Question 3 Start on a new page (17 marks) A.

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Question 3 Start on a new page (17 marks)
A. A plane owned by Fiji Link ATR72 has three engines—a central engine and an engine on each
wing. The plane will crash because it were within the occasion that the central engine fails and
one of the two wing engines fails. The probability of disillusionment in the midst of any given
flight is 0.004 for the central engine and 0.007 for each of the wing engines. Anticipating that
the three engines work independently, what is the probability that the plane will crash in the
midst of a flight? (3 marks)
B. Jackson and Alice work at a firm's office as the boss and secretary, respectively. The probability
that Jackson is in the office at any given time during business hours is 0.72, while the
probability that Alice is in the office is 0.4. Given that Alice is in the office, the probability of
Jackson being there is 0.66. Determine the probability that at any given time during office
hours,
i. Both Jackson and Alice are in the office.
ii. Alice is the office given that Jackson is in the office.
iii.At least one of them is in the office.
(2 + 3+ 2 = 7 marks)
C. In a lottery, you have to select a three-digit number such as 123. During the drawing, there are
three bins, each containing balls numbered 1 through 9. One ball is drawn from each bin to form
the three-digit winning number.
i. You purchase one ticket with one three-digit number. What is the probability that you will
win this lottery?
ii. There are many variations of this lottery. The primary variation allows you to win if the
three digits in your number are selected in any order as long as they are the same three digits
as obtained by the lottery agency. For example, if you pick three digits making the number
123, then you will win if 123, 132, 213, 231, and so forth, are drawn. The variations of the
lottery game depend on how many unique digits are in your number. Consider the following
two different versions of this game. Find the probability that you will win this lottery in
each of these two situations.
a. All three digits are unique (e.g., 123)
b. Exactly one of the digits appears twice (e.g., 122 or 121)
(2+ 2 + 3 = 7 marks)

3
Question 4 Start on a new page (11 marks)
A. The number of masks sold per day at a retail store is shown in the table below, with the
corresponding probabilities. Find the average of the distribution. If the owner of the retail store
wants to be sure that he has enough masks for the next 7 days, how many should the owner
purchase?
Number of masks sold X 18 19 20 21 22
Probability P(X) 0.1 0.2 0.3 0.3 0.1
(3 marks)
B. The average bus fare of a student to travel to USP daily is $5.31. If the distribution of bus fares
is approximately normal with a standard deviation of $0.31, what is the probability that a
randomly selected bus fare is less than $4.50? (4 marks)
C. The average monthly salary of staffs at USP is $4164 in a recent year. If the salaries are
normally distributed with a standard deviation of $360, find the probability that the mean salary
for a random sample of 20 staffs is less than $3900. (4 marks)
Question 5 Start on a new page (20 marks)
A. A survey found that out of 150 citizens, 108 said they have received the first doze of Covid-19
injection. Find the 98% confidence interval of the population proportion of citizens who have
received the first doze of Covid-19 injection. (4 marks)
B. A bakery shop owner wishes to find the 90% confidence interval of the true mean cost of a large
fruit cake. How large should the sample be if he wishes to be accurate to within $0.12? A
previous study showed that the standard deviation of the price was $0.25. (3 marks)
C. The average amount of time a person exercises daily is 22.7 minutes in a population. A random
sample of 20 people showed an average of 29.8 minutes in time with a standard deviation of 9.8
minute. At
? ? 0.01,
can it be concluded that the average differs from the population average?
(6 marks)
D. The average household income for a recent year in Fiji was $30,000. Five years earlier the
average household income was $24,500. Assume sample sizes of 34 were used and the
population standard deviations of both samples were $5928. At 5% level of significance is there
enough evidence to believe that the average household income has increased? (7 marks)
THE EN

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