Chapter 11
Altered marginal probability of an event based on additional information is a ________ probability.
Posterior.
74) Assume X has the following probability distribution : X 1 2 3 4 P(X) .1 .5 .2 .2 Compute the standard deviation of X.
.9219
85) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is more than 24 oz?
0
81) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf of bread is larger than 23 oz?
0.0228
91) What is the area under the normal curve for Z ≥ 1.79?
0.0367
95) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. Assuming the weights of crabs are normally distributed, the probability that a randomly selected crab will weigh less than 1.2 pounds is ________.
0.1587
86) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is more than 22.25 oz?
0.3085
84) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is between 21.75 and 22.25 oz?
0.3830
The standard deviation of the standard normal distribution is equal to ________.
1
The ________ of a random variable is computed by multiplying each possible value of the variable by its probability and summing these products.
Expected value.
A joint probability is the probability that two or more events that are mutually exclusive can occur simultaneously.
F
In a binomial distribution process, there are ________ possible outcomes.
Two.
Almost all of the data from a normal distribution fall within ________ standard deviations of the mean.
±3
The expected value of a random variable is computed by multiplying the sum of each possible value of the variable by the probability of that random variable.
F
The standard normal distribution has a mean of one and a standard deviation of zero.
F
A ________ organizes numerical data to describe the events of an experiment.
Frequency distribution.
The collectively exhaustive set of events for flipping a coin is ________.
Heads and tails.
A succession of events that do not affect each other are ________.
Independent.
The fact that the first toss of a coin has no effect on the outcome of the second toss of the coin suggests that these events are ________.
Independent.
A ________ is the probability of occurence of a single event.
Marginal probability
A Venn diagram depicting two circles that do not overlap or touch in any way represents events that are ________.
Mutually exclusive.
The ________ normal distribution has a mean of 0 and a standard deviation of 1.
Standard.
________ probability is an estimate based on a personal belief, experience, and knowledge of a situation.
Subjective.
A Venn diagram visually displays mutually exclusive and non-mutually exclusive events
T
A binomial probability distribution indicates the probability of r successes in n trials.
T
A marginal probability is the probability of a single event occurring.
T
A set of events is collectively exhaustive when it includes all the events that can occur in an experiment.
T
A succession of events that does not affect other events is independent.
T
An experiment is an activity that results in one of several possible outcomes.
T
An inspector correctly identifies defective products 90% of the time. For the next 10 products, the probability that he makes fewer than two incorrect inspections is 0.736.
T
Probability trees are used only to compute conditional probabilities.
F
For a typical normally distributed random variable, the standard deviation is equal to the variance.
F
75) If x is normally distributed with a mean of 10 and a standard deviation of 3, then P(x ≤ 6) is equal to P( Z ≤ _____)?
-4/3
The expected value of the standard normal distribution is equal to ________.
0
For a standard normal distribution, what is the probability that z is greater than 1.75?
0.0401
93) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. Assuming the weights of the crabs are normally distributed, what is the probability that a randomly selected crab will weigh more than 2.2 pounds?
0.0668
The cumulative probability for all six outcomes of tossing a fair die should end with the value ________.
1.00
An online sweepstakes has the following payoffs and probabilities. Each person is limited to one entry. Payoff Probability $1.00 0.1000 $5.00 0.0100 $10.00 0.0050 $100.00 0.0010 $1,000.00 0.0005 $5,000.00 0,0001 103) The probability that someone wins any money is ________. 104) The probability of winning at least $1000 is ________.
103) 0.1166 104) 0.0006
Horatio Oscar Vineeth Lane (HOV Lane for short) records his commute times for a period of one month and assigned them to five different categories as shown in the table. Commute Length # of Observations commute time <10 minutes 1 10 ≤ commute time < 20 minutes 2 20 ≤ commute time < 30 minutes 4 30 ≤ commute time < 40 minutes 9 40 ≤ commute time < 50 minutes 4 149) What is the duration of Mr. Lane's expected commute? 150) What is the probability that Mr. Lane makes it home in under thirty minutes? 151) What kind of probability is demonstrated if Mr. Lane is asked to predict the duration of his next commute? 152) Which of these events are mutually exclusive?
