The scatter plot shows the number of years of experience, x, and the hourly pay rate, y, for each of 24 cashiers in California.

Question

The scatter plot shows the number of years of experience, x, and the hourly pay rate, y, for each of 24 cashiers in California.(a) Write an approximate equation of the line of best fit for the data. It doesn't have to be the exact line of best fit.(b) Using your equation from part (a), predict the hourly pay rate for a cashier with 18 years of experience.

Answer 1

I can't see your scatter plot, but, in general, this is what you do with this type of question. I've tried to go step by step to make the process clear.

You have a graph, with the x axis (horizontal) representing years of experience, and the y axis (vertical) representing the hourly pay rate. As you can't have negative years of experience, or negative pay, your plot should be only the upper right (both x and y positive) quadrant.

The question has already placed the data points (24 of them for 24 cashiers) on the graph.

Most likely, the expectation is a linear equation (the "line" of best fit) which is a straight line.

The "best fit" line is the line which passes through the most points, or has the best distribution of points above/below the line. If they're being tricksy then there might be one or two points which are "outliers" either because they're earning an unusually high or low wage, or simply because there was a mistake in the data entered (e.g. earning 10 cents, rather than $10). If there's a great fit for 22 or 23 points, but 1 or 2 are way off, you can safely ignore these points in figuring out your line.

It's meant to be a straight line, so draw it with your ruler, preferably.

Even if you're not sure, you should get some marks for actually drawing a line that goes somewhere near the points. So draw a line! :-)

Once you've drawn your line, you can use your line to find expected values, as opposed to the data samples you've been given.

Now, the question asks you to figure out the equation of this line, which I will do shortly.

However, I would quickly find 18 (years of experience) on your x axis. Then draw a line vertically from here. Where does it cross your best fit line? Draw another line horizontally from the intersection point to the y axis. Where it crosses the y axis is the related wage value. That's your answer. Write it down (using correct units of $ per hour).

This method is solving the problem graphically. I would expect you usually to still get some credit for drawing the vertical and horizontal lines showing you know how to find the wage answer. But even without credit, it shows you the approximate answer that you should arrive at.

Note that questions can be about both axes. So a question might ask you, given a wage of y dollars, find the expected experience for this cashier. In this case you find the value on the y axis first, draw a horizontal line to the best fit line, then draw a vertical down to the x axis and read the answer.

Now to the equation.

A straight line conforms to the equation y = mx + c, so you need to derive a function that looks like this.

C is the easiest bit. This is the "y intercept", where your line of best fit crosses the y axis. Try and use the graph values to write the exact value (4.2 or 8.7 or whatever your number is).

Now your equation is y = mx + 8.7.

Hopefully, you remember that m is your gradient. If your best line goes up and to the right, the gradient is positive. If the line goes down and to the right, the gradient is negative.

If it goes up, you have y = mx + 8.7. If it goes down, you have y = 8.7 - mx.

Finally, you need to figure out the gradient of your line. This is how many units it goes up (or down, if it's a negative gradient), for a number of units it goes across. This is essentially a triangle, and you should again get some credit for drawing a triangle somewhere on your line. You will have more accuracy if you have a bigger triangle (more units up) rather than a tiny one.

In maths-speak, the gradient of the line is y/x, the change in y-coordinate divided by the change in x-coordinate.

Let's say you go 12 units across, and 4 units up. So your gradient is 4/12 or 1/3.

So your final equation is y = 1/3 x + 8.7. Or, if it's a negative gradient, y = 8.7 - 1/3x.

And you have one last step to do, which is to put 18 in this equation as the value of x. 1/3x is 6 in my case. 6 + 8.7 = 14.7. So the hourly rate by equation is $14.70. (For the negative gradient 8.7 - 6 = 2.7 or $2.70 per hour.)

You can check your answer because you have the graphical approximation that you found earlier. If it's way off, you've gone wrong in either figuring out the equation of the line, or in plugging in the value of 18.

Answer 2

Note: the scatter plot is not visible.

a.) If the scatter plot follows a linear pattern, the estimated line of best fit would be in the form of y=mx+b. (Otherwise, find a function that more closely resembles the points on the graph). Find m (slope) using two points that fall on your estimated line of best fit and the equation for slope (Y2-Y1)/(X2-X1). Similarly find the y-intercept of that line by drawing where it crosses the y-axis, or by using those same points to solve for b.

b.) Using the new y=mx+b (with m and b solved for), find y, the hourly rate, when x=18 years of experience.