Class examples
I.
(Review for the test 2)
STA 2023
The length of time to complete a door assembly on an automobile factory assembly line is
distributed with mean µ = 7.0 minutes and standard deviation  = 5.0 minutes. What is the
probability that the mean assembly time for a random sample of 100 doors will be:
o At least 6 minutes
o 8 minutes or less
o Between 6.0 and 6.5 minutes
o Between 4.5 and 8 minutes
II.
The length of time to complete a door assembly on an automobile factory assembly line is
normally distributed with mean µ = 7 minutes and standard deviation  = 2.0 minutes. For a
door selected at random, what is the probability the assembly time will be:
o At most 5 minutes
o 10 minutes or more
o Between 4.5 and 6.5 minutes
o Between 6 and 9 minutes
III.
In analyzing hits by V-1 buzz bombs in World War II, Moscow was subdivided into 600 regions,
each with an area of 0.35 km2. A total number of 1200 bombs hit the combined area of 600 regions.
(Use a Poisson Distribution)
1) What is the average number of bombs that hit the randomly selected region?
2)
What is the corresponding variance?
3)
If a region is randomly selected, what is the probability that it was not hit?
III. A small life insurance company has determined that on the average it receives 6 death claims per day.
Find the probability that the company receives at least seven death claims on a randomly selected day. (Use a
Poisson Distribution)
Answer:
P(x ≥ 7) = 1 - P(x ≤ 6) = 0.393697
IV.
Consider the discrete probability distribution when answering the following question.
X
P (x)
V.




1
.15
2
?
4
.35
8
.10

Find the probability that X is equal 2.

Calculate the mean and variance for this distribution.

Find the probability that X is at most 4.

Find the probability that µ + ó < X < µ + 2 ó.
Current estimates suggest that only 60% of the home-based computers have access to on-line
services. Suppose 25 people with home-based computers were randomly and independently
sampled.
Find the probability that more than six but fewer than 9 currently have access to on-line services.
Find the probability that fewer than quarter of those sampled currently have no access to on-line
services.
Find the probability that at least 18 currently have access to on-line services.
How many people do you expect currently do not have access to on-line services?
VI. According to a recent study, 1 in every 5 women has been a victim of domestic abuse at some point
in their lives. Suppose we have randomly and independently sampled twenty women and asked whether
they have been a victim of domestic abuse at some point in their lives.

Find the probability that at most 6 of the sampled women have been the victim of domestic abuse at
some point in their lives.

Find the probability that less than 15 of the sampled women have not been the victim of domestic abuse
at some point in their lives.

How many of the 20 women do we expect have not been the victim of domestic abuse?
VII. Studies for Jet Printers show the lifetime of the printer follows a normal distribution with mean  = 5
years and standard deviation  = 0.8 years. The company will replace any printer that fails during the
guarantee period.

How long should printers be guaranteed if the company wishes to replace no more than 15% of the
printers?

What fractions of these printers will fail before 3 years?
.

If the engineering specifications are 4.0  1.5 years, what is the probability that a randomly selected
jet printer will not meet specifications?

Find the point in the lifetime distribution, which 25% of the printers will not exceed?
VIII. The on-line access computer service industry is growing at an extraordinary rate. Current estimates
suggest that only 45% of the home-based computers have access to on-line services. This number is expected
to grow quickly over the next five years. Suppose 15 people with home-based computers were randomly and
independently sampled. Find the probability that more than two of those sampled currently have access to online services.
n = 15
[the total number of people with computers]
p = .45
q = .55
[the probability that they have access]
[the probability that they do not have access]
Answer:
Formula:
P(x > 2) = 1 - [P(x = 0) + P(x = 1) + P(x = 2)] = 0.989349
15!
P(x = 0)
(.450)(.5515) = 0.00013
=
0!(15)!
15!
P(x = 1) =
(.451)(.5515-1) =
0.00156
(.452)(.5515-2) =
0.00896
1!(15-1)!
15!
P(x = 2) =
2!(15-2)!
15!
(.453)(.5515-3) =
P(x = 3) =
0.031768
3!(15-3)!
Calculations: P(x > 2) = 1 - 0.016065 = 0.989349
IX. As part of a quality control program at a factory, random samples of 36 6-ounce cans of juice are taken, and the
contents carefully measured. When the manufacturing process is working properly, the cans are filled with an
average of 6.04 oz of juice. The standard deviation is 0.02 oz. When the process is working properly, what is the
probability a sample of 36 of the 6-oz cans of juice would contain an average of 6.03 oz or less?
X. The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as
possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of .2 ounces. Suppose 100
bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean
weight of these 100 bags exceeded 10.45 ounces.
XI. An airplane with room for 100 passengers has a total baggage limit of 6,000 lbs. Suppose that the total weight
of the baggage checked by an individual passenger is randomly distributed with a mean of 50 lbs and a standard
deviation of 20 lbs. If 100 passengers board a flight, what is the approximate probability that the total
weight of their baggage will exceed the limit?
XII. A new surgical procedure is said to be successful 80% of the time. Suppose the procedure is performed 10
times.
a. Find the expected number of successes in 10 trials.
b. Find the standard deviation.
c. Find the probability that at most 2 of the surgeries are failures.
XIII. The number of tomatoes processed daily at a plant have a bell-shaped distribution with mean of 11.2 million
and a standard deviation of 2.3 million. The tomatoes are then peeled by some of the peeling machines. Each day
the plant operates between 6 and 10 peeling machines with probabilities given in the table below. (The number
of peeling machines operated daily is represented by the variable x.)
x
p(x)
6 7 8 9 10
.1 .2 .3 .3 .1
a. Find the mean and standard deviation of the distribution of the number of peeling machines operated each day.
b. Find the probability that on any given day 7 or less peeling machines are in operation.