Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used
only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use
a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several
models and compare the values of $\mathrm { R } ^ { 2 }$
-A quadratic regression model is fit to a set of sample data consisting of 6 pairs of data. Given that the sum o squares of residuals is $6.68$ and that the $\mathrm { y }$-values are $11,14,19,22,26,27$, find
$$R ^ { 2 } = 1 - \frac { \sum ( y - \hat { y } ) ^ { 2 } } { \sum ( y - \bar { y } ) ^ { 2 } }$$
A) $0.965$
B) $0.032$
C) $0.968$
D) $0.973$

Question

Quizplus · Accepted Answer

Since a quadratic regression model is fit to the data, we know that the best choice must be either a quadratic, logarithmic, or power model. We can use the coefficient of determination, $R^{2}$, to determine the best fitting model. The value of $R^{2}$ is a measure of how well the selected model explains the variation in the data. The closer the value of $R^{2}$ is to 1, the better the fit of the model to the data.

For the quadratic model, the formula for $R^{2}$ is given by:

$R^{2}=1-\frac{SS_{res}}{SS_{tot}}$

where $SS_{res}$ is the sum of squares of residuals and $SS_{tot}$ is the total sum of squares.

Substituting the given values, we get:

$R^{2}=1-\frac{6.68}{93.2}=0.928$

Since the value of $R^{2}$ is not very close to 1, we can try fitting the data to the logarithmic or power model. However, since the data contains only 6 pairs of points, we should consider using a simpler model. The quadratic model is a good choice because it is relatively simple and provides a reasonable fit to the data. Therefore, the best choice is C.

Construct a Scatterplot and Identify the Mathematical Model That Best R2\mathrm { R } ^ { 2 }R2

Construct a Scatterplot and Identify the Mathematical Model That Best $\mathrm { R } ^ { 2 }$