98. A property of the exponential distribution is that the mean equals the
a.
mode
b.
median
c.
variance
d.
standard deviation
99. The skewness measure for the exponential distributions is
a.
0
b.
1
c.
2
d.
3
100. Exponential distributions
a.
are skewed to the left
b.
are skewed to the right
c.
can be skewed to the left or right
d.
are not skewed
101. 95.4% of the values of a normal random variable are within approximately how many standard
deviations of its mean?
a.
+/– 1.7
b.
+/– 2
c.
+/– 2.5
d.
+/– 3
102. All of the following distributions are symmetric except
a.
uniform
b.
normal
c.
exponential
d.
standard normal
PROBLEM
1. A random variable x is uniformly distributed between 45 and 150.
a.
Determine the probability of x = 48.
b.
What is the probability of x 60?
c.
What is the probability of x 50?
d.
Determine the expected vale of x and its standard deviation.
0.000
2. The price of a bond is uniformly distributed between $80 and $85.
a.
What is the probability that the bond price will be at least $83?
b.
What is the probability that the bond price will be between $81 and $90?
c.
Determine the expected price of the bond.
d.
Compute the standard deviation for the bond price.
a.
0.4
b.
0.8
c.
$82.50
d.
$1.44
3. The price of a stock is uniformly distributed between $30 and $40.
a.
What is the probability that the stock price will be more than $37?
b.
What is the probability that the stock price will be less than or equal to $32?
c.
What is the probability that the stock price will be between $34 and $38?
d.
Determine the expected price of the stock.
e.
Determine the standard deviation for the stock price.
a.
0.3
b.
0.2
d.
$35
e.
$2.89
4. The time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes.
a.
What is the probability that a guitar neck can be carved between 95 and 165 minutes?
b.
What is the probability that the guitar neck can be carved between 120 and 200 minutes?
c.
Determine the expected completion time for carving the guitar neck.
d.
Compute the standard deviation.
b.
.875
c.
150 minutes
d.
23.09 minutes
5. The length of time it takes students to complete a statistics examination is uniformly distributed and
varies between 40 and 60 minutes.
a.
Find the mathematical expression for the probability density function.
b.
Compute the probability that a student will take between 45 and 50 minutes to complete the
c.
0.9524
d.
97.5, 30.31
examination.
c.
Compute the probability that a student will take no more than 40 minutes to complete the
examination.
d.
What is the expected amount of time it takes a student to complete the examination?
e.
What is the variance for the amount of time it takes a student to complete the examination?
6. The advertised weight on a can of soup is 10 ounces. The actual weight in the cans follows a uniform
distribution and varies between 9.3 and 10.3 ounces.
a.
Give the mathematical expression for the probability density function.
b.
What is the probability that a can of soup will have between 9.4 and 10.3 ounces?
c.
What is the mean weight of a can of soup?
d.
What is the standard deviation of the weight?
b.
0.90
c.
9.8 ounces
d.
0.289 ounce
7. The length of time patients must wait to see a doctor in a local clinic is uniformly distributed between
15 minutes and 2 1/2 hours.
a.
Define the random variable in words.
b.
What is the probability of a patient waiting exactly 50 minutes?
c.
What is the probability that a patient would have to wait between 45 minutes and 2 hours?
d.
Compute the probability that a patient would have to wait over 2 hours.
e.
Determine the expected waiting time and its standard deviation.
a.
the length of time patients must wait
b.
0.000
c.
0.556
e.
82.5 minutes, 38.97 minutes
8. For the standard normal distribution, determine the probability of obtaining a z value
a.
greater than zero.
b.
between -2.34 to –2.55
c.
less than 1.86.
d.
between -1.95 to 2.7.
e.
between 1.5 to 2.75.
b.
0.25
c.
0.00
d.
50 minutes
e.
9. Z is a standard normal random variable. Compute the following probabilities.
a.
P(-1.33 z 1.67)
b.
P(1.23 z 1.55)
c.
P(z 2.32)
d.
P(z -2.08)
e.
P(z -1.08)
0.8607
b.
0.0487
c.
