Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Sampling Distributions
Let X1, X2, ..., X6 be a random sample from a gamma distribution with the probability density function
\(f(x \mid \lambda)=\left\{\begin{array}{cl} \frac{\lambda^4}{6} e^{-\lambda x} x^3, & \text { if } x>0 \\ 0, & \text { if } x \leq 0 \end{array},\right.\)
where λ > 0 is unknown. Let \(T=\sum_{i=1}^6 X_i\) and ψ be the uniformly most powerful test of size α = 0.05 for testing null hypothesis H0 : λ = 1 against alternative hypothesis H1 : λ > 1. For any positive integer v, let \(\chi_{v, α}^2\) denote the (1 - α)th quantile of \(\chi_v^2\) distribution. Then the test ψ rejects H0 if and only if
1
\(T \geq \frac{1}{2} \chi_{48,0.05}^2\)
2
\(T \leq \frac{1}{2} \chi_{48,0.95}^2\)
3
\(T \geq \frac{1}{2} \chi_{24,0.05}^2\)
4
\(T \leq \frac{1}{2} \chi_{24,0.95}^2\)