11.4 Scatter diagrams, regression lines and correlation — Practise with help

Work each question closed book first. Describe the correlation in words before you use the line.

Warm-up: remember

Find the gradient of the line through $(2, 3)$ and $(6, 11)$ .
Work out $3x + 4$ when $x = 7$ .
Is the point $(5, 12)$ above or below the line $y = 2x + 1$ ?

Answers

$\dfrac{11 - 3}{6 - 2} = \dfrac{8}{4} = 2$ .
$3(7) + 4 = 25$ .
On the line, $y = 2(5) + 1 = 11$ . The point has $y = 12$ , so it lies above the line.

If question 1 was hard, revise gradients from Section 3.1.

Practise with help

1. A scatter diagram plots the mass of a parcel against the cost of posting it. The points rise from left to right, close to a straight line. Describe the correlation.

Hint

Say whether the pattern is positive, negative or none, and then say whether it is strong or weak.

Next step

The points rise, so the correlation is positive. They lie close to a straight line, so it is strong.

Full working

The correlation is positive and strong. Heavier parcels tend to cost more, and the points lie close to a straight line.

2. A line of best fit is $y = 5x + 20$ , where $x$ is the number of weeks of practice and $y$ is the score in a test. Interpret the gradient and the intercept.

Hint

The gradient is the number multiplying $x$ . The intercept is the number added. Decide what each one means in the context.

Next step

The gradient is 5 and the intercept is 20. Write one sentence about each in the context of weeks and score.

Full working

The gradient is 5, so each extra week of practice is associated with about 5 extra marks. The intercept is 20, so a student with no practice has a predicted score of 20. Both numbers describe the sample, and neither proves that practice causes the score to rise.

3. Using the line $y = 5x + 20$ , predict the score for 8 weeks of practice.

Hint

Substitute $x = 8$ into the equation of the line.

Next step

Work out $5(8) + 20$ .

Full working

y = 5(8) + 20 = 40 + 20 = 60.

The predicted score is 60. The value $x = 8$ lies inside the range of the data, so this is an interpolation.

4. A scatter diagram of income against age has one group of part-time workers and one group of full-time workers. Explain why a single line of best fit is not appropriate.

Hint

Compare the levels of the two groups.

Next step

Ask where a single line would sit relative to each group.

Full working

The two groups are distinct sections of the population, and the full-time group sits at a higher income than the part-time group. A single line would pass between the two groups and fit neither of them well. Each group should be described with its own line, and the groups should be compared separately.

5. A study finds a strong positive correlation between the number of ice creams sold and the number of drowning accidents. Explain why this does not prove that ice cream causes drowning.

Hint

Think of a variable that could affect both numbers.

Next step

Consider the weather. Hot weather increases both ice-cream sales and the number of people swimming.

Full working

The hot weather is a lurking variable. It increases ice-cream sales, and it also increases the number of people who swim, which increases the number of drowning accidents. The two variables move together because both depend on the weather. Correlation does not imply causation.

6. A line of best fit from data with $x$ between 0 and 10 is $y = 2x + 3$ . A student uses it to predict $y$ when $x = 50$ . Explain why the prediction should not be trusted.

Hint

Compare the value $x = 50$ with the range of the data.

Next step

Say whether the prediction is an interpolation or an extrapolation, then explain what can go wrong.

Full working

The prediction is an extrapolation, because $x = 50$ is far outside the range 0 to 10. The straight-line trend may not continue outside the data. The relationship may curve, level off or reverse, so the prediction of $103$ is not reliable.