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The Central Limit Theorem: Homework is part of the collection col10555 written by Barbara Illowsky and Susan Dean.

X ~ N ( 60 , 9 ) size 12{X "~" N \( "60",9 \) } {} . Suppose that you form random samples of 25 from this distribution. Let X ¯ size 12{ {overline {X}} } {} be the random variable of averages. Let ΣX size 12{ΣX} {} be the random variable of sums. For c - f , sketch the graph, shade the region, label and scale the horizontal axis for X ¯ size 12{ {overline {X}} } {} , and find the probability.

  • Sketch the distributions of X size 12{X} {} and X ¯ size 12{ {overline {X}} } {} on the same graph.
  • X ¯ size 12{ {overline {X}} "~" } {} ~
  • P ( x ¯ < 60 ) = size 12{P \( {overline {x}}<"60" \) ={}} {}
  • P ( 56 < x ¯ < 62 ) = size 12{P \( "56"<{overline {x}}<"62" \) ={}} {}
  • P ( 18 < x ¯ < 58 ) = size 12{P \( "18"<{overline {x}}<"58" \) ={}} {}
  • Σx size 12{Σ} {} ~
  • Find the minimum value for the upper quartile for the sum.
  • P ( 1400 < Σx < 1550 ) = size 12{P \( "1400"<Σx<"1550" \) ={}} {}
  • Xbar ~ N ( 60 , 9 25 ) size 12{ ital "Xbar" "~" N \( "60", { {9} over { sqrt {"25"} } } \) } {}
  • 0.5000
  • 0.8536
  • 0.1333
  • 1530.35
  • 0.8536

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

  • When the sample size is large, the mean of X ¯ size 12{ {overline {X}} } {} is approximately equal to the mean of X size 12{X} {} .
  • When the sample size is large, X ¯ size 12{ {overline {X}} } {} is approximately normally distributed.
  • When the sample size is large, the standard deviation of X ¯ size 12{ {overline {X}} } {} is approximately the same as the standard deviation of X size 12{X} {} .

The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about 10. Suppose that 16 individuals are randomly chosen.

Let X ¯ = size 12{ {overline {X}} ={}} {} average percent of fat calories.

  • X ¯ ~ size 12{ {overline {X}} "~" } {} ______ ( ______ , ______ )
  • For the group of 16, find the probability that the average percent of fat calories consumed is more than 5. Graph the situation and shade in the area to be determined.
  • Find the first quartile for the average percent of fat calories.
  • N ( 36 , 10 16 ) size 12{ ital "Xbar" "~" N \( "36", { {"10"} over { sqrt {"16"} } } \) } {}
  • 1
  • 34.31

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

  • If X ¯ = size 12{ {overline {X}} ={}} {} average distance in feet for 49 fly balls, then X ¯ ~ size 12{ {overline {X}} "~" } {} ______ ( ______ , ______ )
  • What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X ¯ size 12{ {overline {X}} } {} . Shade the region corresponding to the probability. Find the probability.
  • Find the 80th percentile of the distribution of the average of 49 fly balls.
  • N ( 250 , 50 49 ) size 12{ ital "Xbar" "~" N \( "250", { {"50"} over { sqrt {"49"} } } \) } {}
  • 0.0808
  • 256.01 feet

Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from 2 to 6 pounds. We randomly survey 64 homes with children.

  • In words, X = size 12{X={}} {}
  • X ~ size 12{X "~" } {}
  • μ X = size 12{μ rSub { size 8{X} } ={}} {}
  • σ X = size 12{σ rSub { size 8{X} } ={}} {}
  • In words, ΣX = size 12{ΣX={}} {}
  • ΣX ~ size 12{ΣX "~" } {}
  • Find the probability that the total weight of open boxes is less than 250 pounds.
  • Find the 35th percentile for the total weight of open boxes of cereal.

Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of 7 days. We randomly sample 9 trials.

  • In words, ΣX = size 12{ΣX={}} {}
  • ΣX ~ size 12{ΣX "~" } {}
  • Find the probability that the total length of the 9 trials is at least 225 days.
  • 90 percent of the total of 9 of these types of trials will last at least how long?
  • The total length of time for 9 criminal trials
  • N ( 189 , 21 ) size 12{ ital "SumX" "~" N \( 9 * "21", sqrt {9} * 7 \) } {}
  • 0.0432
  • 162.09

According to the Internal Revenue Service, the average length of time for an individual to complete (record keep, learn, prepare, copy, assemble and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is 2 hours. Suppose we randomly sample 36 taxpayers.

  • In words, X = size 12{X={}} {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {}
  • Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.
  • Would you be surprised if one taxpayer finished his Form 1040 in more than 12 hours? In a complete sentence, explain why.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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