For a particular breed of dairy cows, the amount of milk produced by an adult cow
in a single day is well modeled by a 𝑁 (32, 4) distribution, in units of kg.
(a) For what proportion of days do such cows produce less than 23 kg of milk?
(b) Now suppose you follow a sample of 9 such cows for one day. What is the probability that
the average amount of milk produced by these 9 cows is between 29 and 31 kg of milk?
(Hint: use the sampling distribution of the sample mean to find 𝑃 (29 ≤ 𝑋 ≤ 31)). Suppose the amounts of milk (in kg) in your sample of 9 such cows are the following:
24 27 31 31 31 31 33 34 36.
Draw a boxplot of these observations. Label 𝑄1, 𝑄3, and include the details of checking the
IQR criterion for suspected outliers.
(d) Suppose the milk production (in kg) for a particular breed of dairy goats is well modeled by a 𝑁 (4, 0.8) distribution. Which is more atypical: a cow producing 38 kg of milk in a day, or a goat producing 1kg of milk in a day? (Hint: Compare the z-scores).