Answer to Question #221228 in Mechanical Engineering for james

Question

A consumer organisation and the manufacturer of a certain brand of margarine are in dispute. The manufacturer states that his packs of margarine contain at least 250g of margarine whereas the consumer organisation claims that the content is at most 249g. A judge has to decide the issue. The judge decides to base her judgement on a random sample of ten packs of margarine.  The consumer organisation will be adjudged to be right if the random sample has a mean weight of less than 249:5g.  The manufacturer will be adjudged to be right if this mean weight is at least 249:5g. 4 STA4820/002/0/2021 The starting point for the judge is that he assumes a normal distribution with mean  and variance 4, that is, n .; 4/ to be a good model for the weight X of a pack of margarine of that brand. Below, X denotes the mean weight (in grams) of the ten packs of margarine.

Answer 1

Since Xʹ_is are independently normally distributed with mean $\mu$ , variance = 4, and $\bar{X}=\frac{\sum^{10}_{i=1}X_i}{10}$ is linear function of Xʹ_is,

$\bar{X}$ ~$N(\mu, \frac{4}{10}=0.4)$

b)$P(\bar{X}<249.5| \mu) = Φ(\frac{249.5- \mu}{\sqrt{0.4}})$

where

$Φ(x) = cdf$ of the standard normal distribution

$\mu=251 \\ P(\bar{X}<249.5) = Φ(\frac{249.5- 251}{\sqrt{0.4}}) \\ = Φ(-2.37) \\ = 1 - Φ(2.37) \\ = 1 -0.9911 \\ =0.0089$