3 The human resources department of a large company is investigating two methods, A and B, for training employees to carry out a certain complicated and intricate task.

1. Two separate random samples of employees who have not previously performed the task are taken. The first sample is of size 10 ; each of the employees in it is trained by method A. The second sample is of size 12; each of the employees in it is trained by method B. After completing the training, the time for each employee to carry out the task is measured, in controlled conditions. The times are as follows, in minutes.

   Employees trained by method A:	35.2	47.8	25.8	38.0	53.6	31.0	33.9
   	35.4	21.6	42.5
   Employees trained by method B:	43.0	57.5	68.6	20.9	31.4	44.9	62.8
   	27.6	41.8	46.1	39.8	61.6

   Stating appropriate assumptions concerning the underlying populations, use a \(t\) test at the \(5 \%\) significance level to examine whether either training method is better in respect of leading, on the whole, to a lower time to carry out the task.
2. A further trial of method B is carried out to see if the performance of experienced and skilled workers can be improved by re-training them. A random sample of 8 such workers is taken. The times in minutes, under controlled conditions, for each worker to carry out the task before and after re-training are as follows.

   Worker	\(W _ { 1 }\)	\(W _ { 2 }\)	\(W _ { 3 }\)	\(W _ { 4 }\)	\(W _ { 5 }\)	\(W _ { 6 }\)	\(W _ { 7 }\)	\(W _ { 8 }\)
   Time before	32.6	28.5	22.9	27.6	34.9	28.8	34.2	31.3
   Time after	26.2	24.1	19.0	28.6	29.3	20.0	36.0	19.2

   Stating an appropriate assumption, use a \(t\) test at the \(5 \%\) significance level to examine whether the re-training appears, on the whole, to lead to a lower time to carry out the task.
3. Explain how the test procedure in part (ii) is enhanced by designing it as a paired comparison.