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]