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Su07 Math 227
FINAL
Name: ______________________
Show all necessary work neatly, clearly, systematically for full-points. Any wrong statement or
understatement may be penalized. Write the combinatoric notation first if necessary then start the
computation. Be reasonable in rounding.
1.
In a large population, 2% of individuals are infected with a virus V. A test is developed and it
was found that it gives 1% false positive and 2% false negative results.
a. (5) Find the probability that at most 2 of 10 randomly selected people are infected.
(Total 45)
Hint: Don’t think too hard! Use the fact that 2% of individuals are infected.
b.
c.
d.
e.
Construct the probability tree.
(4) Find the probability that a randomly selected individual is tested positive.
(8) Find the probability that a tested positive individual is indeed infected.
(10) Find the probability that, of 10 tested positive individuals, at least 7 of them are actually
infected. Hint: Use part (d).
f. (10) Find the probability that, of 1000 tested positive individuals, at least 750 of them are actually
infected.
(8)
2.
(Total 45) A box
a.
b.
c.
d.
e.
contains 5 quarters, 3 dimes, and 2 nickels.
(6) Seven coins are selected without replacement. Find the probability of selecting 4 or more
quarters.
(5) A coin is drawn randomly 8 times with replacement. Find the probability of getting a dime
twice.
(12) Create a probability distribution of the number of nickels selected for a procedure of selecting 3
coins without replacement.
(10) Extend the probability distribution created in part (c) to compute the expected number of
nickels and its standard deviation.
(12) A box with this configuration is given to each of 100 people. Each person is required to select 3
coins without replacement. Find the probability that the average number of nickels selected by
these 100 people is less than 1.2 nickel.
3.
The number of customers arrive at the Teller Desk of Bank of Nowhere is, on average, 12
customers per hour. Suppose the number of arrivals follows Poisson distribution.
e    k
Note: X ~ Poisson(λ)  P( X  k ) 
k!
(Total 35)
a.
b.
c.
d.
Find the probability that there is less than 3 customers arrive in the next hour.
(7) Find the probability that there are 8 customers in the next 30 minutes.
(7) Find the probability that the next customer arrive after 20 minutes later.
(15) Find the probability that, in a week of 40 working-hours, the average number of customers
arrive at the Teller is less than 11 customers per hour. Hint: the standard deviation is 3.4641 customers per hour. Use CLT.
(6)
4.
The following is the sample of size 40 of a normally distributed Mathematics Assessment Test
results for incoming students in a college. Note that the data are sorted.
(Total 50)
94
56
46
38
a.
(10)
84
56
44
38
78
54
42
38
72
54
42
36
66
50
42
36
64
48
42
36
64
48
40
36
62
48
40
36
58
46
38
34
58
46
38
34
Create a frequency distribution. You should follow either Scott-Tereu Rule or Sturge’s Law.
For a sample of size n,
Scott-Tereu : “number of classes” = 3 2n
Sturge’s
: “number of classes” = 1 + 3.3 ln n.
b.
Expand the frequency distribution to compute the approximate sample mean and approximate
sample standard deviation.
c. (10) Construct a 95%-CI for the population mean.
d. (10) Construct a 95%-CI for the population sample deviation.
e. (10) Suppose that scoring above 50 is considered passing the Assessment Test, construct a 95%-CI
for the proportion of incoming students who pass the Assessment Test.
(10)
5.
(Total 30) An
agent for an international relief organization suspects that the recent shipments of flour bags
(which are supposed to weigh 50 kg each) might amount to less than claimed weight. Twenty five
samples are taken and it is found that the sample mean is 49 kg with sample standard deviation 2.5 kg.
Suppose that the weight of the population of flour bags is normally distributed.
a. (6) Formulate claim, the null and alternate hypothesis.
b. (6) Find the critical value at 5%-SL.
c. (6) Find the test statistics.
d. (6) Find the P-value of the test statistics. Or, at least, the interval of the P-value.
e. (6) State the agent’s conclusion after performing the hypothesis testing.
6.
Thomas’ Statistics class: 14 of 19 students are passing, class average score 62 with s.d. 22.
Stephen’s Statistics class: 16 of 22 students are passing, class average score 64 with s.d. 25.
a. (10) At 5%-SL, are the proportion of passing students in their classes equal?
b. (10) At 5%-SL, are their class averages equal?
(Total 20)