IGNOU MST 21 2026 - A pharmaceutical company fills medicine bottles with a target weight o

Question

A pharmaceutical company fills medicine bottles with a target weight of 500 mg. It is known that the filling weights are normally distributed with known variance o² = 16. Derive the likelihood ratio test for testing

Ηο: μ = 500againstH₁: μ ≠ 500

at significance level a.

18 Mar 2026

Answer :

Word Count : 545

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})}, ] where (\hat{\mu}) is the maximum likelihood estimator of (\mu). For a normal distribution with known variance, the MLE of (\mu) is the sample mean (\bar{X} = \frac{1}{n}\sum_{i=1}^{n}X_i). Substituting into the likelihood function, we have [ L(500) = (2\pi \cdot 16)^{-n/2} \exp\left(-\frac{1}{32} \sum_{i=1}^{n}(x_i - 500)^2 \right), ] and [ L(\hat{\mu}) = (2\pi \cdot 16)^{-n/2} \exp\left(-\frac{1}{32} \sum_{i=1}^{n}(x_i - \bar{X})^2 \right). ] Thus, the likelihood ratio becomes [ \Lambda = \exp\left{-\frac{1}{32} \left[ \sum_{i=1}^{n}(x_i - 500)^2 - \sum_{i=1}^{n}(x_i - \bar{X})^2 \right]\right}. ] Using the identity [

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