The farm manager selects a random sample of \(10\) apples and weighs them with the following results, given in grams.\[82, 98, 102, 96, 111, 95, 90, 89, 99, 101\]

  (i)     Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\) .

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

The farm manager claims that the mean weight of apples is \(100\) grams but the buyer from the local supermarket claims that the mean is less than this. To test these claims, they select a random sample of \(100\) apples and weigh them. Their results are summarized as follows, where \(x\) is the weight of an apple in grams.\[\sum {x = 9831;\sum {{x^2} = 972578} } \]

  (i)     State suitable hypotheses for testing these claims.

  (ii)     Determine the \(p\)-value for this test.

  (iii)     At the \(1\%\) significance level, state which claim you accept and justify your answer.

(i)     from the GDC,

unbiased estimate for \(\mu  = 96.3\)     A1

unbiased estimate for \({\sigma ^2} = 8.028{ \ldots ^2} = 64.5\)     (M1)A1

(ii)     \(95\%\) confidence interval is [\(90.6\), \(102\)]     A1A1

Note: Accept \(102.0\) as the upper limit.

[5 marks]

(i)     \({H_0}:\mu  = 100;{H_1}:\mu  < 100\)     A1

(ii)     \(\overline x  = 98.31,{S_{n - 1}} = 7.8446 \ldots \)     (A1)

\(p\)-value \( = 0.0168\)     A1

(iii)     the farm manager’s claim is accepted because \(0.0168 > 0.01\)     A1R1

[5 marks]