0.0102
d.
0.9812
e.
0.1401
10. Z is a standard normal random variable. Compute the following probabilities.
a.
P(-1.23 z 2.58)
b.
P(1.83 z 1.96)
c.
P(z 1.32)
d.
P(z 2.52)
e.
P(z -1.63)
f.
P(z -1.38)
g.
P(-2.37 z –1.54)
h.
P(z = 2.56)
0.8858
b.
0.0086
c.
0.0934
d.
0.9941
e.
0.9484
0.0838
g.
0.0529
h.
0.0000
11. Z is a standard normal variable. Find the value of z in the following.
a.
The area between 0 and z is 0.4678.
b.
The area to the right of z is 0.1112.
c.
The area to the left of z is 0.8554
0.5000
b.
0.0042
c.
0.9686
d.
0.9709
e.
0.0638
d.
The area between –z and z is 0.754.
e.
The area to the left of –z is 0.0681.
f.
The area to the right of –z is 0.9803.
12. The miles-per-gallon obtained by the 2013 model Q cars is normally distributed with a mean of 22
miles-per-gallon and a standard deviation of 5 miles-per-gallon.
a.
What is the probability that a car will get between 13.35 and 35.1 miles-per-gallon?
b.
What is the probability that a car will get more than 29.6 miles-per-gallon?
c.
What is the probability that a car will get less than 21 miles-per-gallon?
d.
What is the probability that a car will get exactly 22 miles-per-gallon?
a.
0.9538
b.
0.0643
c.
0.4207
d.
0.0000
13. The 9-month salaries at a daycare center are normally distributed with a mean of $19,000 and a
standard deviation of $4,000.
a.
What is the probability that an employee will have a salary between $12,520 and $13,480?
b.
What is the probability that an employee will have a salary more than $11,880?
c.
What is the probability that an employee will have a salary less than $28,440?
a.
0.0312
b.
0.9625
c.
0.9909
14. A major department store has determined that its customers charge an average of $500 per month, with
a standard deviation of $80. Assume the amounts of charges are normally distributed.
a.
What percentage of customers charges more than $380 per month?
b.
What percentage of customers charges less than $340 per month?
c.
What percentage of customers charges between $644 and $700 per month?
a.
93.32%
b.
c.
a.
1.85
b.
1.22
c.
1.06
d.
1.16
e.
1.49
f.
2.06
15. The contents of soft drink bottles are normally distributed with a mean of twelve ounces and a standard
deviation of one ounce.
a.
What is the probability that a randomly selected bottle will contain more than ten ounces of
soft drink?
b.
What is the probability that a randomly selected bottle will contain between 9.5 and 11 ounces?
c.
What percentage of the bottles will contain less than 10.5 ounces of soft drink?
a.
0.9772
b.
0.1525
c.
6.68%
16. The life expectancy of point-of-sale (POS) terminals is normally distributed with a mean of 4 years
and a standard deviation of 10 months.
a.
What is the probability that a randomly selected terminal will last more than 5 years?
b.
What percentage of terminals will last between 5 and 6 years?
c.
What percentage of terminals will last less than 4 years?
d.
What percentage of terminals will last between 2.5 and 4.5 years?
e.
If the manufacturer guarantees the terminals for 3 years (and will replace them if they
malfunction), what percentage of terminals will be replaced?
a.
0.1151
b.
10.69%
c.
50%
e.
11.51%
17. Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard
deviation of 6.
a.
What is the probability that a randomly selected exam will have a score of at least 71?
b.
What percentage of exams will have scores between 89 and 92?
c.
If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an
award?
d.
If there were 334 exams with scores of at least 89, how many students took the exam?
a.
.9332
b.
.04
c.
91.76
d.
5000
18. The average starting salary for this year’s graduates of a large community college is $30,000 with a
standard deviation of $8,000. Furthermore, it is known that the starting salaries are normally
distributed.
a.
What is the probability that a randomly selected graduate of this community college will have a
starting salary of at least $30,400?
b.
Individuals with starting salaries of less than $15,600 receive a low income tax break. What
percentage of the graduates will receive the tax break?
c.
