6 The lifetime, \(X\) hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.
- The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows:
$$\begin{array} { l l l l l }
85 & 92 & 100 & 95 & 99
\end{array}$$
Test the members' suspicion at the \(5 \%\) level of significance.
- The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, \(x\) hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where \(\bar { x }\) denotes the sample mean:
$$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$
Test the manufacturer's belief at the \(5 \%\) level of significance.