Suppose an entomologist studying lygus hesperus, an agricultural insect pest, measures the disbursal pattern of the insects. She releases 36 marked insects at the edge of a narrow cotton field, and recaptures them after one day, noting the distances they traveled. It is believed that the distance travelled by this insect in one day is normally distributed with mean 100 meters and variance 25 meters?. Let X and S2 denote the sample average and variance for a sample of size 36. The entomologist observes a sample mean of = 98. a) What is the distribution of X? Be sure to specify the parameter(s) of this distribution. b) In another (independent) sample of 36 insects, what is the probability that X would exceed the observed value of 98? c) Find the mean and variance of S2. [Hint: We know the distribution of cS2 for some constant c.]
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Part A of this Question,
In general what are the steps to finding the distribution of the Xbar?
![Suppose an entomologist studying lygus hesperus, an agricultural insect pest,
measures the disbursal pattern of the insects. She releases 36 marked insects at the edge of a narrow
cotton field, and recaptures them after one day, noting the distances they traveled. It is believed that the
distance travelled by this insect in one day is normally distributed with mean 100 meters and variance 25
meters?. Let X and S2 denote the sample average and variance for a sample of size 36. The entomologist
observes a sample mean of = 98.
a) What is the distribution of X? Be sure to specify the parameter(s) of this distribution.
b) In another (independent) sample of 36 insects, what is the probability that X would exceed the
observed value of 98?
c) Find the mean and variance of S2. [Hint: We know the distribution of cS2 for some constant c.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8969faf9-4abf-4770-b454-6ea313902659%2Fe1147201-baef-48c5-8936-17140bb93c25%2Fvkwpu2r.jpeg&w=3840&q=75)
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