# 10.3 Comparing two independent population proportions  (Page 2/1)

 Page 1 / 1
1. The two independent samples are simple random samples that are independent.
2. The number of successes is at least five and the number of failures is at least five for each of the samples.

Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populationsor it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions $\left({P}_{A}-{P}_{B}\right)$ reflects a difference in the population proportions.

The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, ${H}_{o}:{p}_{A}={p}_{B}$ . To conduct the test, we use a pooled proportion, ${p}_{c}$ .

## The pooled proportion is calculated as follows:

${p}_{c}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}$

## The distribution for the differences is:

${P\text{'}}_{A}-{P\text{'}}_{B}~N\left[0,\sqrt{{p}_{c}·\left(1-{p}_{c}\right)·\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right]$

## The test statistic (z-score) is:

$z=\frac{\left({p\text{'}}_{A}-{p\text{'}}_{B}\right)-\left({p}_{A}-{p}_{B}\right)}{\sqrt{{p}_{c}·\left(1-{p}_{c}\right)·\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}$

## Two population proportions

Two types of medication for hives are being tested to determine if there is a difference in the proportions of adult patient reactions. Twenty out of a random sample of 200 adults given medication A still had hives 30 minutes after taking the medication. Twelve out of another random sample of 200 adults given medication B still had hives 30 minutes after taking the medication. Test at a 1% level of significance.

## Determining the solution

This is a test of 2 population proportions.

How do you know?

The problem asks for a difference in proportions.

Let $A$ and $B$ be the subscripts for medication A and medication B. Then ${p}_{A}$ and ${p}_{B}$ are the desired population proportions.

## Random variable:

${\mathrm{P\text{'}}}_{A}-{\mathrm{P\text{'}}}_{B}=$ difference in the proportions of adult patients who did not react after 30 minutes to medication A and medication B.

${H}_{o}:{p}_{A}={p}_{B}\phantom{\rule{50pt}{0ex}}{p}_{A}-{p}_{B}=0$

${H}_{a}:{p}_{A}\ne {p}_{B}\phantom{\rule{50pt}{0ex}}{p}_{A}-{p}_{B}\ne 0$

The words "is a difference" tell you the test is two-tailed.

Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal:

${p}_{c}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}=\frac{20+12}{200+200}=0.08\phantom{\rule{12pt}{0ex}}1-{p}_{c}=0.92$

Therefore, $\phantom{\rule{10pt}{0ex}}{\mathrm{P\text{'}}}_{A}-{\mathrm{P\text{'}}}_{B}~N\left[0,\sqrt{\left(0.08\right)\cdot \left(0.92\right)\cdot \left(\frac{1}{200}+\frac{1}{200}\right)}\right]$

${\mathrm{P\text{'}}}_{A}-{\mathrm{P\text{'}}}_{B}$ follows an approximate normal distribution.

Calculate the p-value using the normal distribution: p-value = 0.1404.

Estimated proportion for group A: $\phantom{\rule{12pt}{0ex}}{\mathrm{p\text{'}}}_{A}=\frac{{x}_{A}}{{n}_{A}}=\frac{20}{200}=0.1$

Estimated proportion for group B: $\phantom{\rule{12pt}{0ex}}{\mathrm{p\text{'}}}_{B}=\frac{{x}_{B}}{{n}_{B}}=\frac{12}{200}=0.06$

## Graph:

${\mathrm{P\text{'}}}_{A}-{\mathrm{P\text{'}}}_{B}=0.1-0.06=0.04$ .

Half the p-value is below -0.04 andhalf is above 0.04.

Compare $\alpha$ and the p-value: $\alpha =0.01$ and the $\text{p-value}=0.1404$ . $\alpha <$ p-value.

Make a decision: Since $\alpha <\text{p-value}$ , do not reject ${H}_{o}$ .

Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adultpatients who did not react after 30 minutes to medication A and medication B.

TI-83+ and TI-84: Press STAT . Arrow over to TESTS and press 6:2-PropZTest . Arrow down and enter 20 for $\mathrm{x1}$ , 200 for $\mathrm{n1}$ , 12 for $\mathrm{x2}$ , and 200 for $\mathrm{n2}$ . Arrow down to p1 : and arrow to not equal p2 . Press ENTER . Arrow down to Calculate and press ENTER . The p-value is $p=0.1404$ and the test statistic is 1.47. Do the procedure again but instead of Calculate do Draw .

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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.