Answer to Question #214979 in Statistics and Probability for Madimetja

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.

The starting point for the judge is that he assumes a normal distribution with mean and variance 4, that is (u; 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.

a)Determine the probability distribution of X

b)Calculate the probability that the judge decides (incorrectly) that the consumer organisation is right while the true u is equal to 251.

Answer 1

a) 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

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