7 A shop sells cooked chickens in two sizes: medium and large.
The weights, \(X\) grams, of medium chickens may be assumed to be normally distributed with mean \(\mu _ { X }\) and standard deviation 45. The weights, \(Y\) grams, of large chickens may be assumed to be normally distributed with mean \(\mu _ { Y }\) and standard deviation 65. A random sample of 20 medium chickens had a mean weight, \(\bar { x }\) grams, of 936 .
A random sample of 10 large chickens had the following weights in grams: $$\begin{array} { l l l l l l l l l l } 1165 & 1202 & 1077 & 1144 & 1195 & 1275 & 1136 & 1215 & 1233 & 1288 \end{array}$$

1. Calculate the mean weight, \(\bar { y }\) grams, of this sample of large chickens.
2. Hence investigate, at the \(1 \%\) level of significance, the claim that the mean weight of large chickens exceeds that of medium chickens by more than 200 grams.
3. 1. Deduce that, for your test in part (b), the critical value of \(( \bar { y } - \bar { x } )\) is 253.24, correct to two decimal places.
   2. Hence determine the power of your test in part (b), given that \(\mu _ { Y } - \mu _ { X } = 275\).
   3. Interpret, in the context of this question, the value that you obtained in part (c)(ii).
      (3 marks)