The Islamic University of Gaza
Faculty of Commerce
Department of Economics and Political Sciences
An Introduction to Statistics Course (ECOE 1302)
Spring Semester 2011
Chapter 6 - The Normal Distribution Problems and Solutions
Instructors: Dr. Samir Safi
Mr. Ibrahim Abed
SECTION I: MULTIPLE-CHOICE
1- In its standardized form, the normal distribution
a) has a mean of 0 and a standard deviation of 1.
b) has a mean of 1 and a variance of 0.
c) has an area equal to 0.5.
d) cannot be used to approximate discrete probability distributions.
2-
If a particular batch of data is approximately normally distributed, we would find that
approximately
a) 2 of every 3 observations would fall between ± 1 standard deviation around
the mean.
b) 4 of every 5 observations would fall between ± 1.28 standard deviations
around the mean.
c) 19 of every 20 observations would fall between ± 2 standard deviations
around the mean.
d) All the above.
3- For some value of Z, the probability that a standard normal variable is below Z is
0.2090. The value of Z is
a) – 0.81
b) – 0.31
c) 0.31
d) 1.96
4- For some positive value of X, the probability that a standard normal variable is
between 0 and +2X is 0.1255. The value of X is
a) 0.99
b) 0.40
c) 0.32
d) 0.16
5- Given that X is a normally distributed random variable with a mean of 50 and a
standard
deviation of 2, find the probability that X is between 47 and 54. ( 0.9104 )
6-
A company that sells annuities must base the annual payout on the probability
distribution of the
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length of life of the participants in the plan. Suppose the probability distribution of
the lifetimes
of the participants is approximately a normal distribution with a mean of 68 years
and a standard
deviation of 3.5 years. What proportion of the plan recipients die before they reach
the standard
retirement age of 65? (0.1957 using Excel or 0.1949 using Table E.2)
7- If we know that the length of time it takes a college student to find a parking spot in
the library parking lot follows a normal distribution with a mean of 3.5 minutes and a
standard deviation of 1 minute, find the probability that a randomly selected college
student will find a parking spot in the library parking lot in less than 3 minutes.
a) 0.3551
b) 0.3085
c) 0.2674
d) 0.1915
8. If we know that the length of time it takes a college student to find a parking spot in
the library parking lot follows a normal distribution with a mean of 3.5 minutes and a
standard deviation of 1 minute, find the point in the distribution in which 75.8% of
the college students exceed when trying to find a parking spot in the library parking
lot.
e) 2.8 minutes
f) 3.2 minutes
g) 3.4 minutes
h) 4.2 minutes
9. The owner of a fish market determined that the average weight for a catfish is 3.2
pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are
normally distributed, the probability that a randomly selected catfish will weigh
between 3 and 5 pounds is _______?
(0.5865)
10. The owner of a fish market determined that the average weight for a catfish is 3.2
pounds with a
standard deviation of 0.8 pound. Assuming the weights of catfish are normally
distributed, above
what weight (in pounds) do 89.80% of the weights occur?( 2.184 pounds)
11. A food processor packages orange juice in small jars. The weights of the filled jars
are approximately normally distributed with a mean of 10.5 ounces and a standard
deviation of 0.3 ounce. Find the proportion of all jars packaged by this process that
have weights that fall below 10.875 ounces. ( 0.89440)
Section II: TRUE Or FALSE
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1. The probability that a standard normal random variable, Z, falls between – 1.50 and
0.81 is 0.7242. True
2.
The probability that a standard normal random variable, Z, is below 1.96 is 0.4750.
False
3. True or False: The probability that a standard normal random variable, Z, falls
between –2.00 and –0.44 is 0.6472. False
4. True or False: A worker earns $15 per hour at a plant and is told that only 2.5% of all
workers make a higher wage. If the wage is assumed to be normally distributed and
the standard deviation of wage rates is $5 per hour, the average wage for the plant is
$7.50 per hour. False
5. True or False: Any set of normally distributed data can be transformed to its
standardized form.
