Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Collaborative Statistics (with edits: Teegarden) » Homework

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • College Open Textbooks display tagshide tags

    This module is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook CollaborativeAs a part of collection: "Collaborative Statistics"

    Comments:

    "Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"

    Click the "College Open Textbooks" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

  • JVLA Endorsed

    This module is included inLens: Jesuit Virtual Learning Academy Endorsed Material
    By: Jesuit Virtual Learning AcademyAs a part of collection: "Collaborative Statistics"

    Comments:

    "This is a robust collection (textbook) approved by the College Board as a resource for the teaching of AP Statistics. "

    Click the "JVLA Endorsed" link to see all content they endorse.

  • WebAssign display tagshide tags

    This module is included inLens: WebAssign The Independent Online Homework and Assessment Solution
    By: WebAssignAs a part of collection: "Collaborative Statistics"

    Comments:

    "Online homework and assessment available from WebAssign."

    Click the "WebAssign" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Collaborative Statistics"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Bookshare

    This module is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech InitiativeAs a part of collection: "Collaborative Statistics"

    Comments:

    "DAISY and BRF versions of this collection are available."

    Click the "Bookshare" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Collaborative Statistics"

    Comments:

    "Collaborative Statistics was written by two faculty members at De Anza College in Cupertino, California. This book is intended for introductory statistics courses being taken by students at two- […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • statistics display tagshide tags

    This module is included inLens: Statistics
    By: Brylie OxleyAs a part of collection: "Collaborative Statistics"

    Click the "statistics" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Lucy Van Pelt display tagshide tags

    This module is included inLens: Lucy's Lens
    By: Tahiya MaromeAs a part of collection: "Collaborative Statistics"

    Comments:

    "Part of the Books featured on Community College Open Textbook Project"

    Click the "Lucy Van Pelt" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Educational Technology Lens display tagshide tags

    This module is included inLens: Educational Technology
    By: Steve WilhiteAs a part of collection: "Collaborative Statistics"

    Click the "Educational Technology Lens" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Statistics

    This module is included inLens: Mathieu Plourde's Lens
    By: Mathieu PlourdeAs a part of collection: "Collaborative Statistics"

    Click the "Statistics" link to see all content selected in this lens.

  • statf12

    This module is included inLens: Statistics Fall 2012
    By: Alex KolesnikAs a part of collection: "Collaborative Statistics"

    Click the "statf12" link to see all content selected in this lens.

  • UTEP display tagshide tags

    This module is included inLens: Amy Wagler's Lens
    By: Amy WaglerAs a part of collection: "Collaborative Statistics"

    Click the "UTEP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Make Textbooks Affordable

    This module is included inLens: Make Textbooks Affordable
    By: Nicole AllenAs a part of collection: "Collaborative Statistics"

    Click the "Make Textbooks Affordable" link to see all content selected in this lens.

  • BUS204 Homework display tagshide tags

    This module is included inLens: Saylor BUS 204 Homework
    By: David BourgeoisAs a part of collection: "Collaborative Statistics"

    Comments:

    "Homework for Discrete Variables/Probability. "

    Click the "BUS204 Homework" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • crowe

    This module is included in aLens by: Chris RoweAs a part of collection: "Collaborative Statistics"

    Click the "crowe" link to see all content selected in this lens.

  • Bio 502 at CSUDH display tagshide tags

    This module is included inLens: Bio 502
    By: Terrence McGlynnAs a part of collection: "Collaborative Statistics"

    Comments:

    "This is the course textbook for Biology 502 at CSU Dominguez Hills"

    Click the "Bio 502 at CSUDH" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Homework

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides a number of homework exercises related to Continuous Random Variables.

For each probability and percentile problem, DRAW THE PICTURE!

Exercise 1

Consider the following experiment. You are one of 100 people enlisted to take part in a study to determine the percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer “yes” or “no.” You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor.

  • a. What part of the experiment will yield discrete data?
  • b. What part of the experiment will yield continuous data?

Exercise 2

When age is rounded to the nearest year, do the data stay continuous, or do they become discrete? Why?

Exercise 3

Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a Uniform Distribution from 1 – 53 (spread of 52 weeks).

