The company Fresh Water produces one-litre bottles of mineral water. The company wants to determine the amount of magnesium, in milligrams, in these bottles.

A random sample of ten bottles is analysed and the results are as follows:

6.7, 7.2, 6.7, 6.8, 6.9, 7.0, 6.8, 6.6, 7.1, 7.3.

Find unbiased estimates of the mean and variance of the amount of magnesium in the one-litre bottles.

\(\bar m = \frac{{6.7 + 7.2 + \ldots + 7.3}}{{10}} = 6.91\)     (M1)A1

\(s_{n - 1}^2 = \frac{1}{9}\left( {{{(6.7 - 6.91)}^2} + \ldots + {{(7.3 - 6.91)}^2}} \right)\)     (M1)

\( = \frac{{0.489}}{9} = 0.0543{\text{ (3 sf)}}\)     A1

Note: Award M1A0 for 0.233.

[4 marks]

Most candidates used a GDC to answer this question and many scored full marks in this question. However there were a significant number of candidates who showed little understanding of the meaning of unbiased estimate. In some cases, candidates wasted time by attempting to calculate the required values by hand.