ACJC Hypothesis Testing Tutorial Q11

The random variable \(X\) has a normal distribution with mean \(\mu \) (unknown) and variance \({{\sigma }^{2}}\) (known). To test the null hypothesis \({{H}_{0}}:\mu ={{\mu }_{0}}\), a random sample of \(n\) observations of \(X\) is taken, and the sample mean is \(\bar{x}\).

Find in terms of \({{\mu }_{0}}\), \(\sigma \) and \(n\), the set of values of \(\bar{x}\) which will result in each of the following:

1. \({{H}_{0}}\) being rejected in favour of \({{H}_{1}}:\mu \ne {{\mu }_{0}}\) at the \(5%\) level of significance;
2. not being rejected in favour of \({{H}_{1}}:\mu <{{\mu }_{0}}\) at the \(1%\) level of significance.