Is My Schrödinger's Equation Homework Correct?

[Mountebank]

I'm having a little trouble whit this homework assignment. Any help would be greatly appreciated. =)

Homework Statement

Explain if the following statements are true or false:

a) Wavefunctions \phi₁ and \phi₂ are solutions to the time independent Schrödinger's equation, that correspond to the energies E₁ and E₂. Now theta=c₁\phi₁+c₂\phi₂ is the solution to the same schrödinger's equation and the energy corresponding to the state \Theta is 1/2(E₁+E₂)

b)The wavefunction that depicts the particles state is \Theta=-i/3\phi₁+2\sqrt{2}/3\phi₂, where \phi₁ and \phi₂ are the wavefunktions from question a). When measuring the energy of one million of these identical particles' 11,1% of them show the energy E₁

Homework Equations

The Attempt at a Solution

 

[Amok]

Use:

\left < E \right > = \langle \Theta \mid \hat{H} \mid \Theta \rangle

and the fact that the probability of getting E₁ is the square of the modulus of the coefficient that accompanies it in the wave-function expansion.

 

[Mountebank]

Thanks. I think I got it. Could you still tell me if the statements are true or false so that i would be certain that i got them right?