a city needs
to
perform some road maintenance and will rent excavator m
achines from a
company. Each excavator will work at least 1 hour, but no more than 4 hours at a time.
It is known that
the working time of each excavator is evenly distributed.
a)
What is the distribution of one excavator’s working time?
b)
What is the probabi
lity that an excavator works more than 3.5 hours?
c)
Given that an excavator has already worked for 2.5 hours, what’s the probability that it
works
at least
1
more
hour
until it is given a
rest?
d)
What amount of time represents the highest 30% of excavator working time?
e)
The rental fee of each excavator is 200 dollars per hour, if the city rents 3 excavators, what is the expected rental fee
per day?
At a cell phone battery plant, 5% of cell phone batteries produced are defective. A quality control engineer randomly collects a sample of 50 batteries from a large shipment from this plant and inspects them from defects. Find the probability that A. None of the batteries are defective
a) The distribution of one excavator's working time can be modeled as a continuous uniform distribution between 1 and 4 hours. This means that any duration between 1 and 4 hours is equally likely.
b) To find the probability that an excavator works more than 3.5 hours, we need to calculate the area under the probability density function (PDF) curve for the excavator's working time from 3.5 to 4 hours. Since it is a continuous uniform distribution, the PDF is a constant value of 1/3 (1 divided by the range of 4 - 1) over the interval of 1 to 4. The probability can be calculated as the area of the shaded region under the PDF curve:
Probability = Area under the curve between 3.5 and 4
= (4 - 3.5) * (1/3)
= 0.5/3
= 1/6
≈ 0.1667
So, the probability that an excavator works more than 3.5 hours is approximately 0.1667 or 16.67%.
c) Given that an excavator has already worked for 2.5 hours, we need to find the probability that it works at least 1 more hour until it is given a rest. Since we know that the working time of each excavator is evenly distributed, we can consider the remaining possible working time as uniformly distributed between 0.5 and 1.5 hours (4 - 2.5 = 1.5 - 0.5).
The probability can be calculated as:
Probability = (1.5 - 0.5) / (4 - 1)
= 1 / 3
≈ 0.3333
So, the probability that an excavator works at least 1 more hour is approximately 0.3333 or 33.33%.
d) To find the amount of time representing the highest 30% of excavator working time, we need to determine the value at which the cumulative distribution function (CDF) equals 0.7 (1 - 0.3).
Since it is a continuous uniform distribution, the CDF increases linearly from 0 to 1 over the range of 1 to 4 hours. The value we are looking for is the one that corresponds to a cumulative probability of 0.7, which can be calculated as:
Value = CDF * Range + Minimum value
= 0.7 * (4 - 1) + 1
= 0.7 * 3 + 1
= 2.1 + 1
= 3.1
So, the amount of time representing the highest 30% of excavator working time is 3.1 hours.
e) If the rental fee of each excavator is $200 per hour and the city rents 3 excavators, we need to find the expected rental fee per day (assuming the excavators work for a full day).
Since each excavator can work between 1 and 4 hours at a time, and their working time is evenly distributed, the average working hours for each excavator can be calculated as the mean of the interval 1 to 4:
Average working hours per excavator = (1 + 4) / 2 = 5 / 2 = 2.5 hours
The expected rental fee per day for each excavator can be calculated as:
Expected rental fee per excavator per day = Average working hours * Rental fee per hour
= 2.5 * $200
= $500
Since the city rents 3 excavators, the total expected rental fee per day would be:
Total expected rental fee per day = Number of excavators * Expected rental fee per excavator per day
= 3 * $500
= $1500
Therefore, the expected rental fee per day for the city is $1500.