The retail sales of family clothing stores in the United...

The retail sales of family clothing stores in the United States from 2009 through 2013 are shown in the table. The coefficients of the least squares regression parabola $y=a t^{2}+b t+c,$ where $y$ represents the retail sales (in billions of dollars) and $t$ represents the year, with $t=9$ corresponding to $2009,$ can be found by solving the system $\left\{\begin{array}{rr}80,499 a+6985 b+615 c= & 56,453.6 \\ 6985 a+615 b+55 c= & 5004.4 \\ 615 a+\quad 55 b+\quad 5 c= & 450.8\end{array}\right.$ (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) Use the model to predict the retail sales of family clothing stores in the U.S. in the year 2015

The retail sales of family clothing stores in the United States from 2009 through 2013 are shown in the table.
The coefficients of the least squares regression parabola $y=a t^{2}+b t+c,$ where $y$ represents the retail sales (in billions of dollars) and $t$ represents the year, with $t=9$ corresponding to $2009,$ can be found by solving the system $\left\{\begin{array}{rr}80,499 a+6985 b+615 c= & 56,453.6 \\ 6985 a+615 b+55 c= & 5004.4 \\ 615 a+\quad 55 b+\quad 5 c= & 450.8\end{array}\right.$ (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) Use the model to predict the retail sales of family clothing stores in the U.S. in the year 2015

Key Concepts

Prediction Using Regression Models

Prediction using regression models is the process of using the established model to estimate the value of the dependent variable for given values of the independent variable. This involves extrapolating beyond the observed data range, which is a common application of regression analysis for forecasting future trends.

Graphical Evaluation of Models

Graphical evaluation involves using tools such as graphing utilities or software to plot both the original data points and the fitted regression model. This visual check helps in assessing the adequacy of the model by illustrating how well it approximates the underlying data trends and identifying any potential discrepancies.

Cramer’s Rule

Cramer’s Rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, utilizing determinants. It provides a straightforward, formulaic approach to finding the variables in the system, which is particularly useful in analytical solutions of small systems like those arising in quadratic regression.

Least Squares Regression

Least squares regression is a fundamental statistical method used to determine the best-fitting curve or line by minimizing the sum of the squares of the vertical deviations (residuals) between the observed data points and the values provided by the model. This approach is widely used in statistical estimation and modeling to capture the underlying trend of the data.

Quadratic Modeling

Quadratic modeling involves fitting a second-degree polynomial (a parabola) to the data. This model is appropriate when the relationship between the variables is curved rather than linear. The quadratic model generalizes linear regression by adding a squared term, which allows the model to capture curvature in the data.

System of Linear Equations

In regression analysis, especially when determining the coefficients for a quadratic model, a system of linear equations is generated from the conditions created by the data points and the model's parameters. Solving these simultaneous equations yields the coefficients that minimize the error between the model and the data.

Transcript

00:01 According to the question, we have been given the retail cells of family clothing stores in the united states from 2009 to 2013.

00:10 Now the coefficient of the list square regression parabola is given as y equals at squared is b t plus c.

00:16 So this is the equation of the parabola, which is y equals a t square plus b t plus c.

00:26 Now this is the equation where t is a variable and a b c are constants.

00:30 Now we don't know the value of a b and c.

00:32 So in order to find the value of a b c we have been given three system of equations now with that help of that equation we can obtain the value of a b c and finally we can put substitute those values and get the graph of the required parabola now let's solve the first part of the question so we have to find the co coefficients using kramer's rule now at first the first part of solving a problem by kramer's rule is to find the coefficient matrix so the coefficient matrix is nothing but the matrix which is formed by the coefficients of the equation so we will have 80499 then 6985 6985 and then we have 615 then for in the second row we will have 6985 695 695 6985 and 55 55 55 and 55 55 55 55 and 55 55 55 55 and 55 55 55 and 55 55 and 55 and in the third row we have 615, 615, then 55 and then we have 5.

01:50 So this is our coefficient matrix.

01:53 Then in order to find the value of a, b, c, then a will be equals del a upon del.

01:59 The value of b will be equals del of b upon del and the value of c will be equals del of c upon del.

02:08 Where del a, del b and del c are three unique determinants.

02:12 Where in case of del a, it will be a determinant in which we will replace the coefficients of the term having a by their corresponding constant terms.

02:25 So suppose for the first equation which was 849a plus 4985b plus 615c equals 56453 .6.

02:35 So the coefficient of a was 80499.

02:38 So now in the determinant we will replace it by the constant term of the equation which is 56453 .6 so the del a matrix will sorry determinant will become 5653 .6 56453 .6 504 .4 .550 .5 .50 .5 .50.

03:04 And the coefficients of b and c will remain as it is.

03:11 So here we have 6985 -615 and 55 here we have 615 and 5 so this is our daily determinant now let's find del be determined now for del be determinant we are going to replace the coefficients of the terms having b by day or corresponding constant terms so here we have 8496985 and 695 and 615 now we are going to replace the coefficients of b so this will become 56 453 .6 50 4 .4 .4 and 450 .8 and here we are 615 55 and 5...

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