  • a. X ~ X size 12{X} {} ~
  • b. Graph the probability distribution.
  • c. f ( x ) = f ( x ) size 12{f \( x \) } {} =
  • d. μ = μ size 12{μ} {} =
  • e. σ = σ size 12{σ} {} =
  • f. Find the probability that a person is born at the exact moment week 19 starts. That is, find P ( x = 19 ) = P ( x = 19 ) size 12{P \( X="19" \) } {} =
  • g. P ( 2 < x < 31 ) = P ( 2 < x < 31 ) = size 12{P \( 2<X<"31" \) ={}} {}
  • h. Find the probability that a person is born after week 40.
  • i. {} P ( 12 < x x < 28 ) = P ( 12 < x x < 28 ) size 12{P \( "12"<X \lline X<"28" \) } {} =
  • j. Find the 70th percentile.
  • k. Find the minimum for the upper quarter.

Solution

  • a. X ~ U ( 1, 53 ) X ~ U ( 1, 53 ) size 12{X " ~ " U \( 1,"53" \) } {}
  • c. f(x)=152f(x)=152 size 12{f \( x \) = { {1} over { \( b - a \) } } = { {1} over { \( "53" - 1 \) } } = { {1} over {"52"} } } {} where 1x531x53 size 12{1 <= x <= "53"} {}
  • d. 27
  • e. 15.01
  • f. 0
  • g. 29522952
  • h. 13521352
  • i. 16271627
  • j. 37.4
  • k. 40

Exercise 4

A random number generator picks a number from 1 to 9 in a uniform manner.

  • a. X ~ X ~ size 12{X "~" } {}
  • b. Graph the probability distribution.
  • c. f ( x ) = f ( x ) = size 12{f \( x \) ={}} {}
  • d. μ = μ = size 12{μ={}} {}
  • e. σ = σ = size 12{σ={}} {}
  • f. P ( 3 . 5 < x < 7 . 25 ) = P ( 3 . 5 < x < 7 . 25 ) = size 12{P \( 3 "." 5<X<7 "." "25" \) ={}} {}
  • g. P ( x > 5 . 67 ) = P ( x > 5 . 67 ) = size 12{P \( X>5 "." "67" \) ={}} {}
  • h. P ( x > 5 x > 3 ) = P ( x > 5 x > 3 ) = size 12{P \( X>5 \lline X>3 \) ={}} {}
  • i. Find the 90th percentile.

Exercise 5

The time (in minutes) until the next bus departs a major bus depot follows a distribution with f(x)=120f(x)=120 where xx goes from 25 to 45 minutes.

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. X ~ X ~ size 12{X "~" } {}
  • c. Graph the probability distribution.
  • d. The distribution is ______________ (name of distribution). It is _____________ (discrete or continuous).
  • e. μ = μ = size 12{μ={}} {}
  • f. σ = σ = size 12{σ={}} {}
  • g. Find the probability that the time is at most 30 minutes. Sketch and label a graph of the distribution. Shade the area of interest. Write the answer in a probability statement.
  • h. Find the probability that the time is between 30 and 40 minutes. Sketch and label a graph of the distribution. Shade the area of interest. Write the answer in a probability statement.
  • i. P(25<x<55)=P(25<x<55)= size 12{P \( "25"<X<"55" \) ={}} {} _________. State this in a probability statement (similar to g and h ), draw the picture, and find the probability.
  • j. Find the 90th percentile. This means that 90% of the time, the time is less than _____ minutes.
  • k. Find the 75th percentile. In a complete sentence, state what this means. (See j.)
  • l. Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes.

Solution

  • b. X ~ U ( 25 , 45 ) X ~ U ( 25 , 45 ) size 12{X "~" U \( "25","45" \) } {}
  • d. uniform; continuous
  • e. 35 minutes
  • f. 5.8 minutes
  • g. 0.25
  • h. 0.5
  • i. 1
  • j. 43 minutes
  • k. 40 minutes
  • l. 0.3333

Exercise 6

According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let’s suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month. (Source: The McDougall Program for Maximum Weight Loss by John A. McDougall, M.D.)

