Problem 4 - Basic Statistics for Business and Economics with Student

Question

The production department of Celltronics International wants to explore the relationship between the number of employees who assemble a subassembly and the number produced. As an experiment, two employees were assigned to assemble the subassemblies. They produced 15 during a one-hour period. Then four employees assembled them. They produced 25 during a one-hour period. The complete set of paired observations follows. The dependent variable is production; that is, it is assumed that the level of production depends upon the number of employees. a. Draw a scatter diagram. b. Based on the scatter diagram, does there appear to be any relationship between the number of assemblers and production? Explain. c. Compute the coefficient of correlation. d. Evaluate the strength of the relationship by computing the coefficient of determination.

Step-by-Step Solution

Answer

The scatter diagram shows a positive relationship. The correlation coefficient is close to 1, indicating a strong relationship, and the high coefficient of determination suggests most production variance is explained by the number of employees.

Step 1: Gather and Plot Data

Plot the data on a coordinate system, with the number of employees on the x-axis and the number of subassemblies produced on the y-axis. The data points given are: (2, 15) and (4, 25).

Step 2: Analyze the Scatter Diagram

Examine the plotted points. The diagram suggests a positive relationship between the number of employees and production; as the number of employees increases from 2 to 4, the production increases from 15 to 25.

Step 3: Calculate the Correlation Coefficient

Use Pearson's correlation formula: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}\]Substituting with the given data points, with x as the number of employees and y as production, compute the sums and solve for \(r\).

Step 4: Compute the Coefficient of Determination

The coefficient of determination \(r^2\) is calculated by squaring the correlation coefficient \(r\). This value indicates the proportion of the variance in the dependent variable (production) that is predictable from the independent variable (number of employees).

Key Concepts

Scatter Diagram

A scatter diagram is a graphical representation that helps visualize the relationship between two quantitative variables. In the context of production analysis, a scatter diagram can show how the number of employees affects production output.

To construct a scatter diagram, plot the pairs of data points on a Cartesian coordinate system. The x-axis represents the independent variable, in this case, the number of employees. The y-axis represents the dependent variable, which is production quantity. Each point on the graph corresponds to a pair of observations.

For the given data points of (2, 15) and (4, 25), the scatter diagram shows an upward trend. This implies a positive relationship, as an increase in the number of employees corresponds with an increase in production. Utilizing a scatter diagram can help identify the type and strength of a relationship, whether positive, negative, or none.

Coefficient of Correlation

The coefficient of correlation, denoted as \( r \), quantifies the degree to which two variables are linearly related. When dealing with variables such as the number of employees and production, this coefficient helps determine how strongly they are associated.

A value of \( r \) can range from -1 to 1, where:

• An \( r \) value of 1 indicates a perfect positive correlation.
• An \( r \) value of -1 indicates a perfect negative correlation.
• An \( r \) value of 0 suggests no correlation between the variables.

In the exercise, this value is computed using Pearson's formula, which considers the relationship between the covariation of the variables and their standardized variances. A positive value of \( r \) as seen from our earlier analysis aligns with the positive trend in the scatter diagram, affirming a constructive relationship between the number of employees and production output.

Coefficient of Determination

The coefficient of determination \( r^2 \) is an extension of the correlation coefficient. It provides a clearer indication of how well the independent variable explains the variability of the dependent variable.

Mathematically, \( r^2 \) is derived by squaring the correlation coefficient \( r \). This value represents the proportion of the variance in the dependent variable that can be predicted from the independent variable. For example, if \( r^2 \) equates to 0.64, it means 64% of the variance in production can be explained by the number of employees.

In practical terms, a higher \( r^2 \) value signals a better model fit to the data, suggesting that changes in the independent variable are a solid predictor of changes in the dependent variable. This can be crucial in production analysis for optimizing workforce management to enhance productivity.

Production Analysis

Production analysis is an essential aspect of business operations. It involves examining the effectiveness of production inputs in generating outputs. In our context, it means analyzing how varying the number of employees affects production levels.

Through techniques like correlation analysis and scatter diagrams, decision-makers can visualize relationships and make data-driven decisions. Understanding these trends allows managers to assess whether increasing the workforce will linearly increase production, as seen in the exercise.

Effective production analysis can lead to identifying the optimal number of employees needed for maximizing output while minimizing costs. This often involves balancing labor costs against revenue generated from increased production. Armed with this information, businesses can formulate strategies that enhance productivity and efficiency.