Question

A study was carried out to investigate the relationship between the hardness of molded plastic \((y\), in Brinell units) and the amount of time elapsed since termination of the molding process \((x\), in hours). Summary quantities include \(n=15\), SSResid \(=1235.470\), and SSTo = \(25,321.368\). Calculate and interpret the coefficient of determination.

Short answer

The coefficient of determination for this data set is 0.9512 or 95.12%. This means that 95.12% of the variance in the hardness of molded plastic can be explained by the elapsed time since the termination of the molding process. So, there is a very high degree of relationship between these two variables.

Step-by-step solution

Understanding the Coefficient of Determination

The Coefficient of determination, often denoted by \(R^2\), is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. It measures the strength of the relationship between the model and the dependent variable on a convenient 0 – 100% scale.

Applying the Formula for Coefficient of Determination

The formula for calculating the coefficient of determination is given by \(R^2 = 1 - (SS_{Resid} / SS_{To})\). SSResid stands for the sum of the squares of the residuals, and SSTo is the total sum of the squares.

Substituting Values into the formula

Given that SSResid = 1235.470 and SSTo = 25321.368, we substitute these values into the formula. That gives us: \(R^2 = 1 - (1235.470 / 25321.368)\)

Calculating the Coefficient of Determination

Calculating the above expression, we find that \(R^2 =0.9512\) or 95.12% when expressed as a percentage.

Interpretation of the Result

The coefficient of determination, \(R^2 = 0.9512\), means that 95.12% of the variation in the hardness of molded plastic can be explained by the elapsed time since the termination of the molding process. This indicates that there is a very strong relationship between the elapsed time since the termination of the molding process and the hardness of the molded plastic.

Key concepts

Regression Analysis

Regression analysis is a powerful statistical tool used to examine the relationship between two or more variables. It aims to determine the equation that best predicts the value of a dependent (or response) variable based on one or more independent (or predictor) variables. In our exercise, the objective was to understand how the hardness of molded plastics (\( y \text{, in Brinell units)} \) is influenced by the time elapsed since the termination of the molding process (\( x \text{, in hours)} \)).

The analysis is conducted by plotting data points on a graph and finding a line or curve that best fits those points. This 'line of best fit' is then used to predict the dependent variable's values based on new instances of the independent variable(s). The quality of the regression model is quantified using various statistical measures, such as the coefficient of determination, to determine how well the independent variable(s) explain the variation in the dependent variable.

Dependent Variable

The dependent variable is the outcome of interest, the one we are trying to predict or explain through the regression analysis. In the context of our exercise, the dependent variable is the hardness of molded plastic, measured in Brinell units. It is termed 'dependent' because its values depend on the values of the independent variable, which, in this case, is the time elapsed since the termination of the molding process.

Understanding the dependent variable is essential, as the main goal of regression analysis is to explore its patterns and how it changes in response to different independent variables. A solid grasp of the dependent variable allows researchers to form appropriate models to forecast or make sense of its behavior.

Independent Variable

In contrast to the dependent variable, the independent variable is the one we think influences or determines the behavior of the dependent variable. In our exercise, the independent variable is the amount of time elapsed since the termination of the molding process, measured in hours.

The selection of independent variables in a regression model is a critical step, as these variables are used to explain variation in the dependent variable. It's important to consider only those independent variables that are thought to have an impact on the dependent variable to avoid spurious results or overfitting the model with irrelevant data.

Variance

Variance is a statistical measure that describes the spread of data points in a data set. It tells us about the degree to which each number in the set is different from the mean and from each other. In more technical terms, it is the average of the squared differences from the Mean. High variance means that the data points are spread widely around the mean, and low variance indicates that the data points are close to the mean.

In regression analysis, the concept of variance is central; it is used to calculate several other important statistics, including the coefficient of determination. The two key variances in regression analysis are the total sum of squares (SSTo), which quantifies the total variation in the dependent variable, and the sum of the squares of the residuals (SSResid), which measures the variation that is not explained by the regression model.

Statistical Measure

A statistical measure is a value that represents a property of a sample or population and is used for analytical purposes. Examples of statistical measures include mean, median, variance, standard deviation, and the coefficient of determination (\( R^2 \)). These measures are essential for understanding data distribution, reliability of data, degree of variation, and the strength of relationships between variables.

The coefficient of determination discussed in our exercise is one such measure and is particularly important in regression analysis, as it reflects the proportion of the variance in the dependent variable that can be accounted for by the independent variable(s). A high value of the coefficient of determination indicates a strong predictive relationship between the variables.