Question

Suppose that the relationship between two variables \(x\) and \(y\) can be described by the regression line \(y=2.0+0.5 x.\) a. What is the change in \(y\) for a one-unit change in \(x\) ? b. Do the values of \(y\) increase or decrease as \(x\) increases? c. At what point does the line cross the \(y\) -axis? What is the name given to this value? d. If \(x=2.5,\) use the least squares equation to predict the value of \(y .\) What value would you predict if \(x=4.0 ?\)

Short answer

Answer: The coefficient of x in the regression line equation represents the change in y for a one-unit change in x. In this case, the coefficient is 0.5, meaning that for every one-unit increase in x, the value of y increases by 0.5. We can use this equation to predict future values of y by plugging in the desired values of x and calculating the corresponding values of y using the equation.

Step-by-step solution

Part a: Change in Y for a One-Unit Change in X

In the regression line equation \(y = 2.0 + 0.5x\), the coefficient of \(x\) is 0.5. This coefficient represents the change in \(y\) for a one-unit change in \(x\). Therefore, the change in \(y\) for a one-unit change in \(x\) is 0.5.

Part b: Determining if Y Values Increase or Decrease as X Increases

Since the coefficient of \(x\) is 0.5 and this value is positive, the values of \(y\) will increase as \(x\) increases.

Part c: Line Crossing the Y-Axis and Naming this Value

The equation of the regression line is \(y = 2.0 + 0.5x\). To find at which point the line crosses the \(y\)-axis, we should determine the value of \(y\) when \(x = 0\). So, \(y = 2.0 + 0.5(0) = 2.0\). The point is \((0, 2.0)\). The value of 2.0 is called the y-intercept.

Part d: Predicting Y Values for X = 2.5 and X = 4.0

To predict the value of \(y\) when \(x = 2.5\), plug this value into the equation: \(y = 2.0 + 0.5(2.5) = 2.0 + 1.25 = 3.25\) To predict the value of \(y\) when \(x = 4.0\), plug this value into the equation as well: \(y = 2.0 + 0.5(4.0) = 2.0 + 2.0 = 4.0\) So, for \(x = 2.5\), we predict a value of \(y = 3.25\) and for \(x = 4.0\), we predict a value of \(y = 4.0\).

Key concepts

y-intercept

In linear regression, the y-intercept is a very important concept. It is the point where the regression line crosses the y-axis. This happens when the value of the independent variable, often denoted as \(x\), is zero.
In our example with the equation \(y = 2.0 + 0.5x\), when \(x = 0\), the equation becomes \(y = 2.0\).
So, the y-intercept here is 2.0.

• This value indicates the starting point of the line on the y-axis.
• It represents the value of \(y\) when \(x\) is zero.

Understanding the y-intercept helps in analyzing the initial state of the dependent variable \(y\) without any influence from \(x\). It’s crucial in determining how the line is positioned on the graph.

slope

The slope is a fundamental part of understanding linear regression. It measures the steepness of the line and describes the direction and rate of change between two variables.
In the equation \(y = 2.0 + 0.5x\), the slope is represented by the coefficient of \(x\), which is 0.5.

• A positive slope, like 0.5, means that as \(x\) increases, \(y\) also increases.
• Conversely, a negative slope would mean that as \(x\) increases, \(y\) decreases.

In simple terms, the slope tells us how much \(y\) will increase for a one-unit increase in \(x\). It's vital for understanding how changes in \(x\) are expected to impact \(y\).

predictive modeling

Predictive modeling is the process of using mathematical methods to predict future outcomes based on historical data.
Linear regression, like in our equation \(y = 2.0 + 0.5x\), is a common technique used in predictive modeling to estimate the value of one variable based on the value of another.

• In our exercise, we used the regression line to predict the value of \(y\) when \(x = 2.5\) and \(x = 4.0\).
• The predicted values were \(y = 3.25\) and \(y = 4.0\), respectively.

Predictive modeling with linear regression is powerful for making informed decisions, forecasting trends, and understanding relationships between variables.

least squares method

The least squares method is a mathematical technique used to find the best-fitting line through a set of points in regression analysis.
It minimizes the sum of the squares of the differences between the observed and predicted values. This helps to produce a line that best represents the data.

• The goal is to ensure that the distance between the data points and the line of best fit is as small as possible.
• The resulting regression line is called the least squares regression line.

By using the least squares method, we gain a more accurate predictive model, which can help in analyzing data and drawing conclusions about the relationship between variables. It's a cornerstone technique in statistics and predictive modeling.