4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.
- Carry out the test.
- Explain whether it is necessary to use the Central Limit Theorem in your test.