91Ó°ÊÓ

Two different testsAandBare to be given to a student chosen at random from a certain population.Suppose also that the mean score on testAis 85, and the

standard deviation is 10; the mean score on testBis 90,and the standard deviation is 16; the scores on the two tests have a bivariate normal distribution, and the correlation of the two scores is 0.8.

Let A denote the firsttest scores, and let B denote second test scores.

Then,

\(\begin{array}{*{20}{l}}{{\mu _A} = 85\;}\\{{\sigma _A} = 10}\\{{\mu _B} = 90}\\{{\sigma _B} = 16}\\{p = 0.8}\end{array}\)

\(\)\(\)

For a\(BVN{\rm{ }}\left( {85,10,90,16,0.8} \right)\),

Since the score for test B is 100 is known, then the best prediction is the mean\(E\left( {B|A = a} \right)\)of the conditional distribution of B given that A=a and the M.S.E. of this prediction is the variance\(\left( {1 - {p^2}} \right)\sigma _B^2\)of that conditional distribution.

\(\begin{array}{c}\left( {1 - {p^2}} \right)\sigma _B^2 = \left( {1 - {{0.8}^2}} \right)\left( {16} \right)\\ = 5.76\end{array}\)

The M.S.E. value is 5.76.