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...
Question 1 Report
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 diagram to represent the data.
(b) Calculate \(\bar{x}\), the mean of X and \(\bar{y}\), the mean of Y.
(c) Draw the line of best fit to pass through \((\bar{x}, \bar{y})\).
(d) From your graph in (c), determine the (i) relationship between X and Y ; (ii) value of Y when X is 24.
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 |
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).
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