Question

Two judges in a beauty contest rank the ten competitors in the following order.

x 6 4 3 9 2 7 1 5 10 8

y 4 1 6 7 5 8 2 3 9 10

Do the two judges appear to agree in their standards? (6mks)

Answer 1

The differences of ranks $d_i,i=1,...,10$ are:

2 3 -3 2 -3 -1 -1 2 1 -2

Then the Spearman correlation coefficient is

$r=1-\frac{6\sum{{d_i}^2}}{n\left( n^2-1 \right)}=0.7212$

Test this coefficient for significance:

$t=r\sqrt{\frac{n-2}{1-r^2}}=0.7212\sqrt{\frac{10-2}{1-0.7212^2}}=2.94468$

The P-value is

$P\left( \left| T_{n-2} \right|\geqslant 2.94468 \right) =2F_{T,8}\left( -2.94468 \right) =2\cdot 0.00929=0.01858$

Hence the coefficient is significant at $\alpha =0.05$ level. We conclude that there is a strong positive correlation, hence the judges agree in their standards.