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This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

Test about proportions

Tests of statistical hypotheses are a very important topic, let introduce it through an illustration.

Suppose a manufacturer of a certain printed circuit observes that about p =0.05 of the circuits fails. An engineer and statistician working together suggest some changes that might improve the design of the product. To test this new procedure, it was agreed that n =100 circuits would be produced using the proposed method and the checked. Let Y equal the number of these 200 circuits that fail. Clearly, if the number of failures, Y , is such that Y /200 is about to 0.05, then it seems that the new procedure has not resulted in an improvement. On the other hand, If Y is small so that Y /200 is about 0.01 or 0.02, we might believe that the new method is better than the old one. On the other hand, if Y /200 is 0.08 or 0.09, the proposed method has perhaps caused a greater proportion of failures. What is needed is to establish a formal rule that tells when to accept the new procedure as an improvement. For example, we could accept the new procedure as an improvement if Y 5 of Y / n 0.025 . We do note, however, that the probability of the failure could still be about p =0.05 even with the new procedure, and yet we could observe 5 of fewer failures in n =200 trials.

That is, we would accept the new method as being an improvement when, in fact, it was not. This decision is a mistake which we call a Type I error . On the other hand, the new procedure might actually improve the product so that p is much smaller, say p =0.02, and yet we could observe y =7 failures so that y /200=0.035. Thus we would not accept the new method as resulting in an improvement when in fact it had. This decision would also be a mistake which we call a Type II error .

If it we believe these trials, using the new procedure, are independent and have about the same probability of failure on each trial, then Y is binomial b ( 200 , p ) . We wish to make a statistical inference about p using the unbiased p ^ = Y / 200 . We could also construct a confidence interval, say one that has 95% confidence, obtaining p ^ ± 1.96 p ^ ( 1 p ^ ) 200 .

This inference is very appropriate and many statisticians simply do this. If the limits of this confidence interval contain 0.05, they would not say the new procedure is necessarily better, al least until more data are taken. If, on the other hand, the upper limit of this confidence interval is less than 0.05, then they fell 95% confident that the true p is now less than 0.05. Here, in this illustration, we are testing whether or not the probability of failure has or has not decreased from 0.05 when the new manufacturing procedure is used.

The no change hypothesis, H 0 : p = 0.05 , is called the null hypothesis . Since H 0 : p = 0.05 completely specifies the distribution it is called a simple hypothesis ; thus H 0 : p = 0.05 is a simple null hypothesis .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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