Question

The mean per capita consumption of milk per year is 131 liters with a variance of 841. If a sample of 132 people is randomly selected, what is the probability that the sample mean would be less than 133.5 liters? Round your answer to four decimal places.

Answer 1

The probability that a sample of size n will have a mean,
`\bar{x}`, less that a given value, X, where the population mean,
`\mu`, and the population standard deviation,
`\sigma`, is known is given by:


`P(\bar{x}\ \textless \ X)=P\left(z\ \textless \ \frac{\bar{x}-\mu}{\sigma/√(n)} \right)`

Therefore, the required probability is given by:


`P(\bar{x}\ \textless \ 133.5)=P\left(z\ \textless \ (133.5-131)/(√(841/132)) \right) \\ \\ =P\left(z\ \textless \ (2.5)/(√(6.3712)) \right)=P\left(z\ \textless \ (2.5)/(2.524) \right) \\ \\ =P(z\ \textless \ 0.99)=0.8390`