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. X ~ X ~ size 12{X "~" } {}
  • c. Graph the probability distribution.
  • d. f ( x ) = f ( x ) = size 12{f \( x \) ={}} {}
  • e. μ = μ = size 12{μ={}} {}
  • f. σ = σ = size 12{σ={}} {}
  • g. Find the probability that the individual lost more than 10 pounds in a month.
  • h. Suppose it is known that the individual lost more than 10 pounds in a month. Find the probability that he lost less than 12 pounds in the month.
  • i. P ( 7 < x < 13 x > 9 ) = P ( 7 < x < 13 x > 9 ) = size 12{P \( 7<X<"13" \lline X>9 \) ={}} {} __________. State this in a probability question (similar to g and h), draw the picture, and find the probability.

Exercise 7

A subway train on the Red Line arrives every 8 minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. X ~ X ~ size 12{X "~" } {}
  • c. Graph the probability distribution.
  • d. f ( x ) = f ( x ) = size 12{f \( x \) ={}} {}
  • e. μ = μ = size 12{μ={}} {}
  • f. σ = σ = size 12{σ={}} {}
  • g. Find the probability that the commuter waits less than one minute.
  • h. Find the probability that the commuter waits between three and four minutes.
  • i. 60% of commuters wait more than how long for the train? State this in a probability question (similar to g and h), draw the picture, and find the probability.

Solution

  • b. X ~ U ( 0,8 ) X ~ U ( 0,8 ) size 12{X "~" U \( 0,8 \) } {}
  • d. f(x)=18f(x)=18 where 0x80x8
  • e. 4
  • f. 2.31
  • g. 1818
  • h. 1818
  • i. 3.2

Exercise 8

The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. We randomly select one first grader from the class.

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. X ~ X ~ size 12{X "~" } {}
  • c. Graph the probability distribution.
  • d. f ( x ) = f ( x ) = size 12{f \( x \) ={}} {}
  • e. μ = μ = size 12{μ={}} {}
  • f. σ = σ = size 12{σ={}} {}
  • g. Find the probability that she is over 6.5 years.
  • h. Find the probability that she is between 4 and 6 years.
  • i. Find the 70th percentile for the age of first graders on September 1 at Garden Elementary School.

Exercise 9

Let X~X~ size 12{X "~" } {}Exp(0.1)

  • a. decay rate=
  • b. μ = μ = size 12{μ={}} {}
  • c. Graph the probability distribution function.
  • d. On the above graph, shade the area corresponding to P(x<6)P(x<6) size 12{P \( X<6 \) } {} and find the probability.
  • e. Sketch a new graph, shade the area corresponding to P(3<x<6)P(3<x<6) size 12{P \( 3<X<6 \) } {} and find the probability.
  • f. Sketch a new graph, shade the area corresponding to P(x>7)P(x>7) size 12{P \( X>7 \) } {} and find the probability.
  • g. Sketch a new graph, shade the area corresponding to the 40th percentile and find the value.
  • h. Find the average value of xx size 12{X} {}.

Solution

  • a. 0.1
  • b. 10
  • d. 0.4512
  • e. 0.1920
  • f. 0.4966
  • g. 5.11
  • h. 10

Exercise 10

Suppose that the length of long distance phone calls, measured in minutes, is known to have an exponential distribution with the average length of a call equal to 8 minutes.

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. Is XX size 12{X} {} continuous or discrete?
  • c. X ~ X ~ size 12{X "~" } {}
  • d. μ = μ = size 12{μ={}} {}
  • e. σ = σ = size 12{σ={}} {}
  • f. Draw a graph of the probability distribution. Label the axes.
  • g. Find the probability that a phone call lasts less than 9 minutes.
  • h. Find the probability that a phone call lasts more than 9 minutes.
  • i. Find the probability that a phone call lasts between 7 and 9 minutes.
  • j. If 25 phone calls are made one after another, on average, what would you expect the total to be? Why?

Exercise 11

Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025. We are interested in the life of the battery.

