Question:
Problem 5.60: In previous semesters, during the last 1 ½ hours of a 3-hour final test in statistics, examinations were turned in according to a Poisson distribution, with an average of 1.8 tests per 5-minute interval.

a.) What is the probability that exactly four tests will be turned in during a 5-minute interval?
b.) What is the probability that nine or more tests will be turned in during a 15-minute interval?
c.) What is the probability that no one will turn in an examination during a 10-minute interval? During a 5-minute interval?
d.) What is the expected number of examinations to be turned in during a 5-minute interval? Sketch the graph of this distribution and note the expected number. What is the value with the highest probability of occurrence during a 5-minute interval?

A.:
What is the probability that exactly four tests will be turned in during a 5-minute interval?

Solution / Formula:
p(4;1.8) = (1.8)4 (2.71828) -1.8 / 4! = (10.4976) (.165299) / 24 = .0723

Answer:

P (X=4) = 0.723 or 72.30%

B:
What is the probability that nine or more tests will be turned in during a 15-minute interval?

Solution / Formula:


(summation of all probabilities 9 and above)

Answer:

P (X>=9) = 0.09735 or 9.74%

C:
What is the probability that no one will turn in an examination during a 10-minute interval?
During a 5-minute interval?

Solution / Formula:


p(0;3.6) = (3.6) 0 (2.71828) -3.6 / 0! = .0273

p(0;1.8) = (1.8) 0 (2.71828) -1.8 / 0! = .1653

Answer:

P (X=0) = 0.0273 or 2.73% (10 min.)

P (X=0) = 0.1673 or 16.73% (5 min.)

D:
What is the expected number of examinations to be turned in during a 5-minute interval? Sketch the graph of this distribution and note the expected number. What is the value with the highest probability of occurrence during a 5-minute interval?

Solution / Formula:

m = 1.8

Answer:


The probability is highest at X = 1.



9 is the expected number of examinations to be turned in during a 5-minute interval.
.2975 or 29.75% or X = 1 is the value with highest probability of occurrence during a 5-minute interval.
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