the R² value of the linear function is greater than that of the exponential function, the equation that best fits the data is the linear function.

`\hrulefill`

Additional Notes

The function that actually best fits the data is a quadratic function:

`y=-8.696x^2+66.275x+530.1`

The R² value for this function is 0.8892741234, which is considerably closer to 1 than the R² values for the exponential and linear functions.