Test Size Calculation for Composite Hypothesis |Binomial(3,θ)|GATE ST 2025 | Problem-63 | RitwikMath

Master **composite hypothesis testing**! For \(X \sim \text{Binomial}(3,\theta)\), test \(H_0: \theta \in [\frac{1}{4},\frac{3}{4}]\) vs \(H_1: \theta \notin [\frac{1}{4},\frac{3}{4}]\) with rejection region \(\phi(X) = 1\) if \(X \in \{0,3\}\).

**Size calculation:**
\[
\alpha = \sup_{\theta \in [\frac{1}{4},\frac{3}{4}]} P(X=0 \text{ or } 3) = \sup_{\theta \in [\frac{1}{4},\frac{3}{4}]} [\theta^3 + (1-\theta)^3]
\]

**Critical analysis:**
- \(f(\theta) = \theta^3 + (1-\theta)^3\), maximum at endpoints \(\theta = \frac{1}{4}, \frac{3}{4}\)
- \(f(\frac{1}{4}) = f(\frac{3}{4}) = \frac{27}{64} + \frac{1}{64} = \frac{28}{64} = 0.4375 \approx 0.44\)
- Interior critical point \(\theta = \frac{1}{2}\) gives \(f(\frac{1}{2}) = 0.25 0.4375\)

**Test size: \(\alpha = 0.44\)**

Perfect for GATE aspirants mastering **supremum over composite null**!

#GATE2025 #GATEStatistics #CompositeHypothesis #TestSize #BinomialTest #HypothesisTesting #GATESTPYQs #PowerFunction #StatisticalTests

Видео Test Size Calculation for Composite Hypothesis |Binomial(3,θ)|GATE ST 2025 | Problem-63 | RitwikMath канала RitwikMath