<< Chapter < Page Chapter >> Page >

Uniform : Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Let X = the time, in minutes, it takes a nine-year old child to eat a donut. Then X ~ U ( 0.5 , 4 ) .

The probability that a randomly selected nine-year old child eats a donut in at least two minutes is _______.

0.5714

Got questions? Get instant answers now!

Find the probability that a different nine-year old child eats a donut in more than 2 minutes given that the child has already been eating the donut for more than 1.5 minutes.

The second probability question has a conditional (refer to " Probability Topics "). You are asked to find the probability that a nine-year old child eats a donut in more than 2 minutes given that the child has already been eating the donut for more than 1.5 minutes. Solve the problem two different ways (see the first example ). You must reduce the sample space. First way : Since you already know the child has already been eating the donut for more than 1.5 minutes, you are no longer starting at a = 0.5 minutes. Your starting point is 1.5 minutes.

Write a new f(x):

f x = 1 4 - 1.5 = 2 5 for 1.5 x 4 .

Find P ( x 2 | x 1.5 ) . Draw a graph.

f(X)=2/5 graph displaying a boxed region consisting of a horizontal line extending to the right from point 2/5 on the y-axis, a vertical upward line from points 1.5 and 4 on the x-axis, and the x-axis. A shaded region from points 2-4 occurs within this area.

P ( x 2 | x 1.5 ) = ( base ) ( new height ) = ( 4 - 2 ) ( 2 / 5 ) = ?

4 5

Got questions? Get instant answers now!

The probability that a nine-year old child eats a donut in more than 2 minutes given that the child has already been eating the donut for more than 1.5 minutes is 4 5 .

Second way: Draw the original graph for x ~ U ( 0.5 , 4 ) . Use the conditional formula

P ( x 2 | x 1.5 ) P ( x > 2 AND x > 1.5 ) P ( x 1.5 ) P ( x 2 ) P ( x 1.5 ) 2 3.5 2.5 3.5 0.8 = 4 5

Uniform : Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and 4 hours. Let x = the time needed to fix a furnace. Then x ~ U ( 1.5 , 4 ) .

  1. Find the problem that a randomly selected furnace repair requires more than 2 hours.
  2. Find the probability that a randomly selected furnace repair requires less than 3 hours.
  3. Find the 30th percentile of furnace repair times.
  4. The longest 25% of repair furnace repairs take at least how long? (In other words: Find the minimum time for the longest 25% of repair times.) What percentile does this represent?
  5. Find the mean and standard deviation

Find the probability that a randomly selected furnace repair requires longer than 2 hours.

To find f x : f x 1 4 - 1.5 1 2.5 so f x = 0.4

P(x>2) = (base)(height) = (4 − 2)(0.4) = 0.8

Example 4 figure 1

Uniform Distribution between 1.5 and 4 with shaded area between 2 and 4 representing the probability that the repair time x is greater than 2

Got questions? Get instant answers now!

Find the probability that a randomly selected furnace repair requires less than 3 hours. Describe how the graph differs from the graph in the first part of this example.

P ( x 3 ) = (base)(height) = (3 − 1.5)(0.4) = 0.6

The graph of the rectangle showing the entire distribution would remain the same. However the graph should be shaded between x=1.5 and x=3. Note that the shaded area starts at x=1.5 rather than at x=0; since X~U(1.5,4), x can not be less than 1.5.

Example 4 figure 2

Uniform Distribution between 1.5 and 4 with shaded area between 1.5 and 3 representing the probability that the repair time x is less than 3

Got questions? Get instant answers now!

Find the 30th percentile of furnace repair times.

Example 4 figure 3

Uniform Distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30% of repair times.

P ( x k ) 0.30

P ( x k ) ( base ) ( height ) = ( k - 1.5 ) ( 0.4 )

  • 0.3 = (k − 1.5) (0.4) ; Solve to find k:
  • 0.75 = k − 1.5 , obtained by dividing both sides by 0.4
  • k = 2.25 , obtained by adding 1.5 to both sides

The 30th percentile of repair times is 2.25 hours. 30% of repair times are 2.5 hours or less.

Got questions? Get instant answers now!

The longest 25% of furnace repair times take at least how long? (Find the minimum time for the longest 25% of repairs.)

Example 4 figure 4

Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25% of repair times.

P ( x k ) 0.25

P ( x k ) ( base ) ( height ) = ( 4 - k ) ( 0.4 )

  • 0.25 = (4 − k)(0.4) ; Solve for k:
  • 0.625 = 4 − k , obtained by dividing both sides by 0.4
  • −3.375 = −k , obtained by subtracting 4 from both sides
  • k=3.375

The longest 25% of furnace repairs take at least 3.375 hours (3.375 hours or longer).

Note: Since 25% of repair times are 3.375 hours or longer, that means that 75% of repair times are 3.375 hours or less. 3.375 hours is the 75th percentile of furnace repair times.

Got questions? Get instant answers now!

Find the mean and standard deviation

μ a + b 2 and σ ( b - a ) 2 12

μ 1.5 + 4 2 2.75 hours and σ ( 4 - 1.5 ) 2 12 0.7217 hours

Got questions? Get instant answers now!

**Example 5 contributed by Roberta Bloom

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
Nerisha Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics' conversation and receive update notifications?

Ask