The retail sales y (in billions of dollars) of stores...

The retail sales $y$ (in billions of dollars) of stores selling auto parts, accessories, and tires in the United States from 2009 through 2013 are given by the ordered pairs of the form $(t, y(t)),$ where $t=9$ represents $2009 .$ $$\begin{aligned} &(9,74.1) \quad(10,77.7) \quad(11,82.7)\\ &(12,83.9) \quad(13,82.8) \end{aligned}$$ The coefficients of the least squares regression parabola $y=a t^{2}+b t+c$ can be found by solving the system $\left\{\begin{array}{rr}80,499 a+6985 b+615 c= & 49,853.8 \\ 6985 a+615 b+55 c= & 4436.8 \\ 615 a+\quad 55 b+\quad 5 c= & 401.2\end{array}\right.$ (a) Use Cramer's Rule to solve the system and write the least squares regression parabola for the data. (b) Use a graphing utility to graph the parabola with the data. How well does the model fit the data? (c) Is this a good model for predicting retail sales in future years? Explain.

The retail sales $y$ (in billions of dollars) of stores selling auto parts, accessories, and tires in the United States from 2009 through 2013 are given by the ordered pairs of the form $(t, y(t)),$ where $t=9$ represents $2009 .$
$$\begin{aligned}
&(9,74.1) \quad(10,77.7) \quad(11,82.7)\\
&(12,83.9) \quad(13,82.8)
\end{aligned}$$
The coefficients of the least squares regression parabola $y=a t^{2}+b t+c$ can be found by solving the system $\left\{\begin{array}{rr}80,499 a+6985 b+615 c= & 49,853.8 \\ 6985 a+615 b+55 c= & 4436.8 \\ 615 a+\quad 55 b+\quad 5 c= & 401.2\end{array}\right.$ (a) Use Cramer's Rule to solve the system and write the least squares regression parabola for the data.
(b) Use a graphing utility to graph the parabola with the data. How well does the model fit the data?
(c) Is this a good model for predicting retail sales in future years? Explain.

Key Concepts

Extrapolation and Prediction in Models

Extrapolation in modeling refers to using a model to predict values outside the range of data used to create the model. While a model might fit the existing data well, caution must be exercised when predicting future values because factors not captured by the model may influence future outcomes, potentially reducing its predictive accuracy.

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants. It provides explicit formulas for the solution, allowing for the coefficients of a regression model to be computed when the system of normal equations is set up.

Graphical Analysis of Model Fit

Graphical analysis of model fit involves plotting the model's curve along with the original data points to visually assess how well the model represents the data. This helps in identifying patterns, deviations, or outliers, and in understanding the appropriateness of the chosen model for the data.

Least Squares Regression

Least squares regression is a statistical method used to determine the line or curve that best fits a set of data points by minimizing the sum of the squares of the vertical deviations (errors) between the observed values and the values predicted by the model. It provides an objective way to estimate the unknown parameters of a model, ensuring that the overall error is minimized.

Quadratic Regression

Quadratic regression is a form of polynomial regression where the relationship between the independent variable and the dependent variable is modeled as a second-degree polynomial. This type of regression is useful when the data exhibits a curved trend, and the resulting model often takes the form y = at² + bt + c.

Transcript

00:01 According to the question, the retail sales y in billions of dollars of stores selling auto parts, accessories and tires in united states from 2009 to 2013 are given by the ordered pairs t and yt where t equals 9 represents 2009.

00:18 Now the coefficients of the list square regression parabola y equals a t squared plus b t plus c can be found by solving a system of equation.

00:26 Now, the regression parabola is given by the formula y equals a t squared plus b t plus c, where a, b, c are constants and t is a variable, where t represents the ear.

00:42 Now, we have to find the value of a .b .c using a given system of equation.

00:48 Now, we have to solve this system of equation using kramer's rule.

00:51 Now, we know that in order to solve a question using kramer's rule, at first we have to find the coefficient matrix so let del be the coefficient matrix then the terms in the coefficient matrix is nothing but the coefficients of the equation so let's take the first equation which was 849 a plus 985 b plus 615 equals 4985 3 .8 so the coefficients were 8 .499 6985 and 615 so the first row of the coefficient matrix will be composed of these three coefficients so the coefficients are 849 9, then 6985, 6985, and then we have 615.

01:38 Then for this second row we will use the coefficients of the second equation.

01:42 So they are 6985, 695, 695, 615, 615, and then we have 55, and 5.

01:50 And finally we have 615, 615, 55 and 5.

01:59 So this is our coefficient matrix which upon further simplification will just be a 5.

02:03 Give us 700.

02:05 Now in order to find a, b and c, we need to find three new determinants and they are dla, del b and dl c.

02:12 So a is given as delae upon del...