Question

Let X1, X2, X3, X4 be a random sample from a distribution with the probability mass function
$f(x) = \begin{cases} \theta^x(1-\theta)^{1-x}, & x=0,1 \\\ 0, & \text{otherwise}, \end{cases}$
where θ ∈ (0, 1) is unknown. Let 0 < α ≤ 1. To test the hypothesis $H_0:\theta=\frac{1}{2}$ against $H_1:\theta>\frac{1}{2},$, consider the size α test that rejects H0 if and only if $\sum^4_{i=1} 𝑋𝑖 ≥ k_α$, for some 𝑘α ∈ {0, 1, 2, 3, 4}. Then for which one of the following values of α, the size α test does NOT exist ?

• A. $\frac{1}{16}$
• B. $\frac{1}{4}$
• C. $\frac{11}{16}$
• D. $\frac{5}{16}$

Answer 1

Answer: (B)

$\frac{1}{4}$

Answer 2

The correct option is (B) : $\frac{1}{4}$.