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.