Chapter 7.5, Problem 17P

Wildlife: Deer Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in Mesa Verde National Park. Then x has a distribution that is approximately normal, with mean $\mu =63.0$ kg and standard deviation $\sigma =7.1\text{ kg}$ (Source: The Mule Deer of Mesa Verde National Park, by G. W. Mierau and J. L. Schmidt, Mesa Verde Museum Association). Suppose a doe that weighs less than 54 kg is considered undernourished.

(a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished?

(b) If the park has about 2200 does, what number do you expect to be undernourished in December?

(c) Interpretation To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 50 does should be more than 60 kg. If the average weight is less than 60 kg. it is thought that the entire population of does might be undernourished. What is the probability that the average weight $\bar{x}$ for a random sample of 50 does is less than 60 kg (assume a healthy population)?

(d) Interpretation Compute the probability that $\bar{x}<64.2\text{ kg}$ for 50 does (assume a healthy population). Suppose park rangers captured, weighed, and released 50 does in December, and the average weight was $\bar{x}=64.2$. Do you think the doe population is undernourished? Explain.