Answer to Question #332041 in Statistics and Probability for Kokski

Question

The weight of goats at a farm is normally distributed with a mean of 60 kg and a standard deviation of 10 kg. A truck used to transport goats can only accommodate not more than 650 kg. If 10 goats are selected at random from the population, what is the probability that the total weight exceeds the maximum weight?

Answer 1

We have to find the probability that the mean of the weights of the selected at random 10 goats exceeds 65 kg.

We have a normal distribution, 

$μ=60,σ=10,n=10.$

Let's convert it to the standard normal distribution,

$\bar{z}=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}},\\ \bar{z}=\cfrac{65-60}{10/\sqrt{10}}=1.58,\\ P(\bar{X}>65)=P(\bar{Z}>1.58)=\\ =1-P(\bar{Z}<1.58)=\\ =1-0.9429=0.0571\text{ (from z-table).}$