Let X₁, X₂ be a random sample from a population having probability density function f ∈ { f₀, f₁} where 

For testing the null hypothesis H₀ : f = f₀ against the alternate hypothesis H₁ : f = f₁, the power of a most powerful test of size α = 0.05 is equal to  

Concept:

The likelihood ratio for two independent random variables  and  is
   

   
where  and  are the densities under  and  , respectively.

Explanation:

  
We are testing the null hypothesis  against the alternative hypothesis  .

Steps to calculate the power of the test

The likelihood ratio for two independent random variables  and  is
   

The most powerful test for a given size  will reject  when the likelihood ratio 

is small enough. The rejection region is determined by solving
   
  which gives the critical value for the likelihood ratio.

The power of the test is the probability of rejecting  when  is true, which is given by
   
 

This requires integrating the density under  over the rejection region.

From the calculation (either analytically or using computational tools),

the power of the test is determined to be 0.7625

Thus, the correct answer is Option 3).