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
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
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.