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PLEASE USE R CODE. We wish to compare the
expected values, μ_x and μ_y
of two independent normal populations, say X and Y, with known
standard deviations μ_x = 1.1 and
μ_y = 1.3 . We take a random sample of size 12
from X (X₁,X₂, …
,X₁₂) and a random sample of size 9 from Y
(Y₁,Y₂, …
,Y₉) 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 μ_x –
μ_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 μ_x –
μ_y?  (using R
code)

b) Create a 95% confidence interval for μ_x –
μ_y. (using R code)

c) What is the length of your 95% confidence interval for
μ_x –
μ_y?  (using R
code)

d) What would the p value have been if we used this data to test
H₀:μ_x –
μ_y=0
against the alternative H_a:μ_x –
μ_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)