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. Is XX size 12{X} {} continuous or discrete?
  • c. X ~ X ~ size 12{X "~" } {}
  • d. On average, how long would you expect 1 car battery to last?
  • e. On average, how long would you expect 9 car batteries to last, if they are used one after another?
  • f. Find the probability that a car battery lasts more than 36 months.
  • g. 70% of the batteries last at least how long?

Solution

  • c. X ~ Exp ( 0.025 ) X ~ Exp ( 0.025 ) size 12{X "~" "Exp" \( { {1} over {5} } \) } {}
  • d. 40 months
  • e. 360 months
  • f. 0.4066
  • g. 14.27

Exercise 12

The percent of persons (ages 5 and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of 9.848 . Suppose we randomly pick a state. (Source: Bureau of the Census, U.S. Dept. of Commerce)

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. Is XX size 12{X} {} continuous or discrete?
  • c. X ~ X ~ size 12{X "~" } {}
  • d. μ = μ = size 12{μ={}} {}
  • e. σ = σ = size 12{σ={}} {}
  • f. Draw a graph of the probability distribution. Label the axes.
  • g. Find the probability that the percent is less than 12.
  • h. Find the probability that the percent is between 8 and 14.
  • i. The percent of all individuals living in the United States who speak a language at home other than English is 13.8 .
    • i. Why is this number different from 9.848%?
    • ii. What would make this number higher than 9.848%?

Exercise 13

The time (in years) after reaching age 60 that it takes an individual to retire is approximately exponentially distributed with a mean of about 5 years. Suppose we randomly pick one retired individual. We are interested in the time after age 60 to retirement.

  • a. Define the random variable. X = X = size 12{X={}} {}
  • b. Is XX size 12{X} {} continuous or discrete?
  • c. X ~ X ~ size 12{X "~" } {}
  • d. μ = μ = size 12{μ={}} {}
  • e. σ = σ = size 12{σ={}} {}
  • f. Draw a graph of the probability distribution. Label the axes.
  • g. Find the probability that the person retired after age 70.
  • h. Do more people retire before age 65 or after age 65?
  • i. In a room of 1000 people over age 80, how many do you expect will NOT have retired yet?

Solution

  • c. X ~ Exp ( 1 5 ) X ~ Exp ( 1 5 ) size 12{X "~" "Exp" \( { {1} over {5} } \) } {}
  • d. 5
  • e. 5
  • g. 0.1353
  • h. Before
  • i. 18.3

Exercise 14

The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150.

  • a. Define the random variable. X = X = size 12{σ={}} {}
  • b. X ~ X ~ size 12{X "~" } {}
  • c. μ = μ = size 12{μ={}} {}
  • d. σ = σ = size 12{σ={}} {}
  • e. Draw a graph of the probability distribution. Label the axes.
  • f. Find the probability that a car required over $300 for maintenance during its first year.

Try these multiple choice problems

The next three questions refer to the following information. The average lifetime of a certain new cell phone is 3 years. The manufacturer will replace any cell phone failing within 2 years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.

Exercise 15

The decay rate is

  • A. 0.3333
  • B. 0.5000
  • C. 2.0000
  • D. 3.0000

Solution

A

Exercise 16

What is the probability that a phone will fail within 2 years of the date of purchase?

  • A. 0.8647
  • B. 0.4866
  • C. 0.2212
  • d. 0.9997

Solution

B

Exercise 17

What is the median lifetime of these phones (in years)?

  • A. 0.1941
  • B. 1.3863
  • C. 2.0794
  • D. 5.5452

Solution

C

The next three questions refer to the following information. The Sky Train from the terminal to the rental car and long term parking center is supposed to arrive every 8 minutes. The waiting times for the train are known to follow a uniform distribution.

Exercise 18

What is the average waiting time (in minutes)?

  • A. 0.0000
  • B. 2.0000
  • C. 3.0000
  • D. 4.0000

Solution

D

Exercise 19

Find the 30th percentile for the waiting times (in minutes).

  • A. 2.0000
  • B. 2.4000
  • C. 2.750
  • D. 3.000

Solution

B

Exercise 20

The probability of waiting more than 7 minutes given a person has waited more than 4 minutes is?

  • A. 0.1250
  • B. 0.2500
  • C. 0.5000
  • D. 0.7500

Solution

B

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks