# OpenStax_CNX

You are here: Home » Content » Collaborative Statistics » Homework

### 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?

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

#### In these lenses

• Lucy Van Pelt

This collection is included inLens: Lucy's Lens
By: Tahiya Marome

"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 to display tags associated with this content.

• Bio 502 at CSUDH

This collection is included inLens: Bio 502
By: Terrence McGlynn

"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 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.

Inside Collection (Textbook):

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

# Homework

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} {}.

• 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

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

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

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

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

B

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

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

#### 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?

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?

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