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Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standard deviations. The sampling distribution for the difference between the means is normal and both populations must be normal. The random variable is X 1 ¯ X 2 ¯ . The normal distribution has the following format:

Normal distribution Is:

X ¯ 1 X ¯ 2 ~ N [ μ 1 μ 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ]

The standard deviation Is:

( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2

The test statistic ( z -score) is:

z = ( x ¯ 1 x ¯ 2 ) ( μ 1 μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2

Independent groups, population standard deviations known: The mean lasting time of twocompeting floor waxes is to be compared. Twenty floors are randomly assigned to test each wax . Both populations have a normal distributions. The data are recorded in [link] .

Wax Sample Mean Number of Months Floor Wax Lasts Population Standard Deviation
1 3 0.33
2 2.9 0.36

Does the data indicate that wax 1 is more effective than wax 2 ? Test at a 5% level of significance.

This is a test of two independent groups, two population means, population standard deviations known.

Random Variable : X ¯ 1 –  X ¯ 2 = difference in the mean number of months the competing floor waxes last.

H 0 : μ 1 μ 2

H a : μ 1 > μ 2

The words "is more effective" says that wax 1 lasts longer than wax 2 , on average. "Longer" is a “>” symbol and goes into H a . Therefore, this is a right-tailed test.

Distribution for the test: The population standard deviations are known so the distribution is normal. Using the formula, the distribution is:

X ¯ 1 X ¯ 2 ~ N ( 0 , 0.33 2 20 + 0.36 2 20 )

Since μ 1 μ 2 then μ 1 μ 2 ≤ 0 and the mean for the normal distribution is zero.

Calculate the p -value using the normal distribution: p -value = 0.1799

Graph:

This is a normal distribution curve with mean equal to zero. The values 0 and 0.1 are labeled on the horiztonal axis. A vertical line extends from 0.1 to the curve. The region under the curve to the right of the line is shaded to represent p-value = 0.1799.

X ¯ 1 X ¯ 2 = 3 – 2.9 = 0.1

Compare α and the p -value: α = 0.05 and p -value = 0.1799. Therefore, α < p -value.

Make a decision: Since α < p -value, do not reject H 0 .

Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean time wax 1 lasts is longer (wax 1 is more effective) than the mean time wax 2 lasts.

Using a calculator, one finds the p -value is p = 0.1799 and the test statistic is 0.9157.

Try it

The means of the number of revolutions per minute of two competing engines are to be compared. Thirty engines are randomly assigned to be tested. Both populations have normal distributions. [link] shows the result. Do the data indicate that Engine 2 has higher RPM than Engine 1? Test at a 5% level of significance.

Engine Sample Mean Number of RPM Population Standard Deviation
1 1,500 50
2 1,600 60

The p -value is almost zero, so we reject the null hypothesis. There is sufficient evidence to conclude that Engine 2 runs at a higher RPM than Engine 1.

An interested citizen wanted to know if Democratic U. S. senators are older than Republican U.S. senators, on average. On May 26 2013, the mean age of 30 randomly selected Republican Senators was 61 years 247 days old (61.675 years) with a standard deviation of 10.17 years. The mean age of 30 randomly selected Democratic senators was 61 years 257 days old (61.704 years) with a standard deviation of 9.55 years.

Do the data indicate that Democratic senators are older than Republican senators, on average? Test at a 5% level of significance.

This is a test of two independent groups, two population means. The population standard deviations are unknown, but the sum of the sample sizes is 30 + 30 = 60, which is greater than 30, so we can use the normal approximation to the Student’s-t distribution. Subscripts: 1: Democratic senators 2: Republican senators

Random variable: X ¯ 1  –  X ¯ 2 = difference in the mean age of Democratic and Republican U.S. senators.

H 0 : µ 1 µ 2 H 0 : µ 1 µ 2 ≤ 0

H a : µ 1 > µ 2 H a : µ 1 µ 2 >0

The words "older than" translates as a “>” symbol and goes into H a . Therefore, this is a right-tailed test.

Distribution for the test: The distribution is the normal approximation to the Student’s t for means, independent groups. Using the formula, the distribution is: X ¯ 1 X ¯ 2 N [ 0 , ( 9.55 ) 2 30 + ( 10.17 ) 2 30 ]

Since µ 1 µ 2 , µ 1 µ 2 ≤ 0 and the mean for the normal distribution is zero.

(Calculating the p -value using the normal distribution gives p -value = 0.4040)

Graph:

This is a normal distribution curve with mean equal to zero. A vertical line to the right of zero extends from the axis to the curve. The region under the curve to the right of the line is shaded representing p-value = 0.4955.

Compare α and the p -value: α = 0.05 and p -value = 0.4040. Therefore, α < p -value.

Make a decision: Since α < p -value, do not reject H 0 .

Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean age of Democratic senators is greater than the mean age of the Republican senators.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
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
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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