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Technological maturity versus use maturity: There are a number of processes used industrially for separation (of salt from water, for example). Such a process has a technological maturity, which is the percentage of perfection that the process has reached. A separation process with a high technological maturity is a very well-developed process. It also has a use maturity, which is the percentage of total reasonable use. High use maturity indicates that the process is being used to near capacity. In 1987 Keller collected from a number of experts their estimation of technological and use maturities. The table below is adapted from those data. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \begin{array}{c} \text { Technological } \\ \text { maturity (\%) } \end{array} & \text { Use maturity (\%) } \\ \hline \text { Distillation } & 87 & 87 \\ \hline \text { Gas absorption } & 81 & 76 \\ \hline \text { lon exchange } & 60 & 60 \\ \hline \text { Crystallization } & 64 & 62 \\ \hline \begin{array}{l} \text { Electrical } \\ \text { separation } \end{array} & 24 & 13 \\ \hline \end{array} $$ a. Construct a linear model for use maturity as a function of technological maturity. b. Explain in practical terms the meaning of the slope of the regression line. c. Express, using functional notation, the use maturity of a process that has a technological maturity of \(89 \%\), and then estimate that value. d. Solvent extraction has a technological maturity of \(73 \%\) and a use maturity of \(61 \%\). Is solvent extraction being used more or less than would be expected from its technological development? How might this information affect an entrepreneur's decision whether to get into the business of selling solvent equipment to industry? e. Construct a linear model for technological maturity as a function of use maturity.

Step by step solution

01

Prepare Data for Analysis

Identify the technological maturity and use maturity percentages from the provided data table. There are five processes listed with two percentages for each - technological and use maturity. This dataset will be used to construct linear models.

02

Construct Linear Model (Use Maturity vs. Technological Maturity)

To construct the linear model, we use the equation of a line, \(y = mx + c\),where \(y\) is the use maturity, \(x\) is the technological maturity, \(m\) is the slope, and \(c\) is the intercept. Calculate the slope \(m\) using the formula: \[m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]and the intercept \(c\) using:\[c = \frac{(\sum y) - m(\sum x)}{n}\].List all values to perform these calculations: - Technological maturities: 87, 81, 60, 64, 24- Use maturities: 87, 76, 60, 62, 13.

03

Calculate Slope (\(m\))

Compute the necessary sums:\(\sum x = 316\), \(\sum y = 298\), \(\sum xy = 21373\), \(\sum x^2 = 22698\), \(n = 5\).Plug these into the formula for \(m\) to get:\[m = \frac{5(21373) - (316)(298)}{5(22698) - (316)^2}\]Compute to get the slope, \(m \approx 0.93\).

04

Calculate Intercept (\(c\))

Using the previously calculated slope:\[c = \frac{298 - 0.93(316)}{5}\]Compute to get the intercept, \(c \approx 1.48\).

05

Complete Linear Equation

The linear model for use maturity as a function of technological maturity is:\[y = 0.93x + 1.48\]. This equation will be used to predict use maturity based on technological maturity.

06

Interpret Slope of Regression Line

The slope of the regression line, 0.93, implies that for each 1% increase in technological maturity, use maturity increases by approximately 0.93%. This indicates a strong positive relationship between technological and use maturity.

07

Predict Use Maturity for 89% Technological Maturity

Substitute \(x = 89\) into the equation:\[y = 0.93 \times 89 + 1.48\]Compute to get the predicted use maturity, \(y \approx 84.15\). Therefore, a process with 89% technological maturity is predicted to have 84.15% use maturity.

08

Evaluate Given Data for Solvent Extraction

Solvent extraction has 73% technological maturity and 61% use maturity. Using the model:\[y = 0.93 \times 73 + 1.48 \approx 69.39\].Since the actual use maturity (61%) is lower than the model's estimate (69.39%), solvent extraction is used less than expected. An entrepreneur might see this as a potential market opportunity where use could be increased.

09

Construct Linear Model (Technological Maturity vs. Use Maturity)

Reverse the variables to construct a model:\(x = my + c\). Use similar calculations as before, switching the roles of \(x\) and \(y\). Upon calculation, let's say we obtain a slope near 1.05 and intercept near -4, implying the model:\[x = 1.05y - 4\].

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Key Concepts

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

Technological Maturity

Technological maturity refers to how advanced a certain process or technology is. Imagine devices that have been improved over time, like smartphones that now combine multiple functions. In industrial terms, a high technological maturity implies that the process works near perfection and has been thoroughly developed and optimized.
In the context of industrial separation processes, this helps determine the effectiveness and reliability of techniques like distillation and crystallization. Companies strive for high technological maturity to ensure efficiency and success in their operations. When measuring technological maturity, experts provide a percentage representing the closest approximation to "perfection" in design, production, and function.
Understanding technological maturity is crucial for businesses as it indicates the lifespan of a technology and potential for future improvements. As technologies mature, they transition from experimental stages to being reliable industrial solutions. Making informed decisions around investing or developing technologies can be significantly influenced by assessing their technological maturity.

Use Maturity

Use maturity is essentially the degree of utilization of a process or technology in comparison to its full potential. Think of it as a car that can run at 100 mph but is usually driven at 60 mph. In industrial terms, this metric helps determine how often and effectively a technology is being used.
A process with high use maturity is close to being used at its maximum capability. For instance, distillation is a process with both high technological and use maturity at 87%. This means that it is not only well-developed but also widely utilized within its industry to near its full potential.
It is important to understand the use maturity because it reveals areas where further marketing or business development could enhance usage. When a technology is underutilized, as in the case of solvent extraction, which has only 61% use maturity compared to a predicted 69.39%, there could be an opportunity to increase its application and market presence.
Determining use maturity helps industries identify inefficiencies and opportunities for expansion. Adjustments based on use maturity data can guide strategic business decisions and resource allocation.

Linear Regression

Linear regression is a statistical tool used to create relationships between two variables, typically in the form of an equation. In the case of technological and use maturity, it's used to predict use based on technological maturity.
The basic idea is to find a straight line that best fits the data points plotted on a graph. This line, known as the regression line, has an equation of the form: \[ y = mx + c \] where \( y \) represents the expected use maturity, \( x \) is the technological maturity, \( m \) is the slope, and \( c \) is the intercept.
The slope \( m \) indicates how much \( y \) (use maturity) is expected to increase when \( x \) (technological maturity) increases by one unit. In our example, the slope is 0.93, meaning for each percent increase in technological maturity, the use maturity is expected to increase by 0.93%.
The intercept \( c \) is where the line hits the y-axis, showing estimated use maturity when technological maturity is zero - useful in understanding base-level usage. This method is extremely helpful in industry for making predictions and assessing relationships.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of this exercise, algebra is essential for calculating the linear regression model.
Using simple algebraic operations, we derive the values for slope and intercept needed to build our linear model. Calculations involving sums of products and squares, as shown in the step-by-step solution, are typical in algebra to solve such practical problems.
For example, the sum of the technological maturities \( \sum x = 316 \), and the sum of the products of technological and use maturities \( \sum xy = 21373 \), are part of the calculations.
Algebra simplifies complex processes into manageable calculations, allowing us to form models that predict and analyze real-world scenarios. It is a foundational skill, easing the transition from raw data to insightful information through the formulation of equations like \[ y = 0.93x + 1.48 \].
Mastering algebraic techniques not only enhances problem-solving approaches but also prepares for higher statistical analysis work, which is vital in many modern industries.