7 The manufacturer's specification for batteries used in a certain electronic game is that the mean lifetime should be 32 hours. The manufacturer tests a random sample of 10 batteries made in Factory \(A\), and the lifetimes ( \(x\) hours) are summarised by $$n = 10 , \sum x = 289.0 \text { and } \sum x ^ { 2 } = 8586.19 .$$ It may be assumed that the population of lifetimes has a normal distribution.

1. Carry out a one-tail test at the \(5 \%\) significance level of whether the specification is being met.
2. Justify the use of a one-tail test in this context. Batteries made with the same specification are also made in Factory \(B\). The lifetimes of these batteries are also normally distributed. A random sample of 12 batteries from this factory was tested. The lifetimes are summarised by $$n = 12 , \sum x = 363.0 \text { and } \sum x ^ { 2 } = 11290.95 \text {. }$$
3. (a) State what further assumption must be made in order to test whether there is any difference in the mean lifetimes of batteries made at the two factories.
   Use the data to comment on whether this assumption is reasonable.
   (b) Carry out the test at the \(10 \%\) significance level.