The weight loss, in kilograms, of people using the slimming regime SLIM3M for a period of three months is modelled by a random variable X. Experimental data showed that 67 % of the individuals using SLIM3M lost up to five kilograms and 12.4 % lost at least seven kilograms. Assuming that X follows a normal distribution, find the expected weight loss of a person who follows the SLIM3M regime for three months.

$X \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})$

${\text{P}}(X \leqslant 5) = 0.670 \Leftrightarrow \frac{{5 - \mu }}{\sigma } = 0.4399 \ldots$     M1A1

${\text{P}}(X > 7) = 0.124 \Leftrightarrow \frac{{7 - \mu }}{\sigma } = 1.155 \ldots$     A1

solve simultaneously

$\mu + 0.4399\sigma = 5{\text{ and }}\mu + 1.1552\sigma = 7$     M1

$\mu = 3.77{\text{ (3 sf)}}$     A1     N3

the expected weight loss is 3.77 kg

Note: Award A0 for $\mu = 3.78$ (answer obtained due to early rounding).

[5 marks]

Although many candidates were successful in answering this question, a surprising number of candidates did not even attempt it. The main difficulty was in finding the correct z scores. A fairly common error was to misinterpret one of the conditions and obtain one of the equations as $\frac{{7 - \mu }}{\sigma } = - 1.155 \ldots$. In some cases candidates failed to keep the accuracy throughout the question and obtained inaccurate answers.