Suppose we havei.i.drandom variablesX1, , X10from distributionN(1, 2) andi.i.drandom variablesY1, , Y12from distributionN(2, 2), where1, 2and2are unknown, the observed values are:

X: 3.58 6.08 6.52 5.94 8.16 3.24 2.31 3.68 6.23 6.24

Y: 4.89 7.07 7.34 5.89 8.00 6.85 8.86 7.00 9.19 9.16 8.46 8.40

1. (a)Construct a 95% confidence interval for12
2. (b)ConsiderH0:12= 0 v.s.H1:12= 0. Please use the results in (a) and formulate a hypothesis testing procedure at level of significance= 0.05.
3. (c)ConsiderH0:12= 0 v.s.H1:12= 0. Please implement the p-value approach. Do you obtain the same conclusion as (b)? You can set the level of significance at= 0.05.
4. (d)ConsiderH0:12= 0 v.s.H1:12= 0. Please implement the rejection region approach. Do you obtain the same conclusion as (b)?