1. a.) Define the following terms.
i.) Mean square Error
ii.) Unbiased Estimator
ii.) Confidence level
b.) Show that MSE(θˆ)= V(θˆ) + B(θˆ)2
c.) Let X and Y be two jointly continuous random variables with joint pdf
{2, y+x, ≤ 1, x > 0, y > 0 (1)
fXY (x, y) =
{ 0, elsewhere (2)
Find Cov(X,Y) and ρ(X, Y )
d.) Given X ∼ N (µ1, σ12) and Y ∼ N (µ2, σ22). Since E(X) = µ1, Var(X)=σ12, E(Y)=µ2 and Var(X)=σ22. Show that
i.) E(X)=σ1σ2ρ + µ1µ2
ii.) Cov(X,Y)=σ1σ2ρ
e.) Let Y1, Y2,....., Yn be a random sample of size n from a normal distribution with mean µ and variance σ2. Show that
Y‾ = (1/n∑i=1nYi) ~ (µ, σ2/n).
2. Use the following information to construct the confidence interval specified to estimate µ.
i.) 90% confidence interval for x¯ = 25, σ = 3.5 and n = 60
ii.) 95% confidence interval for x¯ = 119.6, σ = 23.89 and n = 25
3. i.) A survey was conducted to get an estimate of the proportion of smokers among the graduate students. Report says 28% of them are smokers. Winnie doubts the result and thinks that the actual proportion is much less than this. Choose the correct choice of null and alternative hypothesis Winnie wants to test.
a.) H0: p=0.28 versus Ha: p ≤0.28.
b.) H0: p=0.28 versus Ha: p > 0.28.
c.) H0: p=0.28 versus Ha: p < 0.28.
d.) None of the above.
ii.) Suppose the p-value for a test is 0.02. Which of the following is true? a.) We will not reject H0 at alpha = 0.05
b.) We will reject H0 at alpha = 0.01
c.) We will reject H0 at alpha = 0.05
d.) We will reject H0 at alpha equals 0.01, 0.05, and 0.10
e.) None of the above.
iii.) The mean height of 50 female students who showed above-average participation in college basketball games was 68.2 inches with a standard deviation of 2.5 inches, while 50 female students who showed no interest in such participation had a mean height of 67.5 inches with a standard deviation of 2.8 inches.
(a) Test the hypothesis that female students who participate in college athletics are taller than other female students.
(b) What is the p-value of the test?