Problems: The Central Limit Theorem Part
2
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to Part 1
11. A sample of size n=49 is taken from a population with
a mean of µ = 100 and a standard deviation of σ =
14. Find the probability that the sample mean will lie
between 96 and 104.
12. Suppose that the time it takes students at Gauss
Normal School to walk from the campus to the movie theater
is normally distributed with a mean equal to 15 minutes and
a standard deviation of 3 minutes.
- a) If we pick a student at random and have him make
this walk, what is the probability that it will take him
less than 19 minutes?
- b) If we take a random sample of nine students, what
is the probability that the total time spent walking will
exceed 144 minutes?
13. Suppose that it is an established fact when
19-year-old males throw a baseball as far as they can, the
results form a normal distribution with a mean of 180 feet
and a standard deviation of 25 feet.
- a. Suppose one nineteen-year-old male is chosen at
random. What is the probability that he will throw over
190 feet?
b. Suppose that a sample of 25 nineteen-year-old males is
chosen at random. What is the probability that the
average throw of these 25 will be greater than 190
feet?
c. Suppose repeated random samplings of 100 19-year-old
males are taken and the means of their throws computed.
What percentage of the sample means would be greater than
190 feet?
14. Suppose it is established through comprehensive and
exhaustive studies that when 50-to-60-year-old men run a 100
yards, their times average 24 seconds with a standard
deviation of 5 seconds. Assume further that the distribution
is normal.
- a) What percentage of these men could run 100 yards
in 20 seconds or less?
b) Suppose random samples of 16 men were tested and their
run times averaged. What percentage of these averages
would be 20 seconds or less?
c) Between what two times would we find the middle 95% of
run times when we are looking at individual runners from
this population of old guys?
d) Between what two times would we find the middle 95% of
all sample means when the sample size is 100?
15. Suppose that a credit card company finds, using its
computer records, that the average customer owes $750 with a
standard deviation of $250. If a random sample of 225
customers is taken, what is the probability of getting a
sample mean that is more than $25 away from the true
mean?
16. An average Quark battery lasts 20 hours with a
standard deviation of 30 minutes.
- a) Assuming a normal distribution, what percentage of
the batteries will last more than 20 hours and 15
minutes?
- b) If a sample of 16 is taken, what is the
probability that the average lifetime will be greater
than 20 hours and 15 minutes?
17. A sugar refiner produces bags of sugar that average
100 pounds with a standard deviation of .5 pounds. The
distribution of bags is normally distributed.
- a) What is the probability that one bag taken at
random will weigh less than 99.6 pounds?
- b) What is the probability that one bag taken at
random will weigh 100.7 pounds or more?
- c) What percentage of bags will weigh between 99.4
and 100.6 pounds?
- c) How heavy must a bag be so that 90% of bags weigh
less?
- d) If a random sample of four bags is taken, what is
the probability that the average weight will be less than
99.5 pounds?
If a random sample of nine bags is taken form this
population, what is the probability that the sample average
will be:
- a) 99.5 pounds or less?
b) 100.3 pounds or more?
c) between 99.9 and 100.1 pounds?
d) How heavy must the sample average be so that fewer
than 10% of sample averages would fall above it?
18. If we flip a coin and consider the number of heads we
get, we have the following probability distribution:
Compute the expected value and standard deviation of this
distribution.
b) If we flip a coin twice and compute the fraction of
the total that are heads, we get the following probability
distribution:
Compute the expected value and standard deviation of this
distribution.
Flipping the coin three times gives the following
probability distribution:
|
x
|
P(x)
|
|
0
|
1/8
|
|
1/3
|
3/8
|
|
2/3
|
3/8
|
|
1
|
1/8
|
Again, compute the expected value and standard deviation
of this distribution.
Flipping the coin four times give the following:
|
x
|
P(x)
|
|
0
|
1/16
|
|
.25
|
4/16
|
|
.5
|
6/16
|
|
.75
|
4/16
|
|
1
|
1/16
|
Compute the expected value and standard deviation of this
distribution.
Finally, flipping the coin five times gives this
distribution:
|
x
|
P(x)
|
|
0
|
1/32
|
|
.2
|
5/32
|
|
.4
|
10/32
|
|
.6
|
10/32
|
|
.8
|
5/32
|
|
1
|
1/32
|
Compute the expected value and standard deviation of this
distribution.
What happens to the mean as we flip the coin more
times?
What happens to the standard deviation as we flip the
coin more times?
|