Hypothesis Testing - Significance Levels and \(p\)-Values

The mean of a random variable \(X\) is denoted by \(\mu\). A sample of 50 random observations of \(X\) is taken and the results are summarised by \(\sum x = 527.1\).

1. It is given that the population variance is 15. Carry out a 2-tail test of the null hypothesis \(\mu = 9.5\), at the 5% significance level.
2. It is given instead that \(\sum x^2 = 6172.31\). In a 1-tail test of the null hypothesis \(\mu = 11.5\), the alternative hypothesis is accepted. State the alternative hypothesis, and find an inequality satisfied by the significance level of the test.