Use R to answer the following questions. Set the seed (order) of your randomizations as 5065.

a) Generate 10,000 random values from the Poisson distribution with mean 8, using the command rpois(10000, 8). Draw a density histogram and in words, briefly describe the shape of the distribution of your values.

b) Generate a random sample of size 16 from the Poisson distribution with mean 8. Find the mean of the sample.

c) Repeat part (b) 9,999 additional times. For your collection of 10,000 sample means,

i) plot a density histogram (using at least 20 breaks/groups) and describe its overall shape.

ii) find the mean

iii) find the standard deviation

iv) theoretically, what do you expect the mean and standard deviation of parts (ii) and (iii) to be?

d) Suppose you were repeat parts (b) and using samples of size 100 instead of size 16, describe in words, how the results will compare. (it is optional to use actual random generations in R to answer this part.)