


Differences between Percentages and Paired Alternatives
Lecture 6

Percentages 
Apply
the same methodology to percentages
Where P is the percentage calculated from the data
while r is the Greek letter that represents the population parameter
Using the same logic, the standard error (SE) is:
The variances are
This test is approximate because we are calculating
the variances from the data
Then calculate the zstatistic
Example:
300 students are randomly chosen (n_{1}) at
Suleyman Demirol
P_{1} = 55% are women
1 – P_{1} = 45% are men
400 students are randomly chosen (n_{2}) from
the business school
P_{2} = 60% are women
1 – P_{2} = 40% are men
Are the same percentage of women studying business is
the same percentage as the student body?
The hypothesis is:
The standard error (SE) is:
The zstatistic
The critical zvalue is 1.96. The pvalue is 0.092669
Excel, the pvalue is calculated from
=normdist(1.3245, 0, 1, 1)
According to both the z and p values, fail to reject
the null hypothesis and conclude both percentages are the same
Both the methods will give the same results. The z
and p values are testing the same hypothesis from different angles
Note – Excel doubles the pvalue for twotail test
You
can also use confidence intervals for hypothesis testing

Poisson Distribution 
The
probability of a number of events that occur in a specific time
period

Counting distribution

Number of events

Number of deaths

Number of births

Number of accidents at a street intersection
The
PDF is
k is the number of occurrences, k = 1, 2, 3, …
l is expected number of occurrences in interval
This
distribution is unique, the mean = variance = l
Example:
Heart disease
In 2008, there were 543 deaths (n_{1})
In 2009, there were 674 deaths (n_{2})
Is this increase in deaths due to chance?
The hypothesis is:
The standard error (SE) is
The zstatistic is:
Using a = 0.05,
the z_{c} = 1.96
Reject the null hypothesis and conclude the heart
attack rate is higher
The zstatistic is approximate, because it came from
a Poisson distribution

McNeman’s Test 
You
will not be tested over this test
It is an interesting test
You have an example where your sample has two
treatments and the results are paired
Your sample had two experimental medications
A matrix of your results

Treatment A 
Treatment B 
Outcome 1 
Responded 
Responded 
Outcome 2 
Responded 
Did not
respond 
Outcome 3 
Did not respond 
Responded 
Outcome 4 
Did not respond 
Did not respond 
You
are interested if Treatment A is better than Treatment B?
Ignore Outcomes 1 and 4
Focus on Outcomes 2 and 3
Observations have to be paired
Example:
Each person gets both treatments
Or the sample is divided by 2 and then randomly pair
one person to another
Example:
200 people with heart problems
Treatment A: Patients have to eat right and exercise
Treatment B: Patients take a drug, Plavix
Randomly pair sample into 100 pairs

Treatment A 
Treatment B 
Observations 
Outcome 1 
Responded 
Responded 
15 
Outcome 2 
Responded 
Did not respond 
30 
Outcome 3 
Did not respond 
Responded 
45 
Outcome 4 
Did not respond 
Did not respond 
10 



100 
The n_{1} = 45 and n_{2} = 30
Calculate the zstatistic
Fail to reject the treatments are the same, because a
= 0.05 and the z_{c} = 1.96
Note – I could give all patients both treatments,
but I have to discern which treatment did what