What are the minimum and the maximum starting salaries of the middle 95% of the graduates?
d.
If 303 of the recent graduates have salaries of at least $43,120, how many students graduated
this year from this community college?
19. The weights of items produced by a company are normally distributed with a mean of 4.5 ounces and a
standard deviation of 0.3 ounces.
a.
What is the probability that a randomly selected item from the production will weigh at least
4.14 ounces?
b.
What percentage of the items weighs between 4.8 and 5.04 ounces?
c.
Determine the minimum weight of the heaviest 5% of all items produced.
d.
If 27,875 of the items of the entire production weigh at least 5.01 ounces, how many items
have been produced?
a.
0.8849
b.
12.28%
c.
4.99
20. The life expectancy of Timely brand watches is normally distributed with a mean of four years and a
standard deviation of eight months.
a.
What is the probability that a randomly selected watch will be in working condition for more
than five years?
b.
The company has a three-year warranty period on their watches. What percentage of their
watches will be in operating condition after the warranty period?
c.
What are the minimum and the maximum life expectancy of the middle 95% of the watches?
d.
Ninety-five percent of the watches will have a life expectancy of at least how many months?
a.
0.0668
b.
93.32%
Min = 32.32 months Max = 63.68 months
d.
34.84 months
21. The weights of the contents of cans of tomato paste produced by a company are normally distributed
with a mean of 6 ounces and a standard deviation of 0.3 ounces.
a.
What percentage of all cans produced contains more than 6.51 ounces of tomato paste?
a.
0.4801
b.
3.59%
minimum = $14320, maximum = $45,680
d.
6000
b.
What percentage of all cans produced contains less than 5.415 ounces?
c.
What percentage of cans contains between 5.46 and 6.495 ounces?
d.
Ninety-five percent of cans will contain at least how many ounces?
e.
What percentage of cans contains between 6.3 and 6.6 ounces?
22. A professor at a local university noted that the grades of her students were normally distributed with a
mean of 78 and a standard deviation of 10.
a.
The professor has informed us that 16.6% of her students received grades of A. What is the
minimum score needed to receive a grade of A?
b.
If 12.1% of her students failed the course and received Fs, what was the maximum score
among those who received an F?
c.
If 33% of the students received grades of B or better (i.e., As and Bs), what is the minimum
score of those who received a B?
a.
87.7
b.
66.3
c.
82.4
23. “DRUGS R US” is a large manufacturer of various kinds of liquid vitamins. The quality control
department has noted that the bottles of vitamins marked 6 ounces vary in content with a standard
deviation of 0.3 ounces. Assume the contents of the bottles are normally distributed.
a.
What percentage of all bottles produced contains more than 6.51 ounces of vitamins?
b.
What percentage of all bottles produced contains less than 5.415 ounces?
c.
What percentage of bottles produced contains between 5.46 and 6.495 ounces?
d.
Ninety-five percent of the bottles will contain at least how many ounces?
e.
What percentage of the bottles contains between 6.3 and 6.6 ounces?
a.
4.46%
b.
2.56%
d.
5.508 ounces
e.
13.59%
24. The daily dinner bills in a local restaurant are normally distributed with a mean of $28 and a standard
deviation of $6.
a.
Define the random variable in words.
b.
What is the probability that a randomly selected bill will be at least $39.10?
c.
What percentage of the bills will be less than $16.90?
a.
4.46%
b.
2.56%
c.
91.46%
d.
5.5067 oz
e.
13.59%
d.
What are the minimum and maximum of the middle 95% of the bills?
e.
If twelve of one day’s bills had a value of at least $43.06, how many bills did the restaurant
collect on that day?
25. The monthly income of residents of Daisy City is normally distributed with a mean of $3000 and a
standard deviation of $500.
a.
Define the random variable in words.
b.
The mayor of Daisy City makes $2,250 a month. What percentage of Daisy City’s residents has
incomes that are more than the mayor’s?
c.
Individuals with incomes of less than $1,985 per month are exempt from city taxes. What
percentage of residents is exempt from city taxes?
d.
