<< Chapter < Page Chapter >> Page >
Homework problems for Central Limit Theorem for the textbook collection Collaborative Statistics by Susan Dean and Dr. Barbara Illowsky; their original module has been modified to include more questions about CLT for means and eliminate questions about CLT for sums.

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. 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 \( {overline {X}}<"60" \) ={}} {}
  • Find the 30th percentile.
  • P ( 56 < X ¯ < 62 ) = size 12{P \( "56"<{overline {X}}<"62" \) ={}} {}
  • P ( 18 < X ¯ < 58 ) = size 12{P \( "18"<{overline {X}}<"58" \) ={}} {}
  • Find the minimum value for the upper quartile.
  • Xbar ~ N ( 60 , 9 25 ) size 12{ ital "Xbar" "~" N \( "60", { {9} over { sqrt {"25"} } } \) } {}
  • 0.5000
  • 59.06
  • 0.8536
  • 0.1333
  • 61.214

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

Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $0.88. Suppose that we randomly pick 25 daytime statistics students.

  • In words, X = size 12{X={}} {}
  • X ~ size 12{X "~" } {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {} ______ ( ______ , ______ )
  • Find the probability that an individual had between $0.80 and $1.00. Graph the situation and shade in the area to be determined.
  • Find the probability that the average of the 25 students was between $0.80 and $1.00. Graph the situation and shade in the area to be determined.
  • Explain the why there is a difference in (e) and (f).

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

Question removed from textbook.

Note: Problem has been changed from original version of textbook.

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 25 trials.

  • Find the probability that the average length of the 25 trials is at least 24 days.
  • Find the 10th percentile for the average length for samples of 25 trials of this type.
  • Answer: 0.0161
  • Answer: 19.2 days

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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
Do you know which machine is used to that process?
s.
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
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
what is the k.e before it land
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Collaborative statistics homework book: custom version modified by r. bloom. OpenStax CNX. Dec 23, 2009 Download for free at http://legacy.cnx.org/content/col10619/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics homework book: custom version modified by r. bloom' conversation and receive update notifications?

Ask