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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
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I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
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Commplementary angles
Idrissa Reply
im all ears I need to learn
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
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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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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