149) A 150) B 151) C 152) D
92) A study of a company's practice regarding the payment of invoices revealed that on the average an invoice was paid 20 days after it was received. The standard deviation equaled 5 days. Assuming that the distribution is normal, what percent of the invoices is paid within 15 days of receipt?
15.87%
Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. 60) What is the probability that Jim will be accepted at both universities? 61) What is the probability that Jim will not be accepted at either university? 62) What is the probability that Jim will be accepted by at least one of the two universities?
60) 0.09 61) 0.44 = (.55) × (.80) 62) 0.56 = 1 - [(.55) × (.80)]
137) The weight of a jar of jelly is normally distributed with a mean of 16 oz and a standard deviation of 0.02 oz. What is the probability that a jar of jelly contains more than 16.03 oz? A) .0668 B) .1587 C) .4332 D) .9332
A
Probabilistic techniques assume that no uncertainty exists in model parameters.
F
A company markets educational software products, and is ready to place three new products on the market. Past experience has shown that for this particular software, the chance of "success" is 80%. Assume that the probability of success is independent for each product. 71) Find the probability that exactly 1 of the 3 products is successful. 72) Find the probability that none of the 3 products is successful.
71) (3)(.8)(.2)(.2) = .096 72) (1)(.2)(.2)(.2) = .008
116) The probability of independent events occurring in succession is computed by ________ the probabilities of each event. A) multiplying B) adding C) subtracting D) dividing
A
118) In Bayesian analysis, additional information is used to alter the ________ probability of the occurrence of an event. A) marginal B) conditional C) binomial D) revised
A
120) A ________ probability is the altered marginal probability of an event based on additional information. A) posterior B) joint C) marginal D) conditional
A
123) In a ________ distribution, for each of n trials, the event always has the same probability of occurring. A) binomial B) joint C) frequency D) standard
A
126) A fair die is rolled nine times. What is the probability that an odd number (1, 3, or 5) will occur less than 3 times? A) .0899 B) .2544 C) .7456 D) .9101
A
129) ________ is a measure of the dispersion of random variable values about the expected value or mean. A) Standard deviation B) Sample mean C) Population mean D) Expected value
A
131) The expected value of the standard normal distribution is equal to: A) 0 B) 1 C) 1.5 D) 2
A
145) Assume that it takes a college student an average of 5 minutes to find a parking spot in the main parking lot. Assume also that this time is normally distributed with a standard deviation of 2 minutes. What time is exceeded by approximately 75% of the college students when trying to find a parking spot in the main parking lot? A) 3.65 minutes B) 5.75 minutes C) 6.36 minutes D) 9.21 minutes
A
148) A professor would like to assign grades such that 7% of students receive Fs. If the exam average is 62 with a standard deviation of 13, what grade should be the cutoff for an F? (Round your answer.) A) 43 B) 49 C) 50 D) 55
A
121) Mutually exclusive events are A) events with identical probabilities. B) events that have no outcomes in common. C) events that have no effect on each other. D) events that are represented in a Venn diagram by two overlapping circles.
B
122) Bayesian analysis involves a(n) ________ probability. A) a priori B) posterior C) joint D) relative frequency
B
124) Experiments with repeated independent trials will be described by the binomial distribution if A) each trial result influences the next. B) each trial has exactly two outcomes whose probabilities do not change. C) the trials are continuous. D) the time between trials is constant.
B
125) In a binomial distribution, for each of n trials, the event A) time between trials is constant. B) always has the same probability of occurring. C) result of the first trial influence the next trial. D) trials are continuous.
B
139) The metropolitan airport commission is considering the establishment of limitations on noise pollution around a local airport. At the present time, the noise level per jet takeoff in one neighborhood near the airport is approximately normally distributed with a mean of 100 decibels and a standard deviation of 3 decibels. What is the probability that a randomly selected jet will generate a noise level of more than 105 decibels? A) 0.0228 B) 0.0475 C) 0.0485 D) 0.0500
B
140) For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.2910. The value of Z is: A) 0.17 B) 0.81 C) 1.25 D) 1.65
B
141) For some value of Z, the probability that a standard normal variable is below Z is 0.3783. The value of Z is: A) -0.81 B) -0.31 C) 0.82 D) 1.55
B
143) 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
B
147) A professor would like to assign grades such that 5% of students receive As. If the exam average is 62 with a standard deviation of 13, what grade should be the cutoff for an A? (Round your answer.) A) 80 B) 83 C) 90 D) 93
B
________ can enable one to improve marginal probabilities of the occurrence of an event by gathering additional information.
