Deck 3: Probability Distributions

Question 1

Larger values of the standard deviation result in a normal curve that is

A)shifted to the right
B)shifted to the left
C)narrower and more peaked
D)wider and flatter

Answer 1

D

Question 2

A probability density function

A)gives the probability that the random variable equals a specific value of x.
B)is used for discrete random variables.
C)describes the graph whose underlying area represents probability over an interval.
D)each of these choices are true.

Answer 2

C

Question 3

Experiments with repeated independent trials will be described by the binomial distribution if

A)the trials are continuous.
B)each trial result influences the next.
C)the time between trials is constant.
D)each trial has exactly two outcomes whose probabilities do not change.

Answer 3

D

Question 4

The uniform distribution defined over the interval from 25 to 40 has the probability density function

A)f(x) = 1/40 for all x
B)f(x) = 5/8 for 25  x  40 and f(x) = 0 elsewhere
C)f(x) = 1/25 for 0  x  25 and f(x) = 1/40 for 26  x  40
D)f(x) = 1/15 for 25  x  40 and f(x) = 0 elsewhere

Question 5

In order to measure the dispersion of a random variable,look at its

A)standard deviation.
B)mean.
C)expected value.
D)average.

Question 6

The weight of an object,measured to the nearest gram,is an example of

A)a continuous random variable
B)a discrete random variable
C)either a continuous or a discrete random variable, depending on the weight of the object
D)either a continuous or a discrete random variable depending on the units of measurement

Question 7

When the exponential distribution probability is given as P(x  9 )= 1  e^9/18 the average value for x is

A)18
B)9
C)9/18
D)1/2

Question 8

z is a standard normal random variable.The P(z  2.11)equals

A)0.4821
B)0.9821
C)0.5
D)0.0174

Question 9

The Poisson probability distribution is used with

A)a discrete random variable
B)a continuous random variable
C)either a discrete or continuous random variable
D)any random variable

Question 10

The binomial probability distribution is most symmetric when

A)n is 30 or greater
B)n equals p
C)p approaches 1
D)p equals 0.5

Question 11

An experiment consists of measuring the speed of automobiles on a highway by the use of radar equipment.The random variable in this experiment is speed,measured in miles per hour.This random variable is a

A)discrete random variable
B)continuous random variable
C)complex random variable
D)None of these alternatives is correct.

Question 12

A numerical description of the outcome of an experiment is

A)a normal variable.
B)a discrete variable.
C)a random variable.
D)an experimental variable.

Question 13

A statement that matches the values of a random variable with the probabilities of those values is

A)the expected value.
B)the variation of the random variable.
C)an experiment.
D)a probability distribution.

Question 14

Which of the following are continuous random variables?
I)the weight of an elephant
II)the time to answer a questionnaire
III)the number of floors in a skyscraper
IV)the square feet of countertop in a kitchen

A)I and II only
B)III and IV only
C)I, II and IV
D)I, II, II, and IV

Question 15

A standard normal distribution is a normal distribution with

A)a mean of 1 and a standard deviation of 0
B)a mean of 0 and a standard deviation of 1
C)any mean and a standard deviation of 1
D)any mean and any standard deviation

Question 16

If x is normally distributed with mean 12 and standard deviation 2,then P(x  9)is

A)P(z  9/10).
B)P(z  3/2)
C)P(z  2/3)
D)P(z  3/4)

Question 17

A probability distribution showing the probability of x successes in n trials,where the probability of success does not change from trial to trial,is termed a

A)uniform probability distribution
B)binomial probability distribution
C)Poisson probability distribution
D)normal probability distribution

Question 18

The expected value for a binomial probability distribution is

A)E(x)  np
B)E(x)  p(1  p)
C)E(x)  np(1  p)
D)E(x)  np(1  n)

Question 19

When dealing with the number of occurrences of an event over a specified interval of time or space and when the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval,the appropriate probability distribution is a

A)binomial distribution
B)Poisson distribution
C)normal distribution
D)hypergeometric distribution

Question 20

In a Poisson probability problem,the rate of errors is one every two hours.To find the probability of three defects in four hours,

A) = 1, x = 4
B) = 2, x = 3
C) = 3, x = 2
D) = 3, x = 6

Question 21

The probability density function matches each value of a continuous random variable with its probability.

Question 22

Whenever the probability is proportional to the length of the interval,the random variable is

A)normally distributed
B)binomially distributed
C)uniformly distributed
D)exponentially distributed

Question 23

The standard normal distribution is a normal distribution with a mean of 1 and a standard deviation of 1/3.

Question 24

If one wanted to find the probability of ten customer arrivals in an hour at a service station,one would generally use the Poisson distribution.

Question 25

The Poisson probability distribution is used with a continuous random variable.

Question 26

Experimental outcomes must occur as numerical values in order to define their probability distribution.

Question 27

The Poisson probability of three events in one hour is the same as the probability of 36 events in one day.

Question 28

If the Poisson distribution describes the number of occurrences per interval,the exponential distribution describes the length of the interval between occurrences.

Question 29

Whenever the probability is proportional to the length of the interval,the random variable is uniformly distributed.

Question 30

The binomial distribution is appropriate to use to find the probability of the elapsed time between successes.

Question 31

The expected value of the discrete random variable x is ,a weighted average of all possible values of x.

Question 32

For an exponentially distributed random variable x with a mean of 8,the probability of x  8 is .5.

Question 33

The probability of a continuous variable having a specific value is 0.

Question 34

A random variable that may assume only a finite or an infinite sequence (e.g.,1,2,3,...)of values is a discrete random variable.

Question 35

An exponential distribution,like the Poisson distribution,can be described by a single parameter.

Question 36

A random variable that has a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a binomial distribution.

