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.

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

[tex]P(\bar{x}\ \textless \ X)=P\left(z\ \textless \ \frac{\bar{x}-\mu}{\sigma/\sqrt{n}} \right)[/tex]

Therefore, the required probability is given by:

[tex]P(\bar{x}\ \textless \ 133.5)=P\left(z\ \textless \ \frac{133.5-131}{\sqrt{841/132}} \right) \\ \\ =P\left(z\ \textless \ \frac{2.5}{\sqrt{6.3712}} \right)=P\left(z\ \textless \ \frac{2.5}{2.524} \right) \\ \\ =P(z\ \textless \ 0.99)=0.8390[/tex]