Bayesian analysis.
One of the properties of the ________ distribution is that the probability of success remains constant over time.
Binomal
105) ________ techniques assume that no uncertainty exists in model parameters. A) Probability B) Probabilistic C) Deterministic D) Distribution
C
107) Objective probabilities that can be stated prior to the occurrence of an event are A) deterministic or probabilistic. B) subjective or objective. C) classical or a priori. D) relative or subjective.
C
109) In a given experiment the probabilities of mutually exclusive events sum to: A) 0 B) 0.5 C) 1 D) This cannot be answered without knowing the probability values of the events.
C
117) A ________ probability is the probability that an event will occur given that another event has already occurred. A) subjective B) objective C) conditional D) binomial
C
119) Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category. Male (M) Female (F) Job Administrative (AD) 110 10 Salaried staff (SS) 30 50 Hourly staff (HS) 60 40 If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member? A) .50 B) .60 C) .625 D) .70
C
127) A fair die is rolled 8 times. What is the probability that an even number (2, 4, or 6) will occur between 2 and 4 times? A) .2188 B) .4922 C) .6016 D) .8204
C
132) The area under the normal curve represents probability, and the total area under the curve sums to: A) 0 B) 0.5 C) 1 D) 2
C
135) For the normal distribution, the mean plus and minus 1.96 standard deviations will include what percent of the observations? A) 84% B) 90% C) 95% D) 97%
C
136) The weight of a jar of jelly is normally distributed with a mean of 16 oz and a standard deviation of 0.02 oz. What is the probability that a jar of jelly contains less than 16 oz? A) .1915 B) .3085 C) .5000 D) .7257
C
138) Under the normal curve, the area between z = 2 and z = -2 includes ________ of the values. A) 98% B) 96% C) 95% D) 93%
C
146) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. What weight is exceeded by 2% of all of the crabs? (Assume the weights are normally distributed.) A) 0.78 pounds B) 1.82 pounds C) 2.42 pounds D) 4.36 pounds
C
The ________ test is a statistical test to see if an observed data fit a particular probability distribution.
Chi-square.
Objective probabilities that can be stated prior to the occurrence of an event are ________.
Classical or a priori
The term for including all possible events that can occur in an experiment is ________.
Collectively exhaustive.
A ________ probability is the probability that an event will occur given that another event has already occurred.
Conditional
115) A ________ probability distribution indicates the probability of r successes in n trials. A) joint B) subjective C) marginal D) binomial
D
128) A company markets educational software products and is ready to place three new products on the market. Past experience has shown that for this particular software, the chance of "success" is 80%. Assume that the probability of success is independent for each product. What is the probability that exactly 1 of the 3 products is successful? A) 0.80 B) 0.032 C) 0.24 D) 0.096
D
130) An automotive center keeps tracks of customer complaints received each week. The probability distribution for complaints can be represented as the table shown below. The random variable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints. xi 0 1 2 3 4 5 6 p(xi) .10 .15 .18 .20 .20 .10 .07 What is the average number of complaints received per week? A) 2.12 B) 3.32 C) 4.12 D) 2.83
D
133) The ________ and variance are derived from a subset of the population data and are used to make inferences about the population. A) population variance B) population standard deviation C) population mean D) sample mean
D
134) Under the normal curve, the area between z = 1 and z = -2 includes approximately ________ of the values. A) 98% B) 95% C) 85% D) 82%
D
142) For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3554. The value of Z is: A) 0.31 B) 0.36 C) 0.95 D) 1.06
D
144) Assume that it takes a college student an average of 5 minutes to find a parking spot in the main parking lot. Assume also that this time is normally distributed with a standard deviation of 2 minutes. Find the probability that a randomly selected college student will take between 2 and 6 minutes to find a parking spot in the main parking lot. A) 0.1950 B) 0.4772 C) 0.4332 D) 0.6247
D
111) A frequency distribution is an organization of ________ data about the events in an experiment. A) quantitative B) integer C) qualitative D) unknown
A
87) A life insurance company wants to estimate its annual payouts. Assume that 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 4 years. What proportion of the plan recipients would receive payments beyond age 75?
