Deck 3: Probability Distributions

Question 1

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.

Answer 1

True

Question 2

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

Answer 2

False

Question 3

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

Answer 3

True

Question 4

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

Question 5

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

Question 6

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 7

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

Question 8

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 9

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

Question 10

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

Question 11

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 12

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

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

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 22

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 23

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 24

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 25

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

Question 26

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 27

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 28

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 29

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 30

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 31

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 32

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

Question 33

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 34

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 35

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 36

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 37

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

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

Question 38

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 39

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.

Question 40

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.

Question 41

If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability of 4 successes in 15 trials,the correct probability function to use is the

A)standard normal probability density function
B)normal probability density function
C)Poisson probability function
D)binomial probability function

Question 42

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

A)​binomial probability distribution
B)​Poisson probability distribution
C)​uniform probability distribution
D)​exponential probability distribution

Question 43

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

A)binomial probability distribution
B)Poisson probability distribution
C)hypergeometric probability distribution
D)exponential probability distribution

Question 44

​Variance is

A)​a measure of the average,or central value of a random variable
B)​a measure of the dispersion of a random variable
C)​the square root of the standard deviation
D)​the sum of the deviation of data elements from the mean

Question 45

Which of the following is not a property of a binomial experiment?​

A)​the experiment consists of a sequence of n identical trials
B)​each outcome can be referred to as a success or a failure
C)​the probabilities of the two outcomes can change from one trial to the next
D)​the trials are independent

Question 46

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 47

Experimental outcomes that are based on measurement scales such as time,weight,and distance can be described by _____ random variables.

A)discrete
B)continuous
C)uniform
D)intermittent

Question 48

Which of the following is not a property of a binomial experiment?

A)the experiment consists of a sequence of n identical trials
B)each outcome can be referred to as a success or a failure
C)the probabilities of the two outcomes can change from one trial to the next
D)the trials are independent

Question 49

A binomial probability distribution with p = .3 is​

A)​negatively skewed
B)​symmetric
C)​positively skewed
D)​bimodal

Question 50

A binomial probability distribution with p = .3 is

A)negatively skewed
B)symmetrical
C)positively skewed
D)bimodal

Question 51

If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability of 4 successes in 15 trials,the correct probability function to use is the

A)​standard normal probability density function
B)​normal probability density function
C)​Poisson probability function
D)​binomial probability function

Question 52

Variance is

A)a measure of the average,or central value of a random variable
B)a measure of the dispersion of a random variable
C)the square root of the standard deviation
D)the sum of the deviation of data elements from the mean

Question 53

Z is a standard normal random variable.The P(1.05 $\leq$ z $\leq$ 2.13)equals​

A)​0.8365
B)​0.1303
C)​0.4834
D)​0.6618

Question 54

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 55

Z is a standard normal random variable.The P(1.20 $\leq$ z $\leq$ 1.85)equals​

A)​0.4678
B)​0.3849
C)​0.8527
D)​0.0829

Question 56

For a standard normal distribution,the probability of obtaining a z value of less than 1.6 is​

A)​0.1600
B)​0.0160
C)​0.0016
D)​0.9452

Question 57

Z is a standard normal random variable.What is the value of z if the area to the right of z is 0.9803?​

A)​-2.06
B)​0.4803
C)​0.0997
D)​3.06

Question 58

During lunch time,customers arrive at Bob's Drugs according to a Poisson distribution with λ = 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 59

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 60

Experimental outcomes that are based on measurement scales such as time,weight,and distance can be described by _____ random variables.​

A)​discrete
B)​continuous
C)​uniform
D)​intermittent

Question 61

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 62

Scores on an endurance test for cardiac patients are normally distributed with μ = 182 and σ = 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?

Question 63

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 64

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 65

A salesperson contacts eight potential customers per day.From past experience,we know that the probability of a potential customer making a purchase is 0.10.
a.Define the random variable in words for this experiment.
b.What is the probability the salesperson will make exactly two sales in a day?
c.What is the probability the salesperson will make at least two sales in a day?
d.What percentage of days will the salesperson not make a sale?
e.What is the expected number of sales per day?
​

Question 66

The time it takes to travel from home to the office is normally distributed with μ = 25 minutes and σ = 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 67

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

Question 68

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 69

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 70

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?
​

Question 71

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 72

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?
​

Question 73

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 74

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 75

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 76

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?
​

Question 77

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 78

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?
c.What is the probability that it will take a worker between 6 and 10 minutes to complete the task?
​

Question 79

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 80

Twenty-five percent of all resumes received by a corporation for a management position are from females.Fifteen resumes will be received tomorrow.
a.Define the random variable in words for this experiment.
b.What is the probability that exactly 5 of the resumes will be from females?
c.What is the probability that fewer than 3 of the resumes will be from females?
d.What is the expected number of resumes from women?
e.What is the variance of the number of resumes from women?
​