### Problem 4 RStudio a)

A certain strain of bacteria occurs in all raw milk. Let x be the bacteria count per milliliter of milk. The health department has found that if the milk is not contaminated, then x has a distribution that is more or less mound-shaped and symmetric. The mean of the x-distribution is ${\bf \mu}=2500$, and the standard deviation is ${\bf \sigma}=300$. In a large comercial dairy, the health inspector takes 42 random samples of the milk produced each day. At the end of the day, the bacteria count in each of the 42 samples is averaged to obtain the sample mean bacteria count ${\bar x}$.

a) Assuming the milk is not contaminated, and keeping in mind the conditions of the Central limit theorem, what is the distribution of ${\bar x}$?

b) Assming the milk is not contaminated, what is the probability that the average bacteria count ${\bar x}$ for one day is between 2350 and 2650 bacteria per milliliter?

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