Question

In a linear regression equation of y on x, what does a slope of 2.5 indicate

• A. For every increase of 2.5 on the y-axis, there is an increase of 5.0 on the x-axis.
• B. For every increase of 2.5 on the x-axis, there is an equivalent increase on the y-axis.
• C. For every increase of 1.00 on the x-axis, there is an increase of 2.5 on the y-axis.
• D. For every increase of 1.00 on the y-axis, there is a decrease of 2.5 on the x-axis.

Hint:

Slope (\( b \)) in \( y = a + bx \) always represents the change in the dependent variable (\( y \)) for a single-unit (1.00) increase in the independent variable (\( x \)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a simple linear regression model, the relationship between the independent variable (\( x \)) and the dependent variable (\( y \)) is expressed by the equation of a straight line:
\[ y = a + bx \]
where \( a \) is the y-intercept and \( b \) is the slope (regression coefficient) of the line.

Step 2: Detailed Explanation:
The slope \( b \) represents the rate of change in the dependent variable (\( y \)) relative to changes in the independent variable (\( x \)):
\[ b = \frac{\Delta y}{\Delta x} \]
If the slope is \( b = 2.5 \), it means that for a one-unit change in \( x \) (\(\Delta x = 1\)), the corresponding change in \( y \) is:
\[ \Delta y = b \times \Delta x = 2.5 \times 1 = 2.5 \]
Since the slope value is positive, it indicates a direct positive relationship: as \( x \) increases by 1.00 unit, \( y \) increases by 2.5 units.
Therefore, Option (C) is the correct interpretation of the slope.

Step 3: Final Answer:
A slope of 2.5 indicates that for every increase of 1.00 on the x-axis, there is an increase of 2.5 on the y-axis.