7 A factory manager wished to compare two methods of assembling a new component, to determine which method could be carried out more quickly, on average, by the workforce. A random sample of 12 workers was taken, and each worker tried out each of the methods of assembly. The times taken, in seconds, are shown in the table.
| Worker | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
| Time in seconds for Method 1 | 48 | 38 | 47 | 59 | 62 | 41 | 50 | 52 | 58 | 54 | 49 | 60 |
| Time in seconds for Method 2 | 47 | 40 | 38 | 55 | 57 | 42 | 42 | 40 | 62 | 47 | 47 | 51 |
- (a) Carry out an appropriate \(t\)-test, using a \(2 \%\) significance level, to test whether there is any difference in the times for the two methods of assembly.
(b) State an assumption needed in carrying out this test.
(c) Calculate a \(95 \%\) confidence interval for the population mean time difference for the two methods of assembly. - Instead of using the same 12 workers to try both methods, the factory manager could have used two independent random samples of workers, allocating Method 1 to the members of one sample and Method 2 to the members of the other sample.
(a) State one disadvantage of a procedure based on two independent random samples.
(b) State any assumptions that would need to be made to carry out a \(t\)-test based on two independent random samples.