91Ó°ÊÓ

Economies of Scale in Plant Size Strategic Management \(^{8}\) relates a study in economies of scale in the machine tool industry. The data is found in the following table. \begin{tabular}{|c|ccccccc|} \hline\(x\) & 70 & 115 & 130 & 190 & 195 & 400 & 450 \\ \hline\(y\) & 1.1 & 1.0 & 0.85 & 0.75 & 0.85 & 0.67 & 0.50 \\ \hline \end{tabular} Here \(x\) is the plant capacity in thousands of units, and \(y\) is the employee- hours per unit. a. Determine the best-fitting line using least squares and the correlation coefficient. b. Is there an advantage in having a large plant? Explain. c. What does this model predict the employee-hours per unit will be when the plant capacity is 300,000 units? d. What does this model predict the plant capacity will be if the employee- hours per unit is \(0.90 ?\)

Step by step solution

01

Organize the Data

Our data set includes seven pairs of values, where \(x\) represents the plant capacity in thousands of units, and \(y\) represents employee-hours per unit. We have the following data points:\[(x, y) = (70, 1.1),(115, 1.0),(130, 0.85),(190, 0.75),(195, 0.85),(400, 0.67),(450, 0.50)\]

02

Calculate the Slope (m) and Intercept (b) Using Least Squares

To find the best-fitting line, we use the formulas for slope \( m \) and intercept \( b \):\[m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\]\[b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{n(\Sigma x^2) - (\Sigma x)^2}\]Calculating the sums:\(\Sigma x = 1550\), \(\Sigma y = 5.72\), \(\Sigma xy = 896.45\), \(\Sigma x^2 = 520650\)With \( n = 7 \) we substitute these into the formulas to find \( m \approx -0.0012 \) and \( b \approx 1.1153 \). The equation of the line is:\[ y = -0.0012x + 1.1153 \]

03

Calculate the Correlation Coefficient (r)

The correlation coefficient \( r \) is a measure of the strength and direction of the linear relationship between \( x \) and \( y \). It is given by:\[r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{ \sqrt{ [n(\Sigma x^2) - (\Sigma x)^2] [n(\Sigma y^2) - (\Sigma y)^2] }}\]Plugging in our previously computed sums and calculating, we find: \( r \approx -0.81 \).This indicates a strong negative linear relationship.

04

Interpret the Line for Large Plant Capacity

Having calculated the slope and correlation coefficient, the strong negative slope and correlation suggest that larger plant capacities require less employee-hours per unit. Yes, there is an advantage in having a large plant as it improves efficiency, lowering the employee-hours required.

05

Predict Employee-Hours for 300,000 Units Capacity

We substitute \( x = 300 \) into our linear equation:\[ y = -0.0012(300) + 1.1153 \]Simplifying gives \( y \approx 0.7553 \). So, the model predicts about 0.7553 employee-hours per unit for a plant capacity of 300,000 units.

06

Predict Plant Capacity for 0.90 Employee-Hours

Letting \( y = 0.90 \), we solve for \( x \):\[ 0.90 = -0.0012x + 1.1153 \]\[ x \approx \frac{1.1153 - 0.90}{0.0012} \]\[ x \approx 179.42 \]Thus, a plant capacity of approximately 179,420 units is predicted to require 0.90 employee-hours per unit.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Plant Capacity

Plant capacity is an important factor that refers to the maximum output that an industrial plant can produce under normal conditions in a given period. Measuring it in thousands of units, as in the example, is common, as it provides a clear understanding of production capabilities.
When considering economies of scale, plant capacity becomes crucial. With larger capacities, companies can reduce the employee-hours per unit, enhancing efficiency. This means that as the production capacity increases, each unit consumes less work from employees, thus lowering costs. Understanding and optimizing plant capacity can significantly impact a business's profitability and competitiveness.

Least Squares Method

The least squares method is a statistical tool used to find the line of best fit for a set of data points. It's based on minimizing the sum of the squared differences between observed and predicted values.
This method is widely used in linear regression analysis. The formula for the slope ( m ) and intercept ( b ) of the best-fit line are derived from this technique:

• The slope ( m ) measures the change in y for a one-unit change in x .
• The intercept ( b ) is the expected value of y when x is zero.

By applying these formulas, one can easily find how different variables correlate with each other, as seen in the original exercise.

Correlation Coefficient

The correlation coefficient, denoted as r , quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1:

• If r is closer to 1 or -1, the relationship is strong.
• A positive r indicates the variables move in the same direction, while a negative r means they move in opposite directions.
• An r close to 0 suggests a weak or no linear relationship.

In our example, r was approximately -0.81, indicating a strong negative correlation. This implies that as plant capacity increases, the employee-hours per unit decrease significantly.

Linear Regression

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables using a linear equation. It predicts the value of the dependent variable based on the values of the independent variables.
When dealing with one independent variable, the relationship is described by the equation: y = mx + b , where y is the predicted variable, x is the independent variable, m is the slope, and b is the intercept.
In the machine tool industry example, we use linear regression to model how employee-hours per unit changes with plant capacity. This approach allows us to make informed predictions, like estimating the employee-hours required for a given capacity or determining the necessary capacity for a specific efficiency level.