Hypothesis Testing - Assumptions and Validity of Mean Tests

A random sample of \(90\) batteries, used in a particular model of mobile phone, is tested and the 'standby-time', \(x\) hours, is measured. The results are summarised by \(\sum x = 3040.8\) and \(\sum x^2 = 115773.66\). Test, at the \(1\%\) significance level, whether the mean standby-time is less than \(36.0\) hours.

State, with a reason, whether in using the above test, it is necessary to assume that the 'standby-time' has a normal distribution.

In a test at the \(5\%\) significance level, it is found that there is significant evidence that the population mean talk-time is less than \(5\) hours. Using only this information, and giving a reason in each case, state whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) neither necessarily true nor false.

1. There is significant evidence at the \(10\%\) significance level that the population mean talk-time is less than \(5\) hours.
2. There is significant evidence at the \(5\%\) significance level that the population mean talk-time is not \(5\) hours.