0.0401
88) A life insurance company wants to estimate its annual payouts. Assume that 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 4 years. What proportion of the participants die before they reach the age of 65?
0.2266
94) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. Assuming the weights of crabs are normally distributed, what is the probability that a randomly selected crab will weigh between 1 and 2 pounds?
0.7745
101) Assume 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 48 and 55.
0.8351
80) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is larger than 21 oz?
0.9772
83) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is between 20.75 and 23.25 oz?
0.9876
82) The weight of a loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is less than 24 oz?
1
73) If X has the following probability distribution X 1 2 3 4 P(X) .1 .5 .2 .2 Compute the expected value of X.
2.5 EV = (1)(.1) + (2)(.5) + (3)(.2) + (4)(.2) = 2.5
Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category. Job Male (M) Female (F) Administrative (AD) 110 10 Salaried staff (SS) 30 50 Hourly staff (HS) 60 40 63) If an employee is selected at random, what is the probability that the employee is male? 64) If an employee is selected at random, what is the probability that the employee is male and salaried staff? 65) If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member? 66) If an employee is selected at random, what is the probability that the employee is female or works as a member of the administration
63) .667 = 200/300 64) .10 = 30/300 65) .625 = 50/80 66) .70 = 100/300 + 120/300 -10/300
The Dean's Office keeps tracks of student complaints received each week. The probability distribution for complaints can be represented as a table as shown below. The random variable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints. xi 0 1 2 3 4 5 6 p(xi) .10 .15 .18 .20 .20 .10 .07 67) What is the probability that they receive less than 3 complaints in a week? 68) What is the average number of complaints received per week? 69) A fair die is rolled nine times. What is the probability that an odd number (1, 3, or 5) will occur less than 3 times? 70) A fair die is rolled 8 times. What is the probability that an even number (2, 4, or 6) will occur between 2 and 4 times?
67) 0.43 68) 2,83 69) 0.0899 70) 0.6016
89) A life insurance company wants to estimate its annual payouts. Assume that 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 4 years. By what age have 80% of the plan participants passed away?
71.36 years old
Two psychology majors, in two different sections of Clinical Psychology, were comparing test scores. The following gives the students' scores, class mean, and standard deviation for each section: Section 1 Section 2 Student score 84 75 Mean 75 60 Standard deviation 7 8 77) What is the z-score of the student from section 1 and what is the probability that a student in section 1 will score higher than 84? 78) What is the z-score of the student from section 2 and what is the probability that a student in section 2 will score higher than 75? 79) Which student scored better compared to the rest of their section?
77) 1.286 and .0985 78) 1.875 and .0301 79) Section 2 student because their z-score is higher.
90) For the normal distribution, the mean plus and minus 1.96 standard deviations will include what percent of the observations?
95%
Horatio Oscar Vineeth Lane (HOV Lane for short) records his commute times for a period of one month and assigned them to five different categories as shown in the table. Commute Length # of Observations commute time <10 minutes 1 10 ≤ commute time < 20 minutes 2 20 ≤ commute time < 30 minutes 4 30 ≤ commute time < 40 minutes 9 40 ≤ commute time < 50 minutes 4 98) What is the likelihood that the commute will take 30 minutes or longer? 99) What is the cumulative probability for the three shortest commute categories? 100) What is the average commute time for HOV Lane?
98) 13/20 = 65% 99) (1+2+4)/20 = 0.35 100) Taking the midpoints of each observation as the value of the random variable, we get .05∗5 + .1∗15 + .2∗25 + .45∗35 + .2∗45 = 31.5 minutes
108) The events in an experiment are ________ if only one can occur at a time. A) mutually exclusive B) non-mutually exclusive C) mutually inclusive D) independent
A
97) What are the differences between deterministic and probabilistic techniques?
