Video: Determining Possible Values of Regression Coefficients from a Scatterplot

Video Transcript

A linear regression model of the form 𝑦 equals 𝑎 plus 𝑏𝑥 has been fitted to the data shown. Which of the following statements are true about the values 𝑎 and 𝑏 in the fitted regression model?

Okay. Looking at this question, we actually have five options. So A) is: 𝑏 is less than zero, 𝑎 is equal to zero; B) 𝑏 is less than zero, 𝑎 is greater than zero; C) 𝑏 is greater than zero, 𝑎 is less than zero; D) 𝑏 is less than zero, 𝑎 is less than zero; or E) 𝑏 is greater than zero, 𝑎 is greater than zero.

To be able to work out what is true about the values 𝑎 and 𝑏, we need to know a little bit about the regression model particularly in this form. The regression model allows us to mathematically predict the relationship between 𝑦 and 𝑥. So we can actually use this regression model to maybe make predictions. So when we have the regression model in this form, we need to know what do the 𝑎 and 𝑏 stand for. So what do they mean? Well, 𝑏 is the slope of our line, so 𝑏 is gonna be how much the 𝑦-direction changes divided by how much the 𝑥-direction changes, and 𝑎 is the 𝑦-intercept. And what this means, this is where our line will cross the 𝑦-axis.

Okay, great. So now we know what the 𝑎 and 𝑏 actually mean. We can go about searching to find what is true about the values 𝑎 and 𝑏 in this model. We’re gonna go about doing this by actually eliminating answers as we go. We’re gonna start by having a look at 𝑎, the 𝑦-intercept. Well straightaway, we can see that actually the line, the regression model line, would not go through the origin. So therefore, 𝑎 could not be equal to zero, which means we can actually get rid of answer A.

So now this has been eliminated. Now let’s have a look at what the actual 𝑦-intercept could be. What I’ve done is I’ve actually extended the line down. And as you see, and where I’ve put the purple dashes, you can see that it actually would cross the 𝑦-axis below zero. So therefore, our 𝑎 value would have to be less than zero. This means that we can now eliminate any answers that have 𝑎 is greater than zero. So two more have been eliminated.

Okay, great. So finally, we need to choose between that final two answers. I’m gonna do that by using the slope. Okay, to help us consider the slope, I’ve done a couple of sketches. My top sketch, we see a line that’s actually going up to the right-hand side. This actually has a positive slope, as 𝑦 increases as 𝑥 increases. So our 𝑦-value would be positive. If we look at the second sketch, this one actually have a negative slope because actually our 𝑦-values will be going down. So the change in 𝑦 would be negative, so it give us a negative value for our slope.

Okay, bearing this in mind, we go back to our graph and our model. And if we see the regression model that we have, we can see that our line is up into the right-hand side. So it’s the same as our positive slope. And this gives us our final bit of information that we need to know. Well therefore, if it’s a positive slope, we know that the value of 𝑏 must be positive. So it’s going to be greater than zero, so we can now eliminate anything where 𝑏 is less than zero and we will found our answer.

So therefore, the correct answer is C 𝑏 is greater than zero and 𝑎 is less than zero.

And we’ll just quickly check that to make sure that it does all look right. So again, we look back at our regression model, we can see that the 𝑦-intercept is below the 𝑥-axis. So it’s gonna be where 𝑦 is actually negative, so 𝑎 will be less than zero. Great, and if we look at the slope of the line, the slope is moving up to the right-hand side. So we’re gonna have a positive change in 𝑦 because it’s going to increase. So therefore, we have a positive slope, so 𝑏 is gonna be greater than zero. There we have it. We found our correct answer.