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...

Answer Details

(a)(i) Scatter diagram. Using a scale of 2 cm to 10 marks on each axis, the nine pairs \((43,54),(46,53),(48,63),(39,30),(30,44),(60,75),(8,20),(45,33),(40,49)\) are plotted with Mathematics \((x)\) on the horizontal axis and Physics \((y)\) on the vertical axis. The line of best fit required in part (a)(iii) is drawn on the same diagram.

(a)(ii) The means.

\[\bar{x}=\frac{43+46+48+39+30+60+8+45+40}{9}=\frac{359}{9}=39.9\] \[\bar{y}=\frac{54+53+63+30+44+75+20+33+49}{9}=\frac{421}{9}=46.8\]

So the mean point is \((\bar{x},\bar{y})=(39.9,\,46.8)\).

(a)(iii) Line of best fit. A single straight line is drawn through the mean point \((39.9,46.8)\), oriented so that the points are balanced roughly equally above and below it (shown as the straight line on the scatter diagram above).

(b) Equation of the line. Set out the sums needed for the least-squares gradient:

\(x\)	\(y\)	\(xy\)	\(x^{2}\)
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
\(\sum x=359\)	\(\sum y=421\)	\(\sum xy=18379\)	\(\sum x^{2}=15979\)

With \(n=9\), the gradient is

\[b=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}=\frac{9(18379)-359(421)}{9(15979)-359^{2}}=\frac{165411-151139}{143811-128881}=\frac{14272}{14930}=0.96.\]

The line passes through \((\bar{x},\bar{y})\), so using \(y-\bar{y}=b(x-\bar{x})\):

\[y-46.8=0.96(x-39.9)\] \[y=0.96x-38.3+46.8\] \[y=0.96x+8.6\]

(c) Physics mark when Mathematics \(=28\).

\[y=0.96(28)+8.6=26.9+8.6=35.5\]

The predicted mark in Physics for a candidate who scored 28 in Mathematics is 35.5 (to one decimal place).