# 7.3 Using the central limit theorem  (Page 3/8)

 Page 3 / 8
(HISTORICAL): Normal Approximation to the Binomial

Historically, being able to compute binomial probabilities was one of the most important applications of the Central Limit Theorem. Binomial probabilities were displayed in a table in a book with a small value for $n$ (say, 20). To calculate the probabilities with large values of $n$ , you had to use the binomial formula which could be very complicated. Using the Normal Approximation to the Binomial simplified the process. To compute the Normal Approximation to the Binomial, take a simple random sample from a population. You must meet the conditionsfor a binomial distribution :

• there are a certain number $n$ of independent trials
• the outcomes of any trial are success or failure
• each trial has the same probability of a success $p$
Recall that if $X$ is the binomial random variable, then $X$ ~ $B\left(n,p\right)$ . The shape of the binomial distribution needs to besimilar to the shape of the normal distribution. To ensure this, the quantities $np$ and $nq$ must both be greater than five ( $np>5$ and $nq>5$ ; the approximation is better if they are both greater than or equal to 10). Then the binomial can be approximated by the normal distribution with mean $\mu =np$ and standard deviation $\sigma =\sqrt{npq}.$ Remember that $q=1-p.$ In order to get the best approximation, add 0.5 to $x$ or subtract 0.5 from $x$ $\left($ use $x+\mathrm{0.5}$ or $x-\mathrm{0.5}$ $\right)$ . The number $\mathrm{0.5}$ is called the continuity correction factor .

Suppose in a local Kindergarten through 12th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K - 5. A simple random sample of 300 is surveyed.

1. Find the probability that at least 150 favor a charter school.
2. Find the probability that at most 160 favor a charter school.
3. Find the probability that more than 155 favor a charter school.
4. Find the probability that less than 147 favor a charter school.
5. Find the probability that exactly 175 favor a charter school.

Let $X=$ the number that favor a charter school for grades K - 5. $X$ ~ $B\left(n,p\right)$ where $n=300$ and $p=\mathrm{0.53}.$ Since $np>5$ and $nq>5,$ use the normal approximation to the binomial. The formulas for the mean and standard deviation are $\mu =np$ and $\sigma =\sqrt{npq}.$ The mean is 159 and the standard deviation is 8.6447. The random variable for the normal distribution is $Y$ . $Y~N\left(159,\mathrm{8.6447}\right)$ . See The Normal Distribution for help with calculator instructions.

For Problem 1., you include 150 so $(P\left(x, 150\right))$ has normal approximation $(P\left(Y, 149.5\right))=\mathrm{0.8641}$ .

normalcdf $\left(149.5,10^99,159,8.6447\right)=0.8641$ .

For Problem 2., you include 160 so $(P\left(x, 160\right))$ has normal approximation $(P\left(Y, 160.5\right))=\mathrm{0.5689}$ .

normalcdf $\left(0,160.5,159,8.6447\right)=0.5689$

For Problem 3., you exclude 155 so $(P\left(x, 155\right))$ has normal approximation $(P\left(y, 155.5\right))=\mathrm{0.6572}$ .

normalcdf $\left(155.5,10^99,159,8.6447\right)=0.6572$

For Problem 4., you exclude 147 so $(P\left(x, 147\right))$ has normal approximation $(P\left(Y, 146.5\right))=\mathrm{0.0741}$ .

normalcdf $\left(0,146.5,159,8.6447\right)=0.0741$

For Problem 5., $P\left(x=175\right)$ has normal approximation $P\left(174.5 .

normalcdf $\left(174.5,175.5,159,8.6447\right)=0.0083$

Because of calculators and computer software that easily let you calculate binomial probabilities for large values of $n$ , it is not necessary to use the the Normal Approximation to the Binomial provided you have access to these technology tools. Most school labs have Microsoft Excel, an example of computer software that calculates binomial probabilities. Many students have access to the TI-83 or 84 series calculators and they easily calculate probabilities for the binomial. In an Internet browser, if you type in "binomial probability distribution calculation," you can find at least one online calculator for the binomial.

For Example 3 , the probabilities are calculated using the binomial ( $n=300$ and $p=0.53$ ) below. Compare the binomial and normal distribution answers. See Discrete Random Variables for help with calculator instructions for the binomial.

$(P\left(x, 150\right))$ : 1 - binomialcdf $\left(300,0.53,149\right)=0.8641$

$(P\left(x, 160\right))$ : binomialcdf $\left(300,0.53,160\right)=0.5684$

$(P\left(x, 155\right))$ : 1 - binomialcdf $\left(300,0.53,155\right)=0.6576$

$(P\left(x, 147\right))$ : binomialcdf $\left(300,0.53,146\right)=0.0742$

$P\left(x=175\right)$ : (You use the binomial pdf.) binomialpdf $\left(175,0.53,146\right)=0.0083$

**Contributions made to Example 2 by Roberta Bloom

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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. By OpenStax By Stephen Voron By Anh Dao By OpenStax By Prateek Ashtikar By Stephen Voron By Briana Knowlton By Laurence Bailen By Madison Christian By OpenStax