Download Some Extra Problems using the Central Limit Theorem

Some Extra Problems using the Central Limit Theorem
Terminal Times
During the last week of the semester, students at a certain college spend on
the average 4.2 hours using the school’s computer terminals with a standard
deviation of 1.8 hours. For a random sample of 36 students at that college,
find the probabilities that the average time spent using the computer
terminals during the last week of the semester is
(a)
at least 4.8 hours;
(b)
between 4.1 and 4.5 hours.
Rusting Cars
A particular make of car is known to show rust when it is 2.4 years old on
the average with a standard deviation of 0.8 year. If a car rental agency
purchases 64 new cars of this kind, what are the probabilities that the
average time it will take for these cars to show rust is
(a)
at most 2.6 years;
(b)
between 1.9 and 2.3 years?
Quota Time
The time it takes salespersons in a certain company to meet their yearly
quotas is 10.6 months on the average with a standard deviation of 1.8
months. Find the probabilities that in a sample of 36 of the company’s
salespersons the average time it will take to meet their yearly quotas is
(a)
at most 11 months;
(b)
between 10.0 and 10.5 months.
3US Suppose x has a distribution with  = 25 and  = 3.5.
a)
If random samples of size n = 9 are selected, can we say anything
about the distribution of sample means?
b)
If the original x distribution is normal, can we say anything about the
distribution from samples of size n = 9? Find P(23  x 26).
7US We have all seen lights flicker momentarily during a storm. At these
times the power went off for an instant, but then came on again. When a
telecommunications computer loses power for an instant, it shuts down and
then reboots itself. During this time important high-speed messages may be
lost. After examining many such cases it was found that the reboot times
had a mound-shaped distribution with mean  = 20.3 msec and standard
deviation  = 6.2 msec.
a)
What is the probability that an average time for 33 reboots will be
less than 19 msec?
b)
What is the probability that an average time for 33 reboots will be
from 19 to 21 msec?
c)
What is the probability that an average of 33 reboot times will be
more than 22 msec?
9US. The Oak Grove College financial aid office did a study showing that
their students spend an average (mean) of $680 in the college bookstore on
books and supplies per year. The standard deviation is $138. If a random
sample of 36 students is surveyed, what is the probability that the mean
amount spent for books and supplies is
a)
less than $600?
b)
more than $700?
c)
between $600 and $700?
13US The diameters of grapefruit in a certain orchard are normally
distributed with mean 4.6 inches and standard deviation 1.3 inches. If a
random sample of ten of these grapefruit are put in a bag and sold in a
grocery store, what is the probability that the mean diameter x will be
a)
larger than 5 in.?
b)
between 4 in. and 5 in.?
c)
smaller than 4 in.?
15US Coal is carried from a mine in West Virginia to a power plant in
New York in hopper cars on a long train. The automatic hopper car loader is
set to put 36 tons of coal in each car. The actual weights of coal loaded into
each car are normally distributed with mean  = 36 tons and standard
deviation  = 0.8 tons.
a)
What is the probability that one car chosen at random will have less
than 35.5 tons of coal?
b)
What is the probability that 20 cars chosen at random will have a
mean load weight of less than 35.5 tons of coal?
c)
Suppose the weight of coal in one car was less than 35.5 tons. Would
that fact make you suspect the loader had slipped out of adjustment?
Suppose the weight of coal in 20 cars selected at random had an average
xbar less than 35.5 tons. Would that fact make you suspect the loader had
slipped out of adjustment? Why?
Tough Problem
A gourmet chef would like to have 50 pounds of shad roe for a special
banquet. Shad roe is sold in “sets” rather than by weight, but the chef knows
that the sets come from a population that has the mean weight  = 0.60 pound
and the standard deviation  = 0.20 pound. How many sets should the chef
order if he wants the probability of reaching the 50 pound objective to be at
least 0.80?
Answers. Have double checked answers, but some inaccuracies may
persist.
Terminal Times (a) 0.0228
Rusting Cars (a) 0.9772
(b) 0.4706
(b) 0.1587
Quota Times (a) 0.9082 (b) 0.3479
3US (a) No. Sample size is too small (b) Yes. The distribution of
sample means will be exactly normal with  x  25; and  x  1.17
P(23  x 26) = 0.7615
7US (a) 0.1151 (b) 0.6271 (c) 0.0571
9US (a) 0.0003 (b) 0.1922 (c) 0.8075
13US (a) 0.1660 (b) 0.7619 (c) 0.0721
15US (a) 0.2643 (b) 0.0026 (c) Yes
Tough Problem Smallest n is 86.