Problem 2 - Basic Statistics for Business and Economics with Student

Question

Thompson Photo Works purchased several new, highly sophisticated processing machines. The production department needed some guidance with respect to qualifications needed by an operator. Is age a factor? Is the length of service as an operator (in years) important? In order to explore further the factors needed to estimate performance on the new processing machines, four variables were listed: \(X_{1}\) \(=\) Length of time an employee was in the industry. \(X_{2}=\) Mechanical aptitude test score. \(X_{3}=\) Prior on-the-job rating. \(X_{3}=\) Age Performance on the new machine is designated \(Y\) Thirty employees were selected at random. Data were collected for each, and their performances on the new machines were recorded. A few results are: The equation is: a. What is this equation called? b. How many dependent variables are there? Independent variables? c. What is the number 0.286 called? d. As age increases by one year, how much does estimated performance on the new machine increase? e. Carl Knox applied for a job at Photo Works. He has been in the business for six years, and scored 280 on the mechanical aptitude test. Carl's prior on- the-job performance rating is \(97,\) and he is 35 years old. Estimate Carl's performance on the new machine.

Step-by-Step Solution

Answer

A) Multiple Regression Equation. B) 1 dependent, 4 independent variables. C) Regression Coefficient. D) Increases by 0.286 units. E) Compute using given values.

Step 1: Understanding the Equation

We are given a multiple regression equation that estimates the performance on new machines. This type of equation helps to determine how different independent variables (factors like previous experience, test scores, etc.) influence a dependent variable, which in this case is the performance on the new machine.

Step 2: Identifying Equation Components

Typically, the equation will be of the form: \[ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 \]where each \( X \) represents an independent variable, \( Y \) is the dependent variable, and the coefficients \( \beta \) represent the impact of each independent variable on \( Y \).

Step 3: Equation Classification

a. The equation given is called a **Multiple Regression Equation**. It is used to predict the value of a dependent variable based on several independent variables.

Step 4: Count Variables

b. Since the dependent variable is \( Y \) (Performance on the new machine), there is **1 dependent variable**. The four independent variables are \( X_1, X_2, X_3, \) and \( X_4 \), so there are **4 independent variables**.

Step 5: Interpreting Coefficient

c. The number 0.286 is a **Regression Coefficient**. It represents the change in the dependent variable (performance on the new machine) for a one-unit change in the corresponding independent variable (Age), holding all other variables constant.

Step 6: Change in Performance with Age

d. As age increases by one year, the estimated performance on the new machine increases by **0.286 units**.

Step 7: Computing Carl Knox's Estimated Performance

e. To estimate Carl's performance, we substitute his details into the equation:Carl has \( X_1 = 6 \) (years in industry), \( X_2 = 280 \) (aptitude test score), \( X_3 = 97 \) (on-the-job rating), \( X_4 = 35 \) (age).Substitute these into the regression equation:\[ Y = \beta_0 + \beta_1(6) + \beta_2(280) + \beta_3(97) + \beta_4(35) \]Replacing the coefficients with the appropriate values, calculate to find Carl's estimated performance.

Key Concepts

Dependent and Independent Variables

In a multiple regression analysis, understanding the distinction between dependent and independent variables is crucial. The

• Dependent Variable is what you are trying to predict or explain. In our exercise, this is the performance on the new machine, denoted by \( Y \). It depends on various factors which are introduced as independent variables.


• Independent Variables are the predictors that provide insight into changes in the dependent variable. They influence the outcome variable, that is \( Y \). In our scenario, they include:
• \( X_1 \): Length of time an employee was in the industry.
• \( X_2 \): Mechanical aptitude test score.
• \( X_3 \): Prior on-the-job rating.
• \( X_4 \): Age.

Recognizing these variables and their roles helps you comprehend how a regression model functions and predicts outcomes.

Regression Coefficient

The regression coefficient is a vital part of multiple regression analysis, representing the impact of each independent variable on the dependent variable. When we look at our equation, each

• \( \beta_0, \beta_1, \beta_2, \beta_3, \) and \( \beta_4 \) denote these coefficients.

It quantifies the change in the dependent variable for a one-unit change in the corresponding independent variable while holding other factors constant.
In the problem at hand:

• The coefficient of \( X_4 \) (age) is 0.286. This means for each additional year in age, the performance on the new machine is expected to increase by 0.286 units, assuming all other variables remain unchanged.

Understanding these coefficients is key to making sense of prediction models and their implications.

Performance Prediction

Predicting performance using a regression equation involves taking values for each independent variable and substituting them into the regression equation to estimate the dependent variable.
In this context, estimating performance on the new processing machines uses the formula:

• \[ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 \]

When Carl Knox applies for a job, his details are inserted into the equation:

• \( X_1 = 6 \): time in the industry.
• \( X_2 = 280 \): aptitude test score.
• \( X_3 = 97 \): previous job rating.
• \( X_4 = 35 \): age.

Plugging these numbers into the equation lets us estimate Carl’s future performance based on existing data, enabling data-driven decision-making.

Regression Equation Interpretation

Interpreting a regression equation enables one to understand how the dependent variable is influenced by changes in independent variables in real-world scenarios.
In our case, the regression equation is a tool for predicting machine performance. The coefficients next to each variable provide insights into:

• How well each independent variable predicts the outcome.
• Whether the relationship is positive or negative.
• The significance of each predictor in isolation and collectively.

The number 0.286 specifically tells us that for every one-year increase in age, the performance measurement expectedly increases by 0.286 units. Such interpretations allow you to gauge the practical implications and significance of different factors in a real-world context.