What are the minimum and the maximum incomes of the middle 95% of the residents?
e.
Two hundred residents have incomes of at least $4,440 per month. What is the population of
Daisy City?
a.
the monthly income of residents of Daisy City
b.
93.32%
c.
2.12%
Min = 2020 Max = 3980
26. The average starting salary of this year’s vocational school graduates is $35,000 with a standard
deviation of $5,000. Furthermore, it is known that the starting salaries are normally distributed. What
are the minimum and the maximum starting salaries of the middle 95% of the graduates?
27. A local bank has determined that the daily balances of the checking accounts of its customers are
normally distributed with an average of $280 and a standard deviation of $20.
a.
What percentage of its customers has daily balances of more than $275?
b.
What percentage of its customers has daily balances less than $243?
c.
What percentage of its customers’ balances is between $241 and $301.60?
a.
59.87%
b.
3.22%
c.
83.43%
a.
the daily dinner bills
b.
0.0322
c.
3.22%
minimum = $16.24 maximum = $39.76
e.
2,000
28. The weekly earnings of fast-food restaurant employees are normally distributed with a mean of $395.
If only 1.1% of the employees have a weekly income of more than $429.35, what is the value of the
standard deviation of the weekly earnings of the employees?
29. The monthly earnings of computer programmers are normally distributed with a mean of $4,000. If
only 1.7 percent of programmers have monthly incomes of less than $2,834, what is the value of the
standard deviation of the monthly earnings of the computer programmers?
30. The Globe Fishery packs shrimp that weigh more than 1.91 ounces each in packages marked” large”
and shrimp that weigh less than 0.47 ounces each into packages marked “small”; the remainder are
packed in “medium” size packages. If a day’s catch showed that 19.77% of the shrimp were large and
6.06% were small, determine the mean and the standard deviation for the shrimp weights. Assume that
the shrimps’ weights are normally distributed.
31. In grading eggs into small, medium, and large, the Linda Farms packs the eggs that weigh more than
3.6 ounces in packages marked “large” and the eggs that weigh less than 2.4 ounces into packages
marked “small”; the remainder are packed in packages marked “medium.” If a day’s packaging
contained 10.2% large and 4.18% small eggs, determine the mean and the standard deviation for the
eggs’ weights. Assume that the distribution of the weights is normal.
32. A manufacturing process produces items whose weights are normally distributed. It is known that
22.57% of all the items produced weigh between 100 grams up to the mean and 49.18% weigh from
the mean up to 190 grams. Determine the mean and the standard deviation.
33. Z is the standard normal random variable. Use Excel to calculate the following:
a.
P(z 2.5)
b.
P(0 z 2.5)
c.
P(-2 z 2)
d.
P(z -0.38)
e.
P(z 1.62)
f.
z value with .05 in the lower tail
g.
z value with .05 in the upper tail
34. X is a normally distributed random variable with a mean of 50 and a standard deviation of 5. Use Excel
to calculate the following:
a.
P(x 45)
b.
P(45 x 55)
c.
P(x 55)
d.
x value with .20 in the lower tail
e.
x value with .01 in the upper tail
P(45 x 55)
d.
=NORM.INV(0.2,50,5)
e.
=NORM.INV(0.99,50,5)
35. The time it takes a mechanic to change the oil in a car is exponentially distributed with a mean of 5
minutes.
a.
What is the probability density function for the time it takes to change the oil?
b.
What is the probability that it will take a mechanic less than 6 minutes to change the oil?
c.
What is the probability that it will take a mechanic between 3 and 5 minutes to change the oil?
b.
0.6988
c.
0.1809
36. The time it takes a worker on an assembly line to complete a task is exponentially distributed with a
mean of 8 minutes.
a.
What is the probability density function for the time it takes to complete the task?
b.
What is the probability that it will take a worker less than 4 minutes to complete the task?
P(0 z 2.5)
=NORM.S.INV(0.05)
g.
=NORM.S.INV(0.95)
c.
What is the probability that it will take a worker between 6 and 10 minutes to complete the
task?
