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When we talk about a linear relationship, we mean that two variables are connected in such a way that when one variable changes, the other variable changes in a consistent, straight-line manner. In this case, we are looking at how mean earnings for women working full-time change with the number of years of education. Based on the exercise:

• For 8 years of education, the earnings are \(19,190.
• For 12 years, the earnings are \)31,190.
• For 16 years, the earnings are $43,190.

When these points are plotted on a graph, we see that as education increases, earnings also increase in a consistent manner. This means we have a linear relationship. In other words, more education leads to higher earnings in a predictable way.
This kind of relationship can be very useful because it allows us to make predictions. If we know the pattern, we can estimate earnings for other educational levels not listed in the data.

Data plotting is basically taking known data points and placing them on a graph to visualize and understand the relationship between variables. In this case, the variables are the years of education (E) and the mean earnings (M) for women working full-time.
Here's how you can plot the data:

• First, plot the point (8, 19190). This means you place a point where the 8 on the x-axis meets 19190 on the y-axis.
• Next, plot the point (12, 31190).
• Finally, plot the point (16, 43190).

Always remember to label your axes correctly with the respective variable names. In this case, the x-axis is years of education (E) and the y-axis is mean earnings (M). Once you have plotted these points, you can draw a line that fits them, showing the trend in the data. This line helps us see the linear relationship between education and earnings at a glance.

To describe our linear relationship mathematically, we use the slope-intercept form of a linear equation. This is generally written as:
The slope (m) is a measure of how steep the line is. It tells us how much the mean earnings change for every additional year of education. You can calculate the slope (m) using two points from our data:
The y-intercept (c) is the value of the mean earnings when the years of education (E) are zero. This is where the line crosses the y-axis. To find the y-intercept (c), we use one of our data points and solve for c.
Putting everything together using the exercise data, our linear equation becomes:
This formula allows us to predict mean earnings for any given years of education. For example, if a woman has 14 years of education, her predicted mean earnings would be:

Always make sure your final equation makes sense by double-checking with the original data points.