# 2.2 Hypothesis testing

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Suppose you measure a collection of scalars ${x}_{1},,{x}_{N}$ . You believe the data is distributed in one of two ways. Your first model, call it ${H}_{0}$ , postulates the data to be governed by the density ${f}_{0}(x)$ (some fixed density). Your second model, ${H}_{1}$ , postulates a different density ${f}_{1}(x)$ . These models, termed hypotheses , are denoted as follows: ${H}_{0}:({x}_{n}, {f}_{0}(x)),n=1N$ ${H}_{1}:({x}_{n}, {f}_{1}(x)),n=1N$ A hypothesis test is a rule that, given a measurement $x$ , makes a decision as to which hypothesis best "explains" the data.

Suppose you are confident that your data is normally distributed with variance 1, but you are uncertain aboutthe sign of the mean. You might postulate ${H}_{0}:({x}_{n}, (-1, 1))$ ${H}_{1}:({x}_{n}, (1, 1))$ These densities are depicted in .

Assuming each hypothesis is a priori equally likely, an intuitively appealing hypothesis test is to compute the sample mean $\langle x\rangle =\frac{1}{N}\sum_{n=1}^{N} {x}_{n}$ , and choose ${H}_{0}$ if $\langle x\rangle \le 0$ , and ${H}_{1}$ if $\langle x\rangle > 0$ . As we will see later, this test is in fact optimal under certain assumptions.

## Generalizations and nomenclature

The concepts introduced above can be extended inseveral ways. In what follows we provide more rigorous definitions, describe different kinds of hypothesis testing, andintroduce terminology.

## Data

In the most general setup, the observation is a collection ${x}_{1},,{x}_{N}$ of random vectors. A common assumption, which facilitates analysis, is that the data are independent and identicallydistributed (IID). The random vectors may be continuous, discrete, or in some cases mixed. It is generally assumedthat all of the data is available at once, although for some applications, such as Sequential Hypothesis Testing , the data is a never ending stream.

## Binary versus m-ary tests

When there are two competing hypotheses, we refer to a binary hypothesis test. When the number of hypotheses is $M\ge 2$ , we refer to an M-ary hypothesis test. Clearly, binary is a special case of $M$ -ary, but binary tests are accorded a special status for certain reasons. These includetheir simplicity, their prevalence in applications, and theoretical results that do not carry over to the $M$ -ary case.

## Phase-shift keying

Suppose we wish to transmit a binary string of length $r$ over a noisy communication channel. We assign each of the $M=2^{r}$ possible bit sequences to a signal ${s}^{k}$ , $k=\{1, , M\}$ where ${s}_{n}^{k}=\cos (2\pi {f}_{0}n+\frac{2\pi (k-1)}{M})$ This symboling scheme is known as phase-shift keying (PSK). After transmitting a signal across the noisy channel, the receiver faces an $M$ -ary hypothesis testing problem: ${H}_{0}:x={s}^{1}+w$  ${H}_{M}:x={s}^{M}+w$ where $(w, (0, ^{2}I))$ .

In many binary hypothesis tests, one hypothesis represents the absence of a ceratinfeature. In such cases, the hypothesis is usually labelled ${H}_{0}$ and called the null hypothesis. The other hypothesis is labelled ${H}_{1}$ and called the alternative hypothesis.

## Waveform detection

Consider the problem of detecting a known signal $s=\left(\begin{array}{c}{s}_{1}\\ \\ {s}_{N}\end{array}\right)$ in additive white Gaussian noise (AWGN). This scenario is common in sonar and radar systems. Denotingthe data as $x=\left(\begin{array}{c}{x}_{1}\\ \\ {x}_{N}\end{array}\right)$ , our hypothesis testing problem is ${H}_{0}:x=w$ ${H}_{1}:x=s+w$ where $(w, (0, ^{2}I))$ . ${H}_{0}$ is the null hypothesis, corresponding to the absence of a signal.

#### Questions & Answers

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
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s.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
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
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s.
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for screen printed electrodes ?
SUYASH
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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I'm interested in nanotube
Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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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
Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
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Azam
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Uday
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how did you get the value of 2000N.What calculations are needed to arrive at it
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