The table below shows the corresponding values of two variables X and Y. X 33 31 28 25 23 22 19 17 16 14 Y 4 6 4 10 12 10 14 15 18 22 (a) Plot a scatter dia...

Answer Details

(a) Scatter diagram. The ten pairs \((33,4),(31,6),(28,4),(25,10),(23,12),(22,10),(19,14),(17,15),(16,18),(14,22)\) are plotted with \(X\) on the horizontal axis and \(Y\) on the vertical axis. The points trend from the upper-left down to the lower-right, showing that \(Y\) falls as \(X\) rises.

(b) Means.

\[\bar{x}=\frac{33+31+28+25+23+22+19+17+16+14}{10}=\frac{228}{10}=22.8\] \[\bar{y}=\frac{4+6+4+10+12+10+14+15+18+22}{10}=\frac{115}{10}=11.5\]

(c) Line of best fit. A single straight line is drawn through the mean point \((\bar{x},\bar{y})=(22.8,\,11.5)\) (marked ★ on the graph) with roughly equal numbers of points on either side. Its least-squares gradient is

\(n\)	\(\sum X\)	\(\sum Y\)	\(\sum XY\)	\(\sum X^2\)
10	228	115	2293	5574

\[b=\frac{n\sum XY-\sum X\sum Y}{n\sum X^{2}-\left(\sum X\right)^{2}}=\frac{10(2293)-228(115)}{10(5574)-228^{2}}=\frac{22930-26220}{55740-51984}=\frac{-3290}{3756}=-0.88\]

The intercept follows from \(\bar{y}=a+b\bar{x}\):

\[a=\bar{y}-b\bar{x}=11.5-(-0.88)(22.8)=11.5+20.06=31.5\]

so the line of best fit is

\[Y=31.5-0.88X.\]

(d)(i) Relationship. The line has a negative gradient, so there is a negative (inverse) correlation between \(X\) and \(Y\): as \(X\) increases, \(Y\) decreases.

(d)(ii) Value of \(Y\) when \(X=24\). Reading up from \(X=24\) to the line of best fit and across to the \(Y\)-axis, or substituting into the equation:

\[Y=31.5-0.88(24)=31.5-21.1=10.4\]

so \(Y\approx 10.5\) (about 10 to 11 from a hand-drawn line).