21. If U is uniformly distributed on (0, 1), show that a + (b- a)U is uniform on (a, b). 22. You arrive at a bus stop at 10 o'clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. What is the probability that will have to wait longer than 10 minutes? If at 10:15 the bus has not yet you arrived, what is the probability that you will have to wait at least an additional 10 minutes? 23. If X is a normal random variable with parameters u = 10, o = 36, compute %3D %3D fa P{X > 5}; Ab) P{4 < X < 16}; (c) P{X < 8}; (d) P{X < 20}; (e) P{X > 16}. 24. The Scholastic Aptitude Test mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that (a) all scored below 600 and (b) exactly three of them scored above 640. 25. The annual rainfall (in inches) in a certain region is normally distributed with = 4. What is the probability that in 2 of the next years the u = 40, o rainfall will exceed 50 inches? Assume that the rainfalls in different years are independent. %3D 26. The weekly demand for a product approximately has a normal distribution with mean 1,000 and standard deviation 200. The current on hand inventory is 2,200 and no deliveries will be occurring in the next two weeks. Assuming that the demands in different weeks are independent, (a) what is the probability that the demand in each of the next 2 weeks is less than 1,100? (b) what is the probability that the total of the demands in the next 2 weeks exceeds 2,200? 27. A certain type of lightbulb has an output that is normally distributed with mean 2,000 end foot candles and standard deviation 85 end foot candles. Determine a lower specification limit L so that only 5 percent of the lightbulbs produced will be below this limit. (That is, determine L so that P{X > L} = .95, where X is the output of a bulb.) 28. A manufacturer produces bolts that are specified to be between 1.19 and 1.21 inches in diameter. If its production process results in a bolt's diameter being normally distributed with mean 1.20 inches and standard deviation .005, what percentage of bolts will not meet specifications?

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Question 28

21. If U is uniformly distributed on (0, 1), show that a + (b- a)U is uniform
on (a, b).
22. You arrive at a bus stop at 10 o'clock, knowing that the bus will arrive at some
time uniformly distributed between 10 and 10:30. What is the probability that
will have to wait longer than 10 minutes? If at 10:15 the bus has not yet
you
arrived, what is the probability that you will have to wait at least an additional
10 minutes?
23. If X is a normal random variable with parameters u = 10, o = 36, compute
%3D
%3D
fa P{X > 5};
Ab) P{4 < X < 16};
(c) P{X < 8};
(d) P{X < 20};
(e) P{X > 16}.
24. The Scholastic Aptitude Test mathematics test scores across the population of
high school seniors follow a normal distribution with mean 500 and standard
deviation 100. If five seniors are randomly chosen, find the probability that
(a) all scored below 600 and (b) exactly three of them scored above 640.
25. The annual rainfall (in inches) in a certain region is normally distributed with
= 4. What is the probability that in 2 of the next
years the
u = 40, o
rainfall will exceed 50 inches? Assume that the rainfalls in different years are
independent.
%3D
26. The weekly demand for a product approximately has a normal distribution with
mean 1,000 and standard deviation 200. The current on hand inventory is 2,200
and no deliveries will be occurring in the next two weeks. Assuming that the
demands in different weeks are independent,
(a) what is the probability that the demand in each of the next 2 weeks is less
than 1,100?
(b) what is the probability that the total of the demands in the next 2 weeks
exceeds 2,200?
27. A certain type of lightbulb has an output that is normally distributed with mean
2,000 end foot candles and standard deviation 85 end foot candles. Determine a
lower specification limit L so that only 5 percent of the lightbulbs produced will
be below this limit. (That is, determine L so that P{X > L} = .95, where X is
the
output
of a bulb.)
28. A manufacturer produces bolts that are specified to be between 1.19 and
1.21 inches in diameter. If its production process results in a bolt's diameter
being normally distributed with mean 1.20 inches and standard deviation .005,
what
percentage of bolts will not meet specifications?
Transcribed Image Text:21. If U is uniformly distributed on (0, 1), show that a + (b- a)U is uniform on (a, b). 22. You arrive at a bus stop at 10 o'clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. What is the probability that will have to wait longer than 10 minutes? If at 10:15 the bus has not yet you arrived, what is the probability that you will have to wait at least an additional 10 minutes? 23. If X is a normal random variable with parameters u = 10, o = 36, compute %3D %3D fa P{X > 5}; Ab) P{4 < X < 16}; (c) P{X < 8}; (d) P{X < 20}; (e) P{X > 16}. 24. The Scholastic Aptitude Test mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that (a) all scored below 600 and (b) exactly three of them scored above 640. 25. The annual rainfall (in inches) in a certain region is normally distributed with = 4. What is the probability that in 2 of the next years the u = 40, o rainfall will exceed 50 inches? Assume that the rainfalls in different years are independent. %3D 26. The weekly demand for a product approximately has a normal distribution with mean 1,000 and standard deviation 200. The current on hand inventory is 2,200 and no deliveries will be occurring in the next two weeks. Assuming that the demands in different weeks are independent, (a) what is the probability that the demand in each of the next 2 weeks is less than 1,100? (b) what is the probability that the total of the demands in the next 2 weeks exceeds 2,200? 27. A certain type of lightbulb has an output that is normally distributed with mean 2,000 end foot candles and standard deviation 85 end foot candles. Determine a lower specification limit L so that only 5 percent of the lightbulbs produced will be below this limit. (That is, determine L so that P{X > L} = .95, where X is the output of a bulb.) 28. A manufacturer produces bolts that are specified to be between 1.19 and 1.21 inches in diameter. If its production process results in a bolt's diameter being normally distributed with mean 1.20 inches and standard deviation .005, what percentage of bolts will not meet specifications?
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