In: Statistics and Probability
Suppose that X1, X2, X3,
X4 is a simple random (independent and identically
distributed) sample of size 4 from a normal distribution with an
unknown mean μ but a known variance 9. Suppose further that
Y1, Y2, Y3, Y4,
Y5 is another simple random sample (independent from
X1, X2, X3, X4 from a
normal distribution with the same mean μand variance
16. We estimate μ with
U = (bar{X}+bar{Y})/2.
where
bar{X} = (X1 + X2 + X3 + X4)/4
bar{Y} = (Y1 + Y2 + Y3 +
Y4+ Y5)/5
a. (6 points) Determine the distribution of U.
b. (4 points) Build a 99% confidence interval for μ.
c. (6 points) Compute the coefficient of correlation between U and
X1 .