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 ?
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 ?
\(f(x) = \begin{cases} \theta^x(1-\theta)^{1-x}, & x=0,1 \\ 0, & \text{otherwise}, \end{cases}\)
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 ?
Updated On: Oct 1, 2024
- \(\frac{1}{16}\)
- \(\frac{1}{4}\)
- \(\frac{11}{16}\)
- \(\frac{5}{16}\)
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The Correct Option is B
Solution and Explanation
The correct option is (B) : \(\frac{1}{4}\).
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