In a large city, the time taken to travel to work is normally distributed with mean $\mu$ and standard deviation $\sigma$ . It is found that $4\%$ of the population take less than 5 minutes to get to work, and $70\%$ take less than 25 minutes.

Find the value of $\mu$ and of $\sigma$ .

correct z-values     (A1)(A1)

$- 1.750686 \ldots$ , $0.524400 \ldots$

attempt to set up their equations, must involve z-values, not %     (M1)

e.g. one correct equation

two correct equations     A1A1

e.g. $\mu - 1.750686\sigma = 5$ , $0.5244005 = \frac{{25 - \mu }}{\sigma }$

attempt to solve their equations     (M1)

e.g. substitution, matrices, one correct value

$\mu = 20.39006 \ldots$ , $\sigma = 8.790874 \ldots$

$\mu = 20.4$ $[20.3{\text{, }}20.4]$, $\sigma = 8.79$ $[8.79{\text{, }}8.80]$     A1A1     N4

[8 marks]

A standard question for which well-prepared candidates frequently earned all eight marks. Common errors included the use of percentages rather than z-values and the inability to find the negative z-value. Others had correct equations but were not able to use their GDC to solve them and ultimately made errors in their algebra.