**PLEASE USE R CODE.** We wish to compare the

expected values, *Î¼*_{x} and *Î¼*_{y}

of two independent normal populations, say X and Y, with known

standard deviations *Î¼*_{x} = 1.1 and

*Î¼*_{y} = 1.3 . We take a random sample of size 12

from X (*X*_{1},*X*_{2}, …

,*X*_{12}) and a random sample of size 9 from Y

(*Y*_{1},*Y*_{2}, …

,*Y*_{9}) as follows:

X: 3.84, 6.18, 5.85, 5.82, 3.66, 3.83, 4.09, 7.25, 5.69, 3.99,

6.17, 5.36

Y: 5.34, 6.43, 5.61, 5.17, 6.93, 3.37, 6.06, 4.15, 5.83

We are interested in examining *Î¼*_{x} –

*Î¼*_{y}. Call the sample means of X and Y, Xbar and

Ybar respectively(xbar and ybar realized values). Assume that all

distributions are normal. **Use R for
computations.**

a) What is the critical value used for a 95% confidence interval

for *Î¼*_{x} –

*Î¼*_{y}? **(using R
code)**

b) Create a 95% confidence interval for *Î¼*_{x} –

*Î¼*_{y}. **(using R code)**

c) What is the length of your 95% confidence interval for

*Î¼*_{x} –

*Î¼*_{y}? **(using R
code)**

d) What would the p value have been if we used this data to test

H_{0}:*Î¼*_{x} –

*Î¼*_{y}=0

against the alternative H_{a}:*Î¼*_{x} –

*Î¼*_{y} > 0? **(using R
code)**

e) Calculate the variance of Xbar **(using R
code)**

f) Calculate the variance of Ybar. **(using R
code)**

g) Calculate the variance of Xbar –

Ybar. **(using R code)**