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. SHOW R WORK

c) Calculate the variance of Xbar

d) Calculate the variance of Ybar.

e) Calculate the variance of Xbar – Ybar.

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

for *Î¼*_{x} –

*Î¼*_{y}?

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

*Î¼*_{y}.

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

*Î¼*_{x} – *Î¼*_{y}?

j) 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

?