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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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