Which point does the regression line always pass through?

The mean point (\bar{x},\bar{y}).

What does 'least squares' mean?

The line minimises the sum of the squares of the vertical distances from the points to the line.

What is the explanatory variable?

The independent variable, used to predict the response; plotted on the horizontal axis.

What is the response variable?

The dependent variable that is predicted; plotted on the vertical axis.

How do you interpret the gradient?

The change in the predicted response for each one-unit increase in the explanatory variable, in context.

What is interpolation?

Predicting for a value inside the range of the data; usually reliable.

What is extrapolation and why is it dangerous?

Predicting outside the range of the data; the relationship may not continue, so the prediction may be unreliable.

How do you convert a line from u=\frac{x-40}{5} back to x?

Substitute u=\frac{x-40}{5} into y=a+bu and simplify.

What do you do first when asked to draw the regression line?

Plot (\bar{x},\bar{y}) and one other point from the equation, then join them.

Scatter diagrams and linear regressionEdexcel International A Level Maths: Flashcards

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Formula for $b$ in the regression line $y=a+bx$?

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Formula for $b$ in the regression line $y=a+bx$ ?: $b=\frac{S_{xy}}{S_{xx}}$
Formula for $a$ ?: $a=\bar{y}-b\bar{x}$
Formula for $S_{xx}$ ?: $S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}$
Formula for $S_{xy}$ ?: $S_{xy}=\sum xy-\frac{\sum x\sum y}{n}$
Which point does the regression line always pass through?: The mean point $(\bar{x},\bar{y})$ .
What does 'least squares' mean?: The line minimises the sum of the squares of the vertical distances from the points to the line.
What is the explanatory variable?: The independent variable, used to predict the response; plotted on the horizontal axis.
What is the response variable?: The dependent variable that is predicted; plotted on the vertical axis.
How do you interpret the gradient?: The change in the predicted response for each one-unit increase in the explanatory variable, in context.
What is interpolation?: Predicting for a value inside the range of the data; usually reliable.
What is extrapolation and why is it dangerous?: Predicting outside the range of the data; the relationship may not continue, so the prediction may be unreliable.
How do you convert a line from $u=\frac{x-40}{5}$ back to $x$ ?: Substitute $u=\frac{x-40}{5}$ into $y=a+bu$ and simplify.
What do you do first when asked to draw the regression line?: Plot $(\bar{x},\bar{y})$ and one other point from the equation, then join them.

Exam questions on Scatter diagrams and linear regression

A teacher records the number of hours of revision per week, $x$ , and the mark out of 40, $y$ , in a test for 8 students. The summary statistics are $\bar{x}=5$ , $\bar{y}=22$ , $S_{xx}=40$ and $S_{xy}=62$ . The regression line of $y$ on $x$ is $y=a+bx$ .
Estimate the mark of a student who revises for 6 hours per week.2 marks
A cafe owner models the daily sales, $s$ pounds, against the midday temperature, $t$ °C, using the regression line $s=120+8.5t$ . The model was fitted to data collected on days when the temperature was between 12 °C and 28 °C.
A student uses the model to predict the sales on a day when the midday temperature is 5 °C. Explain why this prediction may be unreliable.2 marks
A gardener applies $x$ grams of fertiliser to each of five plants of the same type and measures the height, $y$ cm, of each plant after four weeks. The values of $x$ are 2, 4, 6, 8, 10 and the heights are 5, 9, 10, 15, 16. For these data $\sum x=30$ , $\sum y=55$ , $\sum x^2=220$ and $\sum xy=386$ .
Find the values of $S_{xx}$ and $S_{xy}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).