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]