Question 37

Suppose that the arrival time for the next bus at a bus stop is uniformly distributed between 1 and 25 minutes.The probability that the bus will arrive within the next 5 minutes is 1/6.

Question 38

A random variable that can take on any value along a line segment is a continuous random variable.

Question 39

Probabilities for discrete random variables are defined over intervals by a probability density function.

Question 40

The binomial probability distribution is most symmetric when p equals .5.

Question 41

During lunch time,customers arrive at Bob's Drugs according to a Poisson distribution with  = 4 per minute.
a.During a one-minute interval, determine the following probabilities: (1) no arrivals; (2) one arrival; (3) two arrivals; and, (4) three or more arrivals.
b.What is the probability of two arrivals in a two-minute period?
c.What is the probability that no more than 30 seconds elapses between arrivals?

Question 42

Use a graph to illustrate,for a continuous random variable,the relationship among P(x = a),P(a < x < b),and P(a  x  b).

Question 43

The weight of a .5 cubic yard bag of landscape mulch in 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?

Question 44

A video rental store has two video cameras available for customers to rent.Historically,demand for cameras has followed this distribution.The revenue per rental is $40.If a customer wants a camera and none is available,the store gives a $15 coupon for snacks.
a.What is the expected demand for camera rentals?
b.What is the expected revenue from camera rentals?
c.What is the expected cost associated with camera rentals?
d.What is the expected profit from camera rentals?

Question 45

A light bulb manufacturer claims his 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 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?

Question 46

A process follows the binomial distribution with n = 8 and p = .3.Find
a.P(x = 3)
b.P(x > 6)
c.P(x  2)

Question 47

Explain how the binomial distribution could be used in quality control to sample 50 parts in a lot of 500.

Question 48

A uniform distribution is defined on the interval from 1  x  5.Graph it,and explain how to determine the density function.

Question 49

Explain the transformation from any normal distribution to the standard normal.

Question 50

Consider a Poisson probability distribution in a manufacturing process with an average of 3 flaws every 100 feet.Find the probability of
a.no flaws in 100 feet
b.2 flaws in 100 feet
c.1 flaws in 150 feet
d.3 or 4 flaws in 150 feet

Question 51

The number of customers at Winkies Donuts between 8:00a.m.and 9:00a.m.is believed to follow a Poisson distribution with a mean of 2 customers per minute.
a.During a randomly selected one minute interval during this time period, what is the probability of 6 customers arriving to Winkies?
b.What is the probability that at least 2 minutes elapse between customer arrivals?

Question 52

The high school GPAs of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories.
a.Complete the table to make this a valid probability distribution.
b.What is the probability an applicant's GPA will be below 3.0?
c.What is the probability an applicant's GPA will be 2.5 or above?

Question 53

Weight,time,and temperature are examples of discrete random variables.

Question 54

Explain how the Poisson and exponential probability distributions relate to each other and how they relate to waiting line models.

Question 55

For a binomial distribution,compare P(x = 3)when n = 10 and p = .4 to P(x = 7)when n = 10 and p = .6.

Question 56

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 continuous 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?

Question 57

A manufacturer of computer disks has a historical defective rate of .001.What is the probability that in a batch of 1000 disks,2 would be defective?

Question 58

A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained.Following is a record of the number of students in a class of 20 who took the test each number of times.
a.Use the relative frequency approach to construct a probability distribution and show that it satisfies the required condition.
b.Find the expected value of the number of tests taken.
c.Compute the variance.
d.Compute the standard deviation.

Question 59

One way a discrete probability distribution can be portrayed is in a table that lists values for x and f(x).Describe two other ways distributions can be represented.

Question 60

After a severe winter,potholes develop in a state highway at the rate of 5.2 per mile.Thirty-five miles of this highway pass through Washington County.
a.How many potholes would you expect to see in the county?
b.What is the probability of finding 8 potholes in 1 mile of highway?

Question 61

Customers at a popular restaurant that refuses reservations arrive according to the Poisson distribution at a rate of 4 parties every 5 minutes.What is the probability that there will be more than 2 minutes between arriving parties?

Question 62

The time it takes to travel from home to the office is normally distributed with  = 25 minutes and  = 5 minutes.
a.What is the probability the trip takes more than 20 minutes?
b.What is the probability the trip takes less than 15 minutes.
c.What is the probability the trip takes between 30 and 35 minutes?
d.What is the probability the trip takes more than 40 minutes?

Question 63

Water's Edge is a clothing retailer that promotes its products via catalog and accepts customer orders by all of the conventional ways including the Internet.The company has gained a competitive advantage by collecting data about its operations and the customer each time an order is processed.
Among the data collected with each order are: number of items ordered,total shipping weight of the order,whether or not all items ordered were available in inventory,time taken to process the order,customer's number of prior orders in the last 12 months,and method of payment.
For each of the six aforementioned variables,identify which of the variables are discrete and which are continuous.

Question 64

Assume that the time required to download a file from the Internet is exponentially distributed with mean equal to 4 minutes.What is the probability that a download will require at least 2 but not more than 4 minutes?

Question 65

Sandy's Pet Center grooms large and small dogs.It takes Sandy 40 minutes to groom a small dog and 70 minutes to groom a large dog.Large dogs account for 20% of Sandy's business.Sandy has 5 appointments tomorrow.
a. What is the probability that all 5 appointments tomorrow are for small dogs?
b. What is the probability that two of the appointments tomorrow are for large dogs?
c. What is the expected amount of time to finish all five dogs tomorrow?

Question 66

Scores on an endurance test for cardiac patients are normally distributed with  = 182 and  = 24.
a.What is the probability a patient will score above 190?
b.What percentage of patients score below 170?
c.What score does a patient at the 75th percentile receive?