Regression equation?

Question

a production manager has collected data on job times. She wants to learn if there is any relationship between the production time per piece and the number of pieces in a job. The following data were collected from shop records.
Job Time # of Pieces
1 3.1 min 15
2 1.1 min 75
3 2.2. min 45
4 1.6 min 62
5 2.5 min 40
6 1.5 min 70
7 4.2 min 12
8 1.8 min 50
9 1.0 min 90
10 3.4 min 18
11 2.8 min 32

Which variable is the independent variable?
Develop a regression equation for the relationship. What does the slope imply?

Answer 1

We assume y = mx + b is the general linear equation for your set of data. In this and similar equations of the form y = f(x), the y is always the dependent variable because it depends on x. x, then, has to be the independent variable doesn't it?

We call x "independent" because its value does NOT depend on anything else in the equation. In fact, we can assign x any value and then y, which depends on x, has to be a very specific value based on the other parameters in the equation.

So your case wants to know if "production time per piece" depends on "the number of pieces in a job." If time depends on (the key words here) number of pieces, which is time...dependent or independent? Presuming you said "dependent" then what does that make number of pieces?

OK now...you need to use either the manual equations for solving for m and b in the general equation or you could do it like real scientists and engineers do it and use a regression tool. MS Excel has an excellent regression function called LINEST, which comes with an excellent tutorial on how to use it. With it you can easily solve for m, the slope, and b, the intercept, in coming up with a specific equation for your data.

LINEST has the form =LINEST(y's,x's,,) where the y data are the dependent data points and the x data are the independent ones. And after having read the earlier part of this answer, I'm sure you know which series of data, time and pieces, is which in your case.

Have fun, and learn a regression tool...doing these manually just is not done in the real world.

Ooops...almost missed it, slope is just the rate of something. For example, velocity is a slope because it is the rate of distance traveled (dS) per span of time (dt) (v = dS/dt). Or, like in your case, slope is the rate of time spent per piece (m = dt/dN, where dN is the number of pieces finished in the time span dt.) We call this rate a slope because, in a graph, the slope would look like a line at an incline (a slope) with the axes of the graph.