True
6. True or False: A normal probability plot may be used to assess the assumption of
normality for a particular batch of data. True
Section III: FREE RESPONSE PROBLEMS
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1. The probability that a standard normal variable Z is positive is ________.
0.50
2. The amount of pyridoxine (in grams) in a multiple vitamin is normally distributed
with µ = 110 grams and σ = 25 grams. What is the probability that a randomly
selected vitamin will contain between 82 and 100 grams of pyridoxine? 0.2132
3. The amount of pyridoxine (in grams) in a multiple vitamin is normally distributed
with µ = 110 grams and σ = 25 grams. What is the probability that a randomly
selected vitamin will contain between 100 and 120 grams of pyridoxine? 0.3108
4. The amount of pyridoxine (in grams) in a multiple vitamin is normally distributed
with µ = 110 grams and σ = 25 grams. What is the probability that a randomly
selected vitamin will contain less than 100 grams or more than 120 grams of
pyridoxine? 0.6892
5. The true length of boards cut at a mill with a listed length of 10 feet is normally
distributed with a mean of 123 inches and a standard deviation of 1 inch. What
proportion of the boards will be over 125 inches in length? 0.0228
6. You were told that the amount of time lapsed between consecutive trades on the New
York Stock Exchange followed a normal distribution with a mean of 15 seconds.
You were also told that the probability that the time lapsed between two consecutive
trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed
between two consecutive trades would fall below 13 seconds was 7%. What is the
probability that the time lapsed between two consecutive trades will be longer than 17
seconds? 7% or 0.07
7. You were told that the amount of time lapsed between consecutive trades on the New
York Stock Exchange followed a normal distribution with a mean of 15 seconds. You
were also told that the probability that the time lapsed between two consecutive trades
to fall between 16 to 17 seconds was 13%. The probability that the time lapsed
between two consecutive trades would fall below 13 seconds was 7%. What is the
probability that the time lapsed between two consecutive trades will be between 15
and 16 seconds? 30% or 0.30
8. You were told that the amount of time lapsed between consecutive trades on the New
York Stock Exchange followed a normal distribution with a mean of 15 seconds. You
were also told that the probability that the time lapsed between two consecutive trades
to fall between 16 to 17 seconds was 13%. The probability that the time lapsed
between two consecutive trades would fall below 13 seconds was 7%. What is the
probability that the time lapsed between two consecutive trades will be between 13
and 16 seconds? 73% or 0.73
9. You were told that the amount of time lapsed between consecutive trades on the New
York Stock Exchange followed a normal distribution with a mean of 15 seconds. You
were also told that the probability that the time lapsed between two consecutive trades
to fall between 16 to 17 seconds was 13%. The probability that the time lapsed
between two consecutive trades would fall below 13 seconds was 7%. What is the
probability that the time lapsed between two consecutive trades will be between 14
and 17 seconds? 73% or 0.73 d
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10. You were told that the amount of time lapsed between consecutive trades on the New
York Stock Exchange followed a normal distribution with a mean of 15 seconds. You
were also told that the probability that the time lapsed between two consecutive trades
to fall between 16 to 17 seconds was 13%. The probability that the time lapsed
between two consecutive trades would fall below 13 seconds was 7%. The
probability is 20% that the time lapsed will be shorter how many seconds? 14
seconds
11. You were told that the amount of time lapsed between consecutive trades on the New
York Stock Exchange followed a normal distribution with a mean of 15 seconds. You
were also told that the probability that the time lapsed between two consecutive trades
to fall between 16 to 17 seconds was 13%. The probability that the time lapsed
between two consecutive trades would fall below 13 seconds was 7%. The middle
60% of the time lapsed will fall between which two numbers?
