The weights, in grams, of 10 apples were measured with the following results:

     $212.2$     $216.9$     $209.0$     $215.5$     $215.9$     $213.5$     $208.9$     $213.8$     $216.4$     $209.9$

You may assume that this is a random sample from a normal distribution with mean $\mu$ and variance ${\sigma ^2}$.

(a)     Giving all your answers correct to four significant figures,

(i)     determine unbiased estimates for $\mu$ and ${\sigma ^2}$;

(ii)     find a $95\%$ confidence interval for $\mu$.

Another confidence interval for $\mu$, $[211.5, 214.9]$, was calculated using the above data.

(b)     Find the confidence level of this interval.

(a)     (i)     $\bar x = {\text{213.2}}$     A1

$s = 3.0728 \ldots$     (A1)

${s^2} = 9.442$     A1

(ii)     $[211.0, 215.4]$     A1A1

Note: Accept $211$ in place of $211.0$.

Note: Apart from the above note, accept any answers which round to the correct 4 significant figure answers.

[5 marks]

(b)     use of the fact that the width of the interval is $2t \times \frac{s}{{\sqrt n }}$     (A1)

so that $3.4 = 2t \times \frac{{3.0728 \ldots }}{{\sqrt {10} }}$     M1

$t = 1.749$     A1

degrees of freedom $= 9$     (A1)

${\text{P}}(T > 1.749) = 0.0571$     (M1)

confidence level $= 1 - 2 \times 0.0571 = 0.886{\text{ }}(88.6\% )$     A1

Note: Award the ${\text{DF}} = 9$ (A1) mark if the following line has $0.00337$ on the RHS.

Note: Accept any answer which rounds to $88.6\%$.

[6 marks]