<< Chapter < Page Chapter >> Page >

In general, without changing the sample size or the type of the test of the hypothesis, a decrease in α causes an increase in β , and a decrease in β causes an increase in α . Both probabilities α and β 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 α =0.05, then α = P ( X ¯ c ; μ = 60 ) = 0.05 means, since X ¯ is N( μ ,100/100=1) ,

P ( X ¯ 60 1 c 60 1 ; μ = 60 ) = 0.05 and c 60 = 1.645 . Thus c =61.645. The power function is

K ( μ ) = P ( X ¯ 61.645 ; μ ) = P ( X ¯ μ 1 61.645 μ 1 ; μ ) = 1 Φ ( 61.645 μ ) .

In particular, this means that β at μ =65 is = 1 K ( μ ) = Φ ( 61.645 65 ) = Φ ( 3.355 ) 0 ; so, with n =100, both α and β 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 x ¯ c . Further, suppose that we want α =0.025 and, when μ =65, β =0.05. Thus, since X ¯ is N( μ ,100/n) ,

0.025 = P ( X ¯ c ; μ = 60 ) = 1 Φ ( c 60 10 / n ) and 0.05 = 1 P ( X ¯ c ; μ = 65 ) = Φ ( c 65 10 / n ) .

That is, c 60 10 / n = 1.96 and c 65 10 / n = 1.645 .

Solving these equations simultaneously for c and 10 / n , we obtain c = 60 + 1.96 5 3.605 = 62.718 ; 10 n = 5 3.605 .

Thus, n = 7.21 and n = 51.98 . Since n must be an integer, we would use n =52 and obtain α =0.025 and β =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 H 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 H 0 μ =60  against H 1 μ >60 with a sample mean X ¯ based on n =52 observations. Suppose that we obtain the observed sample mean of x ¯ = 62.75 . If we compute the probability of obtaining an x ¯ of that value of 62.75 or greater when μ =60, then we obtain the p -value associated with x ¯ = 62.75 . That is,

p v a l u e = P ( X ¯ 62.75 ; μ = 60 ) = P ( X ¯ 60 10 / 52 62.75 60 10 / 52 ; μ = 60 ) = 1 Φ ( 62.75 60 10 / 52 ) = 1 Φ ( 1.983 ) = 0.0237.

If this p -value is small, we tend to reject the hypothesis H 0 μ =60  . For example, rejection of H 0 μ =60  if the p -value is less than or equal to 0.025 is exactly the same as rejection if x ¯ = 62.718 .That is, 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 H 0 , of the distribution of the statistic, here X ¯ , beyond the observed value of the statistic. See Figure 1 for the p -value associated with x ¯ = 62.75.

The p -value associated with x ¯ = 62.75.

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

K ( μ ) = P ( X ¯ 88.5 ; μ ) = P ( X ¯ μ 3 / 4 88.5 μ 3 / 4 ; μ ) = Φ ( 88.5 μ 3 / 4 ) .

Because 9/16 is the variance of X ¯ . In particular, α = K ( 90 ) = Φ ( 2 ) = 1 0.9772 = 0.0228.

If, in fact, the true mean is equal to μ =88 after the lessons, the power is K ( 88 ) = Φ ( 2 / 3 ) = 0.7475 . If μ =87, then K ( 87 ) = Φ ( 2 ) = 0.9772 . An observed sample mean of x ¯ = 88.25 has a

p v a l u e = P ( X ¯ 88.25 ; μ = 90 ) = Φ ( 88.25 90 3 / 4 ) = Φ ( 7 3 ) = 0.0098 ,

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

Got questions? Get instant answers now!

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to statistics' conversation and receive update notifications?

Ask