Edexcel IAL Mathematics S1.4 correlation and regression

Practise correlation and regression by calculating summary statistics, fitting least-squares lines and judging predictions in context.

Exam points

Calculate Sxx, Sxy or regression coefficients before writing the least-squares line.
Interpret gradient, intercept or r in context using variables, units and direction of change.
Judge prediction reliability by checking correlation strength and whether the input is extrapolated.

Covered syllabus points

S1.4.1 - Scatter diagrams S1.4.2 - Explanatory (independent)S1.4.3 - Product moment correlation

[Maximum number: 10]

Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, p, and the number of marks achieved on the final exam, f, for 34 students and displayed them on the scatter diagram.

The equation of the least squares regression line for these data is found to be
f=10.8+0.748p

For these students, the mean number of marks on the pre-test is 62.4

Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.

Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam.
The summary statistics used for the model f=10.8+0.748p are corrected to include this information and a new least squares regression line is found.
Given the original summary statistics were,
n=34
sum p=2120
sum pf=133486
Spp=15573.76
Spf=11648.35

(a)

Use the regression model to find the mean number of marks on the final exam.

[ 2 ]

(b)

Give an interpretation of the gradient of the regression line.

[ 1 ]

(a)

For each additional mark scored on the pre-test, the average mark on the final exam
increases by 0.748

Figure for Answer to Question (b) — Edexcel A-Level Mathematics AS

B1 (1)

M1
$\mathrm{Al}^{\mathrm{P}_{3}} \mathrm{O}_{2}$
(2)
B1
(1)

(c)

Comment on the reliability of Priya's statement.

[ 1 ]

(d)

Write down the number of marks achieved on the final exam for the student who exceeded the expectation of the regression model by the largest number of marks.

[ 1 ]

(e)

calculate the gradient of the new regression line. Show your working clearly.

[ 5 ]

$S_{pp}=15573.76$ (no change)
$\sum pf=133486-2842+9016=139660$ $\sum pf$ increases by 98(92-29)=6174 $\sum f=57.47\times34+(92-29)\approx2017$ $\frac{\sum p\sum f}{n}$ increases by $\frac{2120(92-29)}{34}=3928.235\ldots$ $S_{pf}$ increases by $6174-3928.235=2245.764\ldots$ $b=\frac{11648.35+2245.764\ldots}{15573.76}=0.89\ldots$
awrt 0.9

M1 M1 dM1 M1 A1

[Maximum number: 14]

A biologist is studying horses.

The biologist records the height, $h \mathrm{~cm}$ , and the weight, $w \mathrm{~kg}$ , of 8 horses.
The table below shows the biologist's results.

Table for Question S1.4 - Correlation and regression question 2 — Edexcel A-Level Mathematics AS

[You may use $\sum h=1160 \sum w=4160 \sum h w=604135 \sum w^{2}=2166600 \quad \mathrm{~S}_{h h}=286$ ]

(a)

Calculate the value of $\mathrm{S}_{h w}$ and the value of $\mathrm{S}_{w w}$

[ 3 ]

$S_{hw}=604135-\frac{1160\times4160}{8}$
or
$S_{ww}=2166600-\frac{4160^2}{8}$ $S_{hw}=935$ $S_{ww}=3400$
M1, A1, A1

(b)

Calculate the value of the product moment correlation coefficient between h and w

[ 2 ]

$r=\frac{935}{\sqrt{286\times3400}}=0.9481\ldots$
awrt 0.948
M1, A1

(c)

Show that the equation of the regression line of w on h can be written as

w=46.0+3.27 h

where the values of the intercept and gradient are given to 3 significant figures.

[ 3 ]

$b=\frac{935}{286}=3.269\ldots$ $a=\frac{4160}{8}-3.269\ldots\times\frac{1160}{8}=45.96\ldots$ w=46.0+3.27h
M1, M1, A1*

(d)

Give an interpretation, in context, of the gradient of the regression line.

[ 1 ]

(e)

Using the equation of the regression line in part (c), estimate the weight of a horse with a height of

[ 4 ]

(i)

153 cm

[ 2 ]

(ii)

170 cm

[ 2 ]

(f)

State, giving a reason, which of the two estimates found in part (e) would give the more reliable estimate.

[ 1 ]

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Covered syllabus points

S1.4 - Correlation and regression question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

S1.4 - Correlation and regression question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (i)

Question (ii)

Question (f)