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Consider a random sample (X_1, X_2, \ldots, X_n) drawn independently from a normal distribution with unknown mean (\mu) and known variance (\sigma^2 = 16). The probability density function of each observation is given by [ f(x_i;\mu) = \frac{1}{\sqrt{2\pi \cdot 16}} \exp\left(-\frac{(x_i - \mu)^2}{2 \cdot 16}\right). ] The joint likelihood function based on the sample is [ L(\mu) = \prod_{i=1}^{n} f(x_i;\mu) = (2\pi \cdot 16)^{-n/2} \exp\left(-\frac{1}{2 \cdot 16} \sum_{i=1}^{n}(x_i - \mu)^2 \right). ] To construct the likelihood ratio test for testing the hypotheses [ H_0: \mu = 500 \quad \text{against} \quad H_1: \mu \ne 500, ] we define the likelihood ratio statistic as [ \Lambda = \frac{L(\mu = 500)}{L(\hat{\mu})}, ] ________ ____ ____ _________ _________ ____ ________ _____ _______ _________ _____ _________.
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