Question

Suppose X, and X, are ramdom samples drawn from a population with mean ji and variance o?. Denote X = (1 - a)X1 +aX2. Find the value of a which minimize the variance of X. O1 O o 2 OO QUESTION 7 Let (X1, X2, ..., X16} be a random sample of size 16 from a normal pop- ulation having a mean of . The sample variance equals to 4. You want to test: HNS 4 versus H : > 4. Suppose the significance level a 0.05. When will you reject the null hypothesis? O 3.1235 < X < 4.8765 OX> 4.8765 or < 3.1235 > 4.8765 > 3.1235

Answer 1

Question 6:

X = X1(1-a) + aX2

Var(X) = (1-a)²Var(X1) + a²Var(X2)

= (1 - a )² σ² + a² σ²

To find mimum value of a

dVar(X)/da = 0

-2(1-a)σ² + 2aσ² = 0

-2+2aσ²+2aσ²= 0

2aσ²+2aσ² = 2

aσ² = 2/4

a = (1/2)σ²

a = σ² / 2

Answer :

• 1/2

  

Question 7:

n = 16 , s = 2 , α = 0.05 , µ = 4

Here test is right tail so if x̄ is greater than 4 then we would reject Ho.

Answer :

• x̄ > 4.8765