Normal Distribution
Sample Problems
1.
2.
3.
4.
5.
6.
7.
8.
Determine the following standard normal probabilities
a.
P (Z≤1.25)
b.
P (Z>1.25)
c.
P (Z≤-1.25)
d.
P (-0.38≤Z≤1.25)
e.
P (Z≤5)
Find the percentile of the standard normal probabilities
a.
99th
b.
67th
c.
95th
d.
5th
In each case, determine the value of the constant c that makes the probability statement correct.
a.
Φ(c) = 0.9838
b.
P (0 ≤ Z ≤ c) = 0.291
c.
P (c ≤ Z) = 0.121
d.
P (-c ≤ Z ≤ c) =0.668
Determine the 𝑍𝛼 for the following values of α
a.
α = 0.0055
b.
α = 0.09
c.
α = 0.663
The time that it takes a driver to react to the brake lights on a decelerating vehicle is critical in
helping to avoid rear-end collisions. The article “Fast-Rise Brake Lamp as a Collision-Prevention
Device” Ergonomics, 1993: 391–395) suggests that reaction time for an in-traffic response to a
brake signal from standard brake lights can be modeled with a normal distribution having mean
value 1.25 sec and standard deviation of .46 sec. What is the probability that reaction time is
between 1.00 sec and 1.75 sec? if we view 2 sec as a critically long reaction time, the probability
that actual reaction time will exceed this value is?
The breakdown voltage of a randomly chosen diode of a particular type is known to be normally
distributed. What is the probability that a diode’s breakdown voltage is within:
a.
1 standard deviation of its mean value?
b.
2 standard deviations of its mean value?
c.
3 standard deviations of its mean value?
Mopeds (small motorcycles with an engine capacity below 50 𝑐𝑚3) are very popular in Europe
because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the
Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (J.
of Automobile Engr., 2008: 1615–1623) described a rolling bench test for determining maximum
vehicle speed. A normal distribution with mean value 46.8 km/h and standard deviation 1.75
km/h is postulated. Consider randomly selecting a single such moped.
a.
What is the probability that maximum speed is at most 50 km/h?
b.
What is the probability that maximum speed is at least 48 km/h?
c.
What is the probability that maximum speed differs from the mean value by at most
1.5 standard deviations?
In a road-paving process, asphalt mix is delivered to the hopper of the paver by trucks that haul
the material from the batching plant. The article “Modeling of Simultaneously Continuous and
Stochastic Construction Activities for Simulation” (J. of Construction Engr. and
Mgmnt.,2013:1037–1045) proposed a normal distribution with mean value 8.46 min and
standard deviation .913 min for the rv X 5 truck haul time.
a.
What is the probability that haul time will be at least 10 min?
b.
What is the probability that haul time will be between 8 and 10 min?
Normal Distribution
Sample Problems
1.
2.
3.
4.
5.
6.
7.
8.
Determine the following standard normal probabilities
a.
P (Z≤1.25)
b.
P (Z>1.25)
c.
P (Z≤-1.25)
d.
P (-0.38≤Z≤1.25)
e.
P (Z≤5)
Find the percentile of the standard normal probabilities
a.
99th
b.
67th
c.
95th
d.
5th
In each case, determine the value of the constant c that makes the probability statement correct.
a.
Φ(c) = 0.9838
b.
P (0 ≤ Z ≤ c) = 0.291
c.
P (c ≤ Z) = 0.121
d.
P (-c ≤ Z ≤ c) =0.668
Determine the 𝑍𝛼 for the following values of α
a.
α = 0.0055
b.
α = 0.09
c.
α = 0.663
The time that it takes a driver to react to the brake lights on a decelerating vehicle is critical in
helping to avoid rear-end collisions. The article “Fast-Rise Brake Lamp as a Collision-Prevention
Device” Ergonomics, 1993: 391–395) suggests that reaction time for an in-traffic response to a
brake signal from standard brake lights can be modeled with a normal distribution having mean
value 1.25 sec and standard deviation of .46 sec. What is the probability that reaction time is
between 1.00 sec and 1.75 sec? if we view 2 sec as a critically long reaction time, the probability
that actual reaction time will exceed this value is?
The breakdown voltage of a randomly chosen diode of a particular type is known to be normally
distributed. What is the probability that a diode’s breakdown voltage is within:
a.
1 standard deviation of its mean value?
b.
2 standard deviations of its mean value?
c.
3 standard deviations of its mean value?
Mopeds (small motorcycles with an engine capacity below 50 𝑐𝑚3) are very popular in Europe
because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the
Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (J.
of Automobile Engr., 2008: 1615–1623) described a rolling bench test for determining maximum
vehicle speed. A normal distribution with mean value 46.8 km/h and standard deviation 1.75
km/h is postulated. Consider randomly selecting a single such moped.
a.
What is the probability that maximum speed is at most 50 km/h?
b.
What is the probability that maximum speed is at least 48 km/h?
c.
What is the probability that maximum speed differs from the mean value by at most
1.5 standard deviations?
In a road-paving process, asphalt mix is delivered to the hopper of the paver by trucks that haul
the material from the batching plant. The article “Modeling of Simultaneously Continuous and
Stochastic Construction Activities for Simulation” (J. of Construction Engr. and
Mgmnt.,2013:1037–1045) proposed a normal distribution with mean value 8.46 min and
standard deviation .913 min for the rv X 5 truck haul time.
a.
What is the probability that haul time will be at least 10 min?
b.
What is the probability that haul time will be between 8 and 10 min?