91Ó°ÊÓ

uarez-Villa and Karlsson \(^{47}\) studied the relationship between the sales in Sweden's electronic industry and production costs (per unit value of product sales). Their data are presented in the following table. $$\begin{array}{|c|ccccc|}\hline x & 7 & 13 & 14 & 15 & 17 \\\\\hline y & 0.91 & 0.72 & 0.91 & 0.81 & 0.72 \\\\\hline x & 20 & 25 & 27 & 35 & 45 \\\\\hline y & 0.90 & 0.77 & 0.65 & 0.73 & 0.70 \\\\\hline x & 45 & 63 & 65 & 82 & \\\\\hline y & 0.78 & 0.82 & 0.92 & 0.96 & \\\\\hline\end{array}$$ Here \(x\) is product sales in millions of krona, and \(y\) is production costs per unit value of product sales. a. Use cubic regression to find the best-fitting cubic to the data and the correlation coefficient. Graph. b. Find the minimum production costs.

Step by step solution

01

Organize the Data

Separate the given data points into two lists, one for product sales (\(x\)) and one for production costs (\(y\)). These lists will be used to perform the cubic regression analysis. - \(x = [7, 13, 14, 15, 17, 20, 25, 27, 35, 45, 45, 63, 65, 82]\)- \(y = [0.91, 0.72, 0.91, 0.81, 0.72, 0.90, 0.77, 0.65, 0.73, 0.70, 0.78, 0.82, 0.92, 0.96]\)

02

Perform Cubic Regression

Using statistical software or a calculator with regression capabilities, input the data lists for \(x\) and \(y\). Select the option for cubic regression analysis. The software will compute the best-fitting cubic polynomial of the form:\[ y = ax^3 + bx^2 + cx + d \]where \(a\), \(b\), \(c\), and \(d\) are coefficients determined by the regression analysis.

03

Determine the Correlation Coefficient

The correlation coefficient, denoted by \(r\), is calculated during the cubic regression analysis. It measures the strength and direction of the relationship between the variables. The closer the correlation coefficient \(r\) is to 1 or -1, the stronger the relationship is.

04

Graph the Cubic Function

Plot the data points and the cubic function on the same graph. Use the equation obtained from the cubic regression to plot the curve, and mark the actual data points for comparison. This visual representation will help in understanding how well the cubic model fits the data.

05

Find the Minimum Production Costs

The minimum production cost corresponds to the minimum value of \(y\) you can obtain from the cubic function given by the regression. Calculate the derivative of the cubic function to find the critical points:\[ y = ax^3 + bx^2 + cx + d \rightarrow \frac{dy}{dx} = 3ax^2 + 2bx + c \]Set \( \frac{dy}{dx} = 0 \) to find critical points, and use the second derivative test or simply evaluate the function at relevant critical points to find the minimum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

The correlation coefficient is a statistical measure used to describe the strength and direction of a relationship between two variables. It is denoted by the symbol \( r \). In the context of the cubic regression problem, it provides insight into how well the cubic function models the relationship between product sales and production costs.
A correlation coefficient of +1 indicates a perfect positive linear relationship, whereas -1 indicates a perfect negative linear relationship. Values closer to 0 suggest a weaker or no linear relationship.
In practical terms:

• An \( r \) value close to 1 suggests that as sales increase, production costs generally trend in a consistent manner predicted by the model.
• An \( r \) value near 0 implies that sales and production costs have little to no predictable linear relationship.

Given its importance, calculating \( r \) in regression analysis helps verify the reliability of the modeled relationship.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world problems. In this scenario, we're employing cubic regression, which fits data to a cubic polynomial. This is of the form \( y = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants derived from the regression analysis.
The goal of modeling in this case is to accurately depict the relationship between sales in Sweden's electronic industry and the associated production costs.
Through a model:

• We gain insights into trends, whether linear or nonlinear, between two phenomena.
• We can predict future behavior based on established patterns from historical data.

By utilizing tools like regression analysis, mathematical modeling acts as a bridge between raw data and actionable insights, allowing businesses to strategize based on expected outcomes.

Data Analysis

Data analysis is the process of inspecting, cleaning, and modeling data to uncover useful information and support decision-making. It is an essential step in tackling any problem, such as understanding Sweden's electronic industry data from the given exercise.
To analyze data effectively, we:

• Initially organize data into understandable formats, such as separating sales and costs lists, helping streamline further analysis.
• Apply statistical techniques, like regression, to interpret complex relationships within data.
• Utilize visual tools such as graphs, to provide clarity and drive insights from the modeled data.

Overall, data analysis empowers us to derive meaningful insights from data, enabling informed financial and managerial decisions. With systematic data analysis, patterns emerge, highlighting crucial areas for potential improvement or optimization.