# 10.3 Hypothesis testing: two population means and two population

 Page 1 / 1

## 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.
• ${\stackrel{̂}{p}}_{1}{\left(n}_{1}\right)$ >10 and ${\left(1-\stackrel{̂}{p}}_{1}\right){\left(n}_{1}\right)$ >10 and
• ${\stackrel{̂}{p}}_{2}{\left(n}_{2}\right)$ >10 and ${\left(1-\stackrel{̂}{p}}_{2}\right){\left(n}_{2}\right)$ >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 $\left({\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}\right)$ 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, ${\stackrel{̂}{p}}_{\mathrm{pooled}}$ .

## The pooled proportion is calculated as follows:

${\stackrel{̂}{p}}_{\mathrm{pooled}}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}$

## The distribution for the differences is:

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}~N\left[0,\sqrt{{\stackrel{̂}{p}}_{\mathrm{pooled}}·\left(1-{\stackrel{̂}{p}}_{\mathrm{pooled}}\right)·\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right]$

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

$z=\frac{\left({\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{A}\right)-\left({p}_{A}-{p}_{B}\right)}{\sqrt{{\stackrel{̂}{p}}_{\mathrm{pooled}}·\left(1-{\stackrel{̂}{p}}_{\mathrm{pooled}}\right)·\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}$

## Two population proportions

Two types of medication for hives are being tested to determine if there is a 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:

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

${H}_{o}:{p}_{A}={p}_{B}\phantom{\rule{25pt}{0ex}}$ or $\phantom{\rule{25pt}{0ex}}{p}_{A}-{p}_{B}=0$

${H}_{a}:{p}_{A}>{p}_{B}\phantom{\rule{25pt}{0ex}}$ or $\phantom{\rule{25pt}{0ex}}{p}_{A}-{p}_{B}>0$

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: ${p}_{c}={\stackrel{̂}{p}}_{\mathrm{pooled}}$

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

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

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}$ follows an approximate normal distribution.

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

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

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

## Graph:

${\stackrel{̂}{p}}_{A}-{\stackrel{̂}{p}}_{B}=0.1-0.06=0.04$ .

The p-value is above 0.1404.

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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
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
Got questions? Join the online conversation and get instant answers!