KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STA 253: Probability and Statistics
(Aero III, Mech III, Industrial III, Geomatic III)
Worksheet 2
First semester (2023/2024)
Instructions: Answer ALL Questions. Make sure you include the name of group
and the INDEX NUMBERS of all members who participated in the solutions.
1. Upon studying low bids for shipping contracts, a microcomputer manufacturing company
finds that intrastate contracts have low bids that are uniformly distributed between 20 and
25, in units of thousands of dollars. Find the probability that the low bid on the next
intrastate shipping contract
(a) is below $22, 000.
(b) is in excess of $24, 000.
2. Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs
to fix a furnace is uniformly distributed between 1.5 and four hours. Let X = the time needed
to fix a furnace. Then X ∼ U (1.5, 4).
(a) Find the probability that a randomly selected furnace repair requires more than two
hours.
(b) Find the 30th percentile of furnace repair times.
(c) The longest 25% of furnace repair times take at least how long? (In other words: find the
minimum time for the longest 25% of repair times.) What percentile does this represent?
(d) Find the mean and standard deviation
3. Suppose that an average of 30 customers per hour arrive at a store and the time between
arrivals is exponentially distributed.
(a) On average, how many minutes elapse between two successive arrivals?
(b) After a customer arrives, find the probability that it takes more than five minutes for
the next customer to arrive.
(c) After a customer arrives, find the probability that it takes less than one minute for the
next customer to arrive.
(d) Seventy percent of the customers arrive within how many minutes of the previous customer?
4. Suppose that X has a Weibull distribution with scale parameter δ = 8.6 and shape parameter
β = 2. Determine the following:
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a) P (X < 10)
b) P (X > 9)
c) P (8 < X < 11)
d) Value for x such that P (X > x) = 0.9
5. In a certain city, the daily consumption of electric power, in millions of kilowatt-hours, is a
random variable X having a gamma distribution with mean µ = 6 and variance σ 2 = 12.
(a) Find the values of α and λ.
(b) Find the probability that on any given day the daily power consumption will exceed 12
million kilowatt- hours.
6. One method of arriving at economic forecasts is to use a consensus approach. A forecast is
obtained from each of a large number of analysts; the average of these individual forecasts
is the consensus forecast. Suppose that the individual 1996 January prime interest–rate
forecasts of all economic analysts are approximately normally distributed with mean7% and
standard deviation 2.6%. If a single analyst is randomly selected from among this group,
what is the probability that the analyst’s forecast of the prime interest rate will
(a) Exceed 11%?
(b) Be less than 9%?
7. The finished inside diameter of a piston ring is normally distributed with a mean of 10
centimeters and a standard deviation of 0.03 centimeter.
(a) What proportion of rings will have inside diameters exceeding 10.075 centimeters?
(b) What is the probability that a piston ring will have an inside diameter between 9.97 and
10.03 centimeters?
(c) Below what value of inside diameter will 15% of the piston rings fall?
8. Suppose that the survival time, in weeks, of an animal subjected to a certain exposure of
gamma radiation in a biomedical research is determined to have a gamma distribution with
α = 5 and λ = 1/10.
(a) What is the mean survival time of a randomly selected animal of the type used in the
experiment?
(b) What is the standard deviation of survival time?
(c) What is the probability that an animal survives more than 30 weeks?
9. The number of miles that a particular car can run before its battery wears out is exponentially
distributed with an average of 10,000 miles. The owner of the car needs to take a 5000-mile
trip. What is the probability that he will be able to complete the trip without having to
replace the car battery?
10. The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution
with δ = 0.986 and β = 50.
(a) Find the probability that the bit will fail before 10 hours of usage.
(b) What is the mean failure time?
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