Suppose a scatter plot indicates that there is a positive linear association between two quantitative variables. Which of these statements MUST be correct about the line of best fit for the data in the scatter plot?
A.
The line must have a positive x-intercept.
B.
The line must have a positive y-intercept.
C.
The line must have a positive slope.
D.
The line must have no slope.
C. The line must have a positive slope.
To determine which statement is correct about the line of best fit for a scatter plot indicating positive linear association between two quantitative variables, we need to understand the properties of such a line.
The line of best fit represents the trend or relationship between the variables in the scatter plot. It is designed to minimize the distance between the actual data points and the line itself. In the context of a positive linear association, it means that as one variable increases, the other variable also tends to increase.
With that understanding, let's evaluate the statements:
A. The line must have a positive x-intercept.
The x-intercept is the point where the line crosses the x-axis. In a scatter plot with a positive linear association, the line may or may not have a positive x-intercept. The x-intercept would be positive only if the line passes through the positive side of the x-axis. Therefore, statement A is not necessary.
B. The line must have a positive y-intercept.
The y-intercept is the point where the line crosses the y-axis. In a scatter plot with a positive linear association, the line will typically have a positive y-intercept. This is because when both variables are zero, the line will intersect the y-axis at a positive value. Therefore, statement B is likely to be correct.
C. The line must have a positive slope.
The slope of the line defines its steepness or inclination. In a scatter plot with a positive linear association, the line must have a positive slope. A positive slope indicates that as the x-variable increases, the y-variable also increases. Therefore, statement C is necessary and correct.
D. The line must have no slope.
The slope of a line represents its inclination. In a scatter plot showing a positive linear association, the line cannot have no slope since the variables are positively related. Therefore, statement D is incorrect.
Based on the analysis above, the correct statement about the line of best fit for a scatter plot indicating positive linear association is:
C. The line must have a positive slope.