
Hypothesis Testing
Let X have a binomial distribution with the number of trails n = 10 and with p either 0.25 or 0.5. The simple null hypothesis p = 0.5 is rejected and the alternate hypothesis p = 0.25 is accepted if the observed value of X1, a random sample of size 1, is less than or equal to 3. Find the significance level and the power of the test.
I believe that the power function is P(X<3) = P(Bin(10,0.25) < 3)and the significane leve would be $\displaystyle = \alpha = P_{H_0}(_{10}C_i 0.5^{10}0.5^{10i}<3) = 0.171875$
Is this correct?

You said .... is less than or equal to 3, but you only summed less than
And you don't even say what your i is.
$\displaystyle \alpha=\sum_{i=0}^3 {10\choose i} (.5)^{10}$

$\displaystyle \alpha=\sum_{i=0}^3 {10\choose i} (.5)^{10} = 0.171875$
I just forgot the equal part (and Sigma), but I have included the '3' in my calculation to calculate alpha.
Is my power function P(X<=3) = P(Bin(10,0.25) < =3) correct?