Page 3 / 3

In general, without changing the sample size or the type of the test of the hypothesis, a decrease in $\alpha$ causes an increase in $\beta$ , and a decrease in $\beta$ causes an increase in $\alpha$ . Both probabilities $\alpha$ and $\beta$ of the two types of errors can be decreased only by increasing the sample size or, in some way, constructing a better test of the hypothesis.

## Example

If n =100 and we desire a test with significance level $\alpha$ =0.05, then $\alpha =P\left(\overline{X}\ge c;\mu =60\right)=0.05$ means, since $\overline{X}$ is $\text{N(}\mu \text{,100/100=1)}$ ,

$P\left(\frac{\overline{X}-60}{1}\ge \frac{c-60}{1};\mu =60\right)=0.05$ and $c-60=1.645$ . Thus c =61.645. The power function is

$K\left(\mu \right)=P\left(\overline{X}\ge 61.645;\mu \right)=P\left(\frac{\overline{X}-\mu }{1}\ge \frac{61.645-\mu }{1};\mu \right)=1-\Phi \left(61.645-\mu \right).$

In particular, this means that $\beta$ at $\mu$ =65 is $=1-K\left(\mu \right)=\Phi \left(61.645-65\right)=\Phi \left(-3.355\right)\approx 0;$ so, with n =100, both $\alpha$ and $\beta$ have decreased from their respective original values of 0.1587 and 0.0668 when n =25. Rather than guess at the value of n , an ideal power function determines the sample size. Let us use a critical region of the form $\overline{x}\ge c$ . Further, suppose that we want $\alpha$ =0.025 and, when $\mu$ =65, $\beta$ =0.05. Thus, since $\overline{X}$ is $\text{N(}\mu \text{,100/n)}$ ,

$0.025=P\left(\overline{X}\ge c;\mu =60\right)=1-\Phi \left(\frac{c-60}{10/\sqrt{n}}\right)$ and $0.05=1-P\left(\overline{X}\ge c;\mu =65\right)=\Phi \left(\frac{c-65}{10/\sqrt{n}}\right).$

That is, $\frac{c-60}{10/\sqrt{n}}=1.96$ and $\frac{c-65}{10/\sqrt{n}}=-1.645$ .

Solving these equations simultaneously for c and $10/\sqrt{n}$ , we obtain $c=60+1.96\frac{5}{3.605}=62.718;$ $\frac{10}{\sqrt{n}}=\frac{5}{3.605}.$

Thus, $\sqrt{n}=7.21$ and $n=51.98$ . Since n must be an integer, we would use n =52 and obtain $\alpha$ =0.025 and $\beta$ =0.05, approximately.

For a number of years there has been another value associated with a statistical test, and most statistical computer programs automatically print this out; it is called the probability value or, for brevity, p -value . The p -value associated with a test is the probability that we obtain the observed value of the test statistic or a value that is more extreme in the direction of the alternative hypothesis, calculated when ${\text{H}}_{\text{0}}$ is true. Rather than select the critical region ahead of time, the p -value of a test can be reported and the reader then makes a decision.

Say we are testing against with a sample mean $\overline{X}$ based on n =52 observations. Suppose that we obtain the observed sample mean of $\overline{x}=62.75$ . If we compute the probability of obtaining an $\overline{x}$ of that value of 62.75 or greater when $\mu$ =60, then we obtain the p -value associated with $\overline{x}=62.75$ . That is,

$\begin{array}{l}p-value=P\left(\overline{X}\ge 62.75;\mu =60\right)=P\left(\frac{\overline{X}-60}{10/\sqrt{52}}\ge \frac{62.75-60}{10/\sqrt{52}};\mu =60\right)\\ =1-\Phi \left(\frac{62.75-60}{10/\sqrt{52}}\right)=1-\Phi \left(1.983\right)=0.0237.\end{array}$

If this p -value is small, we tend to reject the hypothesis . For example, rejection of if the p -value is less than or equal to 0.025 is exactly the same as rejection if $\overline{x}=62.718$ .That is, $\overline{x}=62.718$ has a p -value of 0.025. To help keep the definition of p -value in mind, we note that it can be thought of as that tail-end probability , under ${\text{H}}_{\text{0}}$ , of the distribution of the statistic, here $\overline{X}$ , beyond the observed value of the statistic. See Figure 1 for the p -value associated with $\overline{x}=62.75.$

Suppose that in the past, a golfer’s scores have been (approximately) normally distributed with mean $\mu$ =90 and ${\sigma }^{2}$ =9. After taking some lessons, the golfer has reason to believe that the mean $\mu$ has decreased. (We assume that ${\sigma }^{2}$ is still about 9.) To test the null hypothesis against the alternative hypothesis , the golfer plays 16 games, computing the sample mean $\overline{x}$ .If $\overline{x}$ is small, say $\overline{x}\le c$ , then ${H}_{0}$ is rejected and ${H}_{1}$ accepted; that is, it seems as if the mean $\mu$ has actually decreased after the lessons. If c =88.5, then the power function of the test is

$K\left(\mu \right)=P\left(\overline{X}\le 88.5;\mu \right)=P\left(\frac{\overline{X}-\mu }{3/4}\le \frac{88.5-\mu }{3/4};\mu \right)=\Phi \left(\frac{88.5-\mu }{3/4}\right).$

Because 9/16 is the variance of $\overline{X}$ . In particular, $\alpha =K\left(90\right)=\Phi \left(-2\right)=1-0.9772=0.0228.$

If, in fact, the true mean is equal to $\mu$ =88 after the lessons, the power is $K\left(88\right)=\Phi \left(2/3\right)=0.7475$ . If $\mu$ =87, then $K\left(87\right)=\Phi \left(2\right)=0.9772$ . An observed sample mean of $\overline{x}=88.25$ has a

$p-value=P\left(\overline{X}\le 88.25;\mu =90\right)=\Phi \left(\frac{88.25-90}{3/4}\right)=\Phi \left(-\frac{7}{3}\right)=0.0098,$

and this would lead to a rejection at $\alpha$ =0.0228 (or even $\alpha$ =0.01).

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
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
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Got questions? Join the online conversation and get instant answers!