8 The tensile strength of rope is measured in kilograms. The standard deviation of the tensile strength of a particular design of 10 mm diameter rope is known to be 285 kilograms. A retail organisation, which buys such rope from two manufacturers, A and B , wishes to compare their ropes for mean tensile strength. The mean tensile strength, \(\bar { x }\), of a random sample of 80 lengths from manufacturer A was 3770 kilograms. The mean tensile strength, \(\bar { y }\), of a random sample of 120 lengths from manufacturer B was 3695 kilograms.

1. 1. Test, at the \(5 \%\) level of significance, the hypothesis that there is no difference between the mean tensile strength of rope from manufacturer A and that of rope from manufacturer B.
   2. Why was it not necessary to know the distributions of tensile strength in order for your test in part (a)(i) to be valid?
2. 1. Deduce that, for your test in part (a)(i), the critical values of \(( \bar { x } - \bar { y } )\) are \(\pm 80.63\), correct to two decimal places.
   2. In fact, the mean tensile strength of rope from manufacturer A exceeds that of rope from manufacturer B by 125 kilograms. Determine the probability of a Type II error for a test of the hypothesis in part (a)(i) at the \(5 \%\) level of significance, based upon a random sample of 80 lengths from manufacturer A and a random sample of 120 lengths from manufacturer B. (4 marks)