Question (Pivots, \( z \)-test and t-test [5 marks]). Consider the following observations which were taken from a normally distributed random variable \( X \),
\[
5.2,5.6,7.6,6.8,4.8,5.7,9.0,6.0,4.9,7.4,6.5,7.9,6.8,4.3,8.5,3.6,6.1,5.8,6.4,4.0 \text {. }
\]
Answer the following questions:
1. Assume \( X \sim N\left(\mu, \sigma^{2}\right) \) and \( \sigma^{2}=2.2 \), find the pivot for \( \mu \) and construct a \( 95 \% \) confidence interval;
2. Assume \( X \sim N\left(\mu, \sigma^{2}\right) \) and \( \sigma^{2} \) is unknown, find the pivot for \( \mu \) and construct a \( 95 \% \) confidence interval;
3. Assume \( X \sim N\left(\mu, \sigma^{2}\right), \mu \) is known and \( \sigma^{2} \) is unknown, find the pivot for \( \sigma^{2} \) and construct a \( 95 \% \) confidence interval;