Answer: Deterministic techniques assume that no uncertainty exists in the model parameters. Probabilistic techniques include uncertainty and assume that there can be more than one solution.
106) ________ probability is an estimate based on personal belief, experience, or knowledge of a situation. A) Binomial B) Subjective C) Marginal D) Joint
B
110) A ________ probability is the probability of a single event occurring. A) subjective B) binomial C) marginal D) joint
C
113) Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Jim will not be accepted at either university? A) .30 B) .36 C) .44 D) .56
C
114) Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category. Male (M) Female (F) Job Administrative (AD) 110 10 Salaried staff (SS) 30 50 Hourly staff (HS) 60 40 If an employee is selected at random, what is the probability that the employee is female or works as a member of the administration? A) .17 B) .67 C) .70 D) .73
C
112) Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Jim will be accepted at both universities? A) .65 B) .25 C) .20 D) .09
D
A continuous random variable may assume only integer values within a given interval.
F
A normally distributed random variable has a mean of zero and a standard deviation of one.
F
Conditional probabilities are shown in Venn diagrams.
F
In Bayesian analysis, additional information is used to alter the conditional probability of the occurrence of an event.
F
Seventy-two percent of all observations fall within one standard deviation of the mean if the data is normally distributed.
F
The expected value of a discrete random variable is greater than or equal to zero.
F
The variance of a random variable is computed by multiplying each possible value of the variable by its probability and summing these products.
F
A continuous random variable can take on a(n) ________ number of values within a given interval.
Infinite.
The events in an experiment are ________ if only one can occur at a time.
Mutually exclusive.
If events A and B are independent, then P(A|B) = ________.
P(A)
If two events A and B are not mutually exclusive, then P(A or B) = ________.
P(A) + P(B) - P(AB)
If events A and B are independent, then P(AB) = ________.
P(A) ∗ P(B)
A conditional probability is the probability that an event occurs given that another event has already occurred.
T
Another name for the mean of a probability distribution is its expected value.
T
Deterministic techniques assume that no uncertainty exists in model parameters.
T
In a given experiment, the probabilities of all mutually exclusive events sum to one.
T
Objective probabilities that are stated after the outcomes of an event have been observed are relative frequencies.
T
Objective probabilities that can be stated prior to the occurrence of an event are classical or a priori.
T
Relative frequency is the more widely used definition of objective probability.
T
Subjective probability is an estimate based on personal belief, experience, or knowledge of a situation.
T
The area under the normal curve represents probability.
T
The events in an experiment are mutually exclusive if only one can occur at a time.
T
The variance of a discrete random variable is always greater than or equal to zero.
T
There is just as great a chance of a normally distributed random variable being over one standard deviation above the mean as there is being one standard deviation below the mean.
T
96) A research scientist has observed the monkeys of the Nandi Hills outside of Bangalore for the last twenty years, carefully cataloging their preferences for a number of food items. An unsuspecting tourist leaves his can of soda unattended. Describe the type of probability the research scientist can assign to the likelihood that the soda will become the monkey's next meal. Then contrast this type of probability with the other of the two basic types.
The two types of probability are objective and subjective. Objective probabilities can be stated prior to the occurrence of an event and since the scientist has observed these monkeys for twenty years, there are relative frequency probabilities associated with common items such as a can of soda. The other type of probability is subjective, which is an estimate based on personal belief. Since the scientist is experienced, this would not be an appropriate characterization of the event. We could assume the tourist is unfamiliar with the behavior of these monkeys, thus the tourist's estimate would most likely be subjective (unless the tourist was a primate research scientist on holiday).
________ is a measure of dispersion of random variable values about the expected value.
Variance.
102) A paint manufacturer's production process is normally distributed with a mean of 100,000 gallons and a standard deviation of 10,000 gallons. Management wants to create an incentive bonus for the production crew when the daily production exceeds the 94th percentile of the distribution. At what level of production should management pay the incentive bonus?
z = 1.56, so the incentive level is 115,600 gallons.
