The Regression Line

A regression line (line of best fit) is a straight line drawn through a scatter graph that best represents the linear trend in the data. It's written in the form $y=a+bx$.

In the exam, you'll typically use your calculator's statistical mode to find the equation of the regression line directly from raw data (giving you the values of $a$ and $b$), rather than deriving it by hand. You need to be able to interpret and use the equation once found.

The gradient, $b$, tells you how much $y$ changes for every 1 unit increase in $x$. The intercept, $a$, tells you the predicted value of $y$ when $x=0$ — though this may not always be meaningful in context (e.g. if $x=0$ is outside the sensible range of the data).

The regression line always passes through the mean point, $(\bar{x},\bar{y})$ — the point representing the mean of the x-values and the mean of the y-values. This is a useful fact for checking or sketching a regression line by hand.

(a) The gradient (−4) means that for every 1°C increase in average outdoor temperature, the predicted monthly heating cost decreases by £4 — sensible, since warmer weather needs less heating.

(b) y = 150 − 4(8) = 150 − 32 = £118.

1. Gradient = 3, y-intercept = 20.
2. For every extra item sold, revenue is predicted to increase by £3.
3. y = 15 + 4(12) = 15+48 = 63 minutes.
4. It means the regression line always passes exactly through the point $(\bar x,\bar y)$, where $\bar x$ is the mean of all the x-values and $\bar y$ is the mean of all the y-values in the data set.
5. Because x=200 is far outside the range of the original data (10 to 50) used to create the regression line — this is extrapolation, and there's no guarantee the same linear relationship continues to hold that far beyond the data.