A student sits a national test and is told that the marks follow a normal distribution with mean 100. The student receives a mark of 124 and is told that he is at the ${68^{{\text{th}}}}$ percentile.

Calculate the variance of the distribution.

$X:{\text{N}}(100,{\text{ }}{\sigma ^2})$

${\text{P}}(X < 124) = 0.68$     (M1)(A1)

$\frac{{24}}{\sigma } = 0.4676 \ldots$     (M1)

$\sigma  = 51.315 \ldots$     (A1)

variance = 2630     A1

Notes:     Accept use of ${\text{P}}(X < 124.5) = 0.68$ leading to variance = 2744.

[5 marks]