<< Chapter < Page Chapter >> Page >

Assumptions and conditions: two sample proportions test

When constructing a two sample proportions hypothesis test the assumptions and conditions must be met in order to use the normal model.

  • Randomization Condition: The data must be sampled randomly. Is one of the good sampling methodologies discussed in the Sampling and Data chapter being used?
  • Independence Assumption: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event. Usually, if we know that people or items were selected randomly we can assume that the independence assumption is met.
  • 10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population. This must be true for both groups.
  • Success/ failure Condition: must be met for both groups. When working with proportions we need to be especially concerned about sample size when the proportion is close to zero or one. To check that the sample size is large enough calculate the successes by multiplying the sample percentages by the sample sizes and calculate failure by multiplying one minus the sample percentages by the sample sizes. If all of these products are larger than ten then the condition is met.
  • p ̂ 1 ( n 1 ) >10 and (1- p ̂ 1 ) ( n 1 ) >10 and
  • p ̂ 2 ( n 2 ) >10 and (1- p ̂ 2 ) ( n 2 ) >10
  • Independent Groups: The two groups you are working with must be independent of one another. Is there reason to believe that one group would have influence on the other group? A hypothesis test that is comparing a group of respondent’s pre-test and post-test scores would not be independent. Likewise, groups made by splitting married couples into husband and wife groups would not be independent.

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 ( p ̂ A p ̂ B ) 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 ̂ pooled .

The pooled proportion is calculated as follows:

p ̂ pooled = x A + x B n A + n B

The distribution for the differences is:

p ̂ A p ̂ B ~ N [ 0 , p ̂ pooled · ( 1 p ̂ pooled ) · ( 1 n A + 1 n B ) ]

The test statistic (z-score) is:

z = ( p ̂ A p ̂ A ) ( p A p B ) p ̂ pooled · ( 1 p ̂ pooled ) · ( 1 n A + 1 n B )

Two population proportions

Two types of medication for hives are being tested to determine if there is a medication A is more effective than medication B. 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:

p ̂ A p ̂ 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 or p A - p B = 0

H a : p A > p B or p A - p B > 0

The words "is more effective" tell you the test is one-tailed.

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

p ̂ pooled = x A + x B n A + n B = 20 + 12 200 + 200 = 0.08 1 p ̂ pooled = 0.92

Therefore, p ̂ A p ̂ B ~ N [ 0 , ( 0.08 ) ( 0.92 ) ( 1 200 + 1 200 ) ]

p ̂ A p ̂ B follows an approximate normal distribution.

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

Estimated proportion for group A: p ̂ A = x A n A = 20 200 = 0.1

Estimated proportion for group B: p ̂ B = x B n B = 12 200 = 0.06


Normal distribution curve of the difference in the percentages of adult patients who don't react to medication A and B after 30 minutes. The x-axis has values of  0.01.

p ̂ A p ̂ B = 0.1 - 0.06 = 0.04 .

The p-value is above 0.1404.

Compare α and the p-value: α = 0.01 and the p-value = 0.1404 . α < p-value.

Make a decision: Since α < 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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics using spreadsheets' conversation and receive update notifications?