3. (24\%) Let \( X \) have the pmf \( f(x ; \theta)=\theta^{x}(1-\theta)^{1-x}, x=0,1 \), zero elsewhere. We test \( H_{0}: \theta=1 / 2 \) against \( H_{1}: \theta<1 / 2 \) by taking an iid sample \( X_{1}, \ldots, X_{5} \) of size \( n=5 \) and rejecting \( H_{0} \) if \( Y=\sum_{i=1}^{5} X_{i} \) is observed to be less than or equal to a constant \( c \). (a) Show that this is a UMP test. (b) Find the significance level when \( c=1 \).
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