The table shows the marks obtained by some candidates in Physics (y) and Mathematics (x) tests. (a)(i) Represent this information on a scatter diagram. (ii) Find $\bar{x}$ and $\bar{y}$, the mean of x and y respectively. (iii) Draw the line of best fit to pass through (x, y). (b) Find the equation of the line in a(iii). (c) Use your equation in (b) to find, correct to one decimal place, the mark in Physics for a candidate who scored 28 in Mathematics.

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The table shows the marks obtained by some candidates in Physics (y) and Mathematics (x) tests.

Mathematics	43	46	48	39	30	60	8	45	40
Physics	54	53	63	30	44	75	20	33	49

(a)(i) Represent this information on a scatter diagram.
(ii) Find

\bar{x}

and

\bar{y}

, the mean of x and y respectively.
(iii) Draw the line of best fit to pass through (x, y).
(b) Find the equation of the line in a(iii).
(c) Use your equation in (b) to find, correct to one decimal place, the mark in Physics for a candidate who scored 28 in Mathematics.

Explanation

Mathematics (x)	43	46	48	39	30	60	8	45	40
Physics (y)	54	53	63	30	44	75	20	33	49

(a)(i) and (iii) Scale: On each axis, 2 cm represents 10 marks.
(ii)

\bar{x} = \frac{43 + 46 + 48 + 39 + 30 + 60 + 8 + 45 + 40}{9} = \frac{359}{9} = 39.89 ≊ 39.9

\bar{y} = \frac{54 + 53 + 63 + 30 + 44 + 75 + 20 + 33 + 48}{9} = \frac{421}{9} = 46.78 ≊ 46.8

(\bar{x}, \bar{y}) = (39.9, 46.8)

(b)

$x$	$y$	$x y$	$x^{2}$
43	54	2322	1849
46	53	2438	2116
48	63	3024	2304
39	30	1170	1521
30	44	1320	900
60	75	4500	3600
8	20	160	64
45	33	1485	2025
40	49	1960	1600
Total		18379	15979

The equation of the line :

y - \bar{y} = m (x - \bar{x})

m = \frac{\sum x y - N (\bar{x}) (\bar{y})}{\sum x^{2} - N (\bar{x})^{2}}

\frac{18379 - 9 \times 39.9 \times 46.8}{15979 - (39.9)^{2}} = \frac{18379 - 16805.88}{15979 - 9 \times (39.9)^{2}}

\frac{1573.12}{1650.91} ≊ 0.95

Substituting in the equation,

y - \bar{y} = m (x - \bar{x})

y - 46.8 = 0.95 (x - 39.9)

y = 0.95 x + 8.895

y = (0.95 \times 28) + 8.895

26.6 + 8.895 ≊ 35.5 m a r k s

.
For a student who scored 28 in Mathematics, his mark in Physics is 35.5.

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