14 seconds and 16 seconds
12. You were told that the mean score on a statistics exam is 75 with the scores normally
distributed. In addition, you know the probability of a score between 55 and 60 is
4.41% and that the probability of a score greater than 90 is 6.68%. What is the
probability of a score between 90 and 95? 4.41% or 0.0441
13. You were told that the mean score on a statistics exam is 75 with the scores normally
distributed. In addition, you know the probability of a score between 55 and 60 is
4.41% and that the probability of a score greater than 90 is 6.68%. What is the
probability of a score lower than 55?
2.27% or 0.0227
14. You were told that the mean score on a statistics exam is 75 with the scores normally
distributed. In addition, you know the probability of a score between 55 and 60 is
4.41% and that the probability of a score greater than 90 is 6.68%. What is the
probability of a score between 60 and 75? 43.32% or 0.4332
15. You were told that the mean score on a statistics exam is 75 with the scores normally
distributed. In addition, you know the probability of a score between 55 and 60 is
4.41% and that the probability of a score greater than 90 is 6.68%. What is the
probability of a score between 55 and 90? 91.05% or 0.9105
16. You were told that the mean score on a statistics exam is 75 with the scores normally
distributed. In addition, you know the probability of a score between 55 and 60 is
4.41% and that the probability of a score greater than 90 is 6.68%. The middle
86.64% of the students will score between which two scores? 60 and 90
17. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. The probability that Z is less than 1.15 is __________. 0.8749
18. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. The probability that Z is less than -2.20 is __________. 0.0139
19. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. The probability that Z is between -2.33 and 2.33 is __________. 0.9802
20. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. The probability that Z is between -0.88 and 2.29 is __________. 0.7996
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21. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. The probability that Z values are larger than __________ is 0.6985. -0.52
22. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. So 85% of the possible Z values are smaller than __________. 1.04
23. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation
of 1. So 50% of the possible Z values are between __________ and __________
(symmetrically distributed about the mean). -0.67 and 0.67 or -0.68 and 0.68
24. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2
minutes. The probability is __________ that a product is assembled in between 14
and 16 minutes.
0.3829 using Excel or 0.3830 using Table E.2
25. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2
minutes. The probability is __________ that a product is assembled in between 15
and 21 minutes.0.49865
26. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2
minutes. The probability is __________ that a product is assembled in more than 11
minutes.0.9772
27. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2
minutes. The probability is __________ that a product is assembled in less than 20
minutes.0.9938
28. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2
minutes. So, 90% of the products require more than __________ minutes for
assembly. 12.44
29. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2
minutes. So, 17% of the products would be assembled within __________ minutes.
13.1
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TABLE 6-3
The number of column inches of classified advertisements appearing on Mondays in a
certain daily newspaper is normally distributed with population mean 320 and population
standard deviation 20 inches.
30. Referring to Table 6-3, for a randomly chosen Monday, what is the probability there
will be less than 340 column inches of classified advertisement? 0.8413
31. Referring to Table 6-3, for a randomly chosen Monday the probability is 0.1 that
there will be less than how many column inches of classified advertisements? 294.4
TABLE 6-4
John has two jobs. For daytime work at a jewelry store he is paid $200 per month, plus a
commission. His monthly commission is normally distributed with mean $600 and
standard deviation $40. At night he works as a waiter, for which his monthly income is
normally distributed with mean $100 and standard deviation $30. John's income levels
from these two sources are independent of each other.
32. Referring to Table 6-4, for a given month, what is the probability that John's
commission from the jewelry store is less than $640? 0.8413
33. Referring to Table 6-4, the probability is 0.9 that John's income as a waiter is less
than how much in a given month? $138.45 using Excel or $138.40 using Table E.2
34. Referring to Table 6-4, for a given month, what is the probability that John's total
income from these two jobs is less than $825? 0.0668
35. The owner of a fish market determined that the average weight for a catfish is
3.2 pounds. He also knew that the probability of a randomly selected catfish
that would weigh more than 3.8 pounds is 20% and the probability that a
randomly selected catfish that would weigh less than 2.8 pounds is 30%. The
probability that a randomly selected catfish will weigh between 2.6 and 3.6
pounds is ______. 50% or 0.5
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