Find the standard deviation of the scores.

A student is chosen at random. This student has the same probability of having a score less than 25.4 as having a score greater than b.

(i)     Find the probability the student has a score less than 25.4.

(ii)    Find the value of b.

evidence of approach     (M1)

e.g. finding $0.84 \ldots$, using  $\frac{{23.7 - 21}}{\sigma }$

correct working     (A1)

e.g. $0.84 \ldots = \frac{{23.7 - 21}}{\sigma }$ , graph     A1

$\sigma = 3.21$

[3 marks]

(i) evidence of attempting to find ${\text{P}}(X < 25.4)$     (M1)

e.g. using $z = 1.37$   

${\text{P}}(X < 25.4) = 0.915$     A1     N2

(ii) evidence of recognizing symmetry     (M1)

e.g. $b = 21 - 4.4$ , using $z = - 1.37$     A1     N2

[4 marks]

Candidates who clearly understood the nature of normal probability answered this question cleanly. A common misunderstanding was to use the value of 0.8 as a z-score when finding the standard deviation.

Many correctly used their GDC to find the probability in part (b). Fewer used some aspect of the symmetry of the curve to find a value for b.