Let X1, X2, ..., X, be a random sample drawn from a distribution with mean u and variance o? and let a1,... , an be real numbers such that a; = 1. Define X %3D i=1 i=1

Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in a national park. Then x has a distribution that is approximately normal with mean ? = 62.0 kg and standard deviation ? = 9.0 kg. Suppose a doe that weighs less than 53 kg is considered undernourished. (c) To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 70 does should be more than 59 kg. If the average weight is less than 59 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight x for a random sample of 70 does is less than 59 kg (assuming a healthy population)? (Round your answer to four decimal places.) Suppose park rangers captured, weighed, and released 70 does in December, and the average weight was x = 63.8 kg. Do you think the doe population is undernourished or not? Explain. A. Since the sample average is above the mean, it…

Find the approximate mean and variance of Y = √X, where X is a randomvariable following a Poisson distribution.

Let x¡, denote a series of n numbers {x;: i=1,2,..,n}: Assume that x; is drawn from a distribution that is N(ux, ož). Show that the sample mean x = E, xị has a variance of ož/n carefully stating any required assumptions i. 1 at each step. ii. Show that the sample mean is an unbiased estimator of ux.