37. The time between arrivals of customers at the drive-up window of a bank follows an exponential
probability distribution with a mean of 10 minutes.
a.
What is the probability that the arrival time between customers will be 7 minutes or less?
b.
What is the probability that the arrival time between customers will be between 3 and 7
minutes?
a.
0.5034
b.
0.2442
38. The time required to assemble a part of a machine follows an exponential probability distribution with
a mean of 14 minutes.
a.
What is the probability that the part can be assembled in 7 minutes or less?
b.
What is the probability that the part can be assembled between 3.5 and 7 minutes?
a.
0.3935
b.
0.1723
39. The time it takes to completely tune an engine of an automobile follows an exponential distribution
with a mean of 40 minutes.
a.
Define the random variable in words.
b.
What is the probability of tuning an engine in 30 minutes or less?
c.
What is the probability of tuning an engine between 30 and 35 minutes?
a.
the time it takes to completely tune an engine
b.
0.5276
c.
0.0555
40. X is a exponentially distributed random variable with a mean of 10. Use Excel to calculate the
following:
a.
P(x 15)
b.
P(8 x 12)
c.
P(x 8)
b.
0.3935
c.
0.1859
41. The Harbor Island Ferry leaves on the hour and at 15-minute intervals. The time, x, it takes John to
drive from his house to the ferry has a uniform distribution with x between 10 and 20 minutes. One
morning John leaves his house at precisely 8:00a.m.
a. What is the probability John will wait less than 5 minutes for the ferry?
b. What is the probability John will wait less than 10 minutes for the ferry?
c. What is the probability John will wait less than 15 minutes for the ferry?
d. What is the probability John will not have to wait for the ferry?
e. Suppose John leaves at 8:05a.m. What is the probability John will wait (1) less than 5 minutes for
the ferry; (2) less than 10 minutes for the ferry?
f. Suppose John leaves at 8:10a.m. What is the probability John will wait (1) less than 5 minutes for
the ferry; (2) less than 10 minutes for the ferry?
g. What appears to be the best time for John to leave home if he wishes to maximize the probability
of waiting less than 10 minutes for the ferry?
42. Delicious Candy markets a two-pound box of assorted chocolates. Because of imperfections in the
candy making equipment, the actual weight of the chocolate has a uniform distribution ranging from
31.8 to 32.6 ounces.
a. Define a probability density function for the weight of the box of chocolate.
b. What is the probability that a box weighs (1) exactly 32 ounces; (2) more than 32.3 ounces; (3)
less than 31.8 ounces?
c. The government requires that at least 60% of all products sold weigh at least as much as the stated
weight. Is Delicious violating government regulations?
43. The time at which the mailman delivers the mail to Ace Bike Shop follows a normal distribution with
mean 2:00 PM and standard deviation of 15 minutes.
a. What is the probability the mail will arrive after 2:30 PM?
b. What is the probability the mail will arrive before 1:36 PM?
c. What is the probability the mail will arrive between 1:48 PM and 2:09 PM?
44. The township of Middleton sets the speed limit on its roads by conducting a traffic study and
determining the speed (to the nearest 5 miles per hour) at which 80% of the drivers travel at or below.
A study was done on Brown’s Dock Road that indicated driver’s speeds follow a normal distribution
with a mean of 36.25 miles per hour and a variance of 6.25.
a. What should the speed limit be?
b. What percent of the drivers travel below that speed?
45. A light bulb manufacturer claims its light bulbs will last 500 hours on the average. The lifetime of a
light bulb is assumed to follow an exponential distribution.
a. What is the probability that the light bulb will have to be replaced within 500 hours?
b. What is the probability that the light bulb will last more than 1000 hours?
c. What is the probability that the light bulb will last between 200 and 800 hours.
46. The weight of a .5 cubic yard bag of landscape mulch is uniformly distributed over the interval from
38.5 to 41.5 pounds.
a. Give a mathematical expression for the probability density function.
b. What is the probability that a bag will weigh more than 40 pounds?
c. What is the probability that a bag will weigh less than 39 pounds?
d. What is the probability that a bag will weigh between 39 and 40 pounds?