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

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 approxeq 39.9)

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

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

(b) 

(x) (y) (xy) (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 {xy} – 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} approxeq 0.95)

Substituting in the equation, (y – bar{y} = m(x – bar{x}))

(y – 46.8 = 0.95(x – 39.9))

= (y = 0.95x + 8.895)

(c) y = 0.95x + 8.895

when x = 28,

(y = (0.95 times 28) + 8.895)

= (26.6 + 8.895 approxeq 35.5 marks).

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