91Ó°ÊÓ

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 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. \(\quad X_{3}=\) Prior on-the-job rating. \(X_{2}=\) Mechanical aptitude test score. \(X_{4}=\) 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: $$ Y^{\prime}=11.6+0.4 X_{1}+0.286 X_{2}+0.112 X_{3}+0.002 X_{4} $$ a. What is the full designation of the equation? 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

01

Identify the Components of the Equation

The given equation is a linear regression model used to predict the performance on the new machine (\(Y\)) based on four independent variables: \(X_1\), \(X_2\), \(X_3\), and \(X_4\). This type of equation is known as a multiple linear regression equation.

02

Determine the Number of Variables

In the equation \(Y' = 11.6 + 0.4X_1 + 0.286X_2 + 0.112X_3 + 0.002X_4\), there is one dependent variable, \(Y\), and four independent variables: \(X_1\), \(X_2\), \(X_3\), and \(X_4\).

03

Understand the Coefficient Meaning

The number 0.286 is a coefficient of the variable \(X_2\). In a regression equation, it represents the change in the dependent variable \(Y\) for a one-unit change in \(X_2\), holding other variables constant.

04

Calculate Increase in Performance from Age

The coefficient of \(X_4\) (which represents age) is 0.002. This means if age increases by one year, the estimated performance on the new machine increases by 0.002 units.

05

Estimate Performance for Carl Knox

Substitute Carl Knox's information into the equation:- \(X_1 = 6\) (years in the business)- \(X_2 = 280\) (mechanical aptitude score)- \(X_3 = 97\) (prior job rating)- \(X_4 = 35\) (age)Substitute these into the equation:\[Y' = 11.6 + 0.4(6) + 0.286(280) + 0.112(97) + 0.002(35)\]Calculate each term:\[Y' = 11.6 + 2.4 + 80.08 + 10.864 + 0.07 = 104.994\]Therefore, Carl's estimated performance on the new machine is approximately 105.

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Key Concepts

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

Multiple Linear Regression

Multiple linear regression is a statistical technique that models the relationship between one dependent variable and multiple independent variables. Imagine you are trying to predict how well an employee will perform based on various factors. These factors, like age or previous job experience, are the independent variables. The predicted performance is the dependent variable. This method extends simple linear regression, which only uses one independent variable, by allowing you to analyze more complex situations.
In the case of Thompson Photo Works, we're using four different variables to predict an employee's performance on new machines. The equation takes the form:

• \(Y' = b_0 + b_1X_1 + b_2X_2 + b_3X_3 + b_4X_4\)

In this equation, \(Y'\) represents the predicted value, while \(b_0, b_1, b_2, \ldots \) are the regression coefficients that indicate the effect of each independent variable on \(Y'\).
This technique is useful in determining which variables have a significant impact on the outcome, helping to make informed decisions.

Dependent Variable

The dependent variable is an essential concept in regression analysis. Simply put, it's the outcome you want to predict or explain. In the context of the Thompson Photo Works exercise, the dependent variable is the performance on the new machine, denoted by \(Y\).
The main goal of the regression model is to estimate or predict this value using other information, which are the independent variables. The value of the dependent variable changes as the values of the independent variables change. This means that when any of the factors like industry experience, aptitude test scores, job rating, or age change, so does the predicted performance \(Y'\).
Understanding the dependent variable is crucial for correctly interpreting the results of a regression analysis.

Independent Variable

In multiple linear regression, independent variables are the predictors or factors that influence the dependent variable. These variables are also known as explanatory variables. They are crucial because they provide the necessary information to make predictions.
For the regression equation given in the exercise:

• \(X_1\): 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

Each of these factors can affect the predicted performance \(Y'\) when they change. The more relevant information you have as independent variables, the more accurate your predictions will tend to be. However, it's also crucial to ensure these variables are truly affecting the dependent variable and not just correlated by coincidence.

Regression Coefficient

The regression coefficient in a multiple linear regression equation is a critical value that describes the size and direction of the relationship between an independent variable and the dependent variable. It answers the question: "How much does the dependent variable change with a one-unit change in an independent variable, while keeping all other variables constant?"
Taking the equation from the exercise:

• For instance, the coefficient 0.286 before \(X_2\) (the mechanical aptitude score) indicates that for each additional point on the aptitude test, performance increases by 0.286 units, assuming other factors stay the same.
• The coefficient 0.002 for \(X_4\) (age) suggests that if age goes up by one year, the estimated performance sees a minor increase of 0.002 units.

Regression coefficients are fundamental because they quantify the impact of each independent variable on the dependent variable, providing insight into which factors are more influential in the regression model.