# 6.2 The central limit theorem for sample means (averages)

 Page 1 / 5

Suppose $X$ is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose:

• ${\mu }_{X}$ = the mean of $X$
• ${\sigma }_{X}$ = the standard deviation of $X$
If you draw random samples of size $n$ , then as $n$ increases, the random variable $\overline{X}$ which consists of sample means, tends to be normally distributed and

$\overline{X}$ ~ $N\left({\mu }_{X},\frac{{\sigma }_{X}}{\sqrt{n}}\right)$

The Central Limit Theorem for Sample Means says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by $n$ , the sample size. $n$ is the number of values that are averaged together not the number of times the experiment is done.

To put it more formally, if you draw random samples of size $n$ ,the distribution of the random variable $\overline{X}$ , which consists of sample means, is called the sampling distribution of the mean . The sampling distribution of the mean approaches a normal distribution as $n$ , the sample size, increases.

The random variable $\overline{X}$ has a different z-score associated with it than the random variable $X$ . $\overline{x}$ is the value of $\overline{X}$ in one sample.

$z=\frac{\overline{x}-{\mu }_{X}}{\left(\frac{{\sigma }_{X}}{\sqrt{n}}\right)}$

${\mu }_{X}$ is both the average of $X$ and of $\overline{X}$ .

${\sigma }_{\overline{X}}=\frac{{\sigma }_{X}}{\sqrt{n}}=$ standard deviation of $\overline{X}$ and is called the standard error of the mean.

An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size $n$ = 25 are drawn randomly from the population.

Find the probability that the sample mean is between 85 and 92.

Let $X$ = one value from the original unknown population. The probability question asks you to find a probability for the sample mean .

Let $\overline{X}=$ the mean of a sample of size 25. Since ${\mu }_{X}=90$ , ${\sigma }_{X}=15$ , and ${n}_{}=25$ ;

then $\overline{X}$ ~ $N\left(90,\frac{15}{\sqrt{25}}\right)$

Find $((P\left(85, \overline{x}), 92\right))\phantom{\rule{20pt}{0ex}}$ Draw a graph.

$((P\left(85, \overline{x}), 92\right))=0.6997$

The probability that the sample mean is between 85 and 92 is 0.6997.

TI-83 or 84: normalcdf (lower value, upper value, mean, standard error of the mean)

The parameter list is abbreviated (lower value, upper value, $\mu$ , $\frac{\sigma }{\sqrt{n}}$ )

normalcdf $\mathrm{\left(85,92,90,}\frac{15}{\sqrt{25}}\mathrm{\right) = 0.6997}$

Find the value that is 2 standard deviations above the expected value (it is 90) of the sample mean.

To find the value that is 2 standard deviations above the expected value 90, use the formula

value = ${\mu }_{X}+\text{(#ofSTDEVs)}\left(\frac{{\sigma }_{X}}{\sqrt{n}}\right)$

value = $90+2\cdot \frac{15}{\sqrt{25}}=96$

So, the value that is 2 standard deviations above the expected value is 96.

The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours . A sample of size $n$ = 50 is drawn randomly from the population.

Find the probability that the sample mean is between 1.8 hours and 2.3 hours.

Let $X$ = the time, in hours, it takes to play one soccer match.

The probability question asks you to find a probability for the sample mean time, in hours , it takes to play one soccer match.

Let $\overline{X}$ = the mean time, in hours, it takes to play one soccer match.

If ${\mu }_{X}=$ _________, ${\sigma }_{X}=$ __________, and $n=$ ___________, then $\overline{X}~N\text{(______, ______)}$ by the Central Limit Theorem for Means.

${\mu }_{X}=$ 2 , ${\sigma }_{X}=$ 0.5 , $n=$ 50 , and $X~N\left(2,\frac{0.5}{\sqrt{50}}\right)$

Find $((P\left(1.8, \overline{x}), 2.3\right))$ . $\phantom{\rule{20pt}{0ex}}$ Draw a graph.

$((P\left(1.8, \overline{x}), 2.3\right))=0.9977$

normalcdf $\mathrm{\left(1.8,2.3,2,}\frac{.5}{\sqrt{50}}\mathrm{\right) = 0.9977}$

The probability that the mean time is between 1.8 hours and 2.3 hours is ______.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
Got questions? Join the online conversation and get instant answers!