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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

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
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 ?
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
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
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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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