I want the answer for following Probability problem:
In a sample of 1000 representing a survey from entire population,650
people were from city A and the rest of the people were from city B.
Out of the sample,19 people had cancer. Thirteen of these people were
from city A.
Are the events of living in city A and having cancer independent?

The definition of two events being independent (in the meaning of
probability) is that the probability of their conjunction (living in
city A and having cancer) is equal to the product of their separate
probabilities.

You should be able to apply the numeric values shown to see if this is satisfied:

Pr( lives in A & has cancer ) = Pr( lives in A ) * Pr( has cancer )

regards, mathtalk-ga

Hello,

Since this question deals with categorical information a Pearson
Chi-Square test is appropriate. This is a widely used technique (also
known as just Chi-Square test).

Using this technique you simply arrange the known information into a
2x2 grid. Below is a link to a handy calculator which will do the
arithmetic for you.

http://www.georgetown.edu/faculty/ballc/webtools/web_chi.html

One thing you need to watch out for is the magnitude of the numbers in
each box. When you have small numbers such as only 6 people from city
B with cancer, Fishers Exact test is more appropriate.
(http://www4.stat.ncsu.edu/~berger/tables.html)  The general guideline
is if any of the four boxes has a number of 5 or less you should use
Fishers Exact test. In practice, the Pearson Chi-Square test and
Fishers Exact test will usually give the same answer for independence
(that is, significance) but the p-values of Fishers Exact test will be
more accurate.

Since you are only asking for independence and you have 6 people, the
Pearson Chi-Square test is fine to use and much easier to calculate
and understand. The fact that you numbers are highly non-significant
(that is independent) makes the question moot.

Good luck,
Dr. Hinkle

Hi, dr_hinkle-ga:

Are we reading the same Question?  Since the Question is simply and
specifically whether the two events are independent, invoking a
chi-square test is not necessary.

Note that if the total number of people with cancer were 20 rather
than 19, and all other figures in the problem remain as they are, then
the definition of independence would be satisfied exactly.

regards, mathtalk-ga

mathtalk,

I have to agree with dr_hinkle here. The question explicitly states
that a sample has been taken from some underlying population.
Therefore I would assume that statistical inference is appropriate.
dhan76 can probably determine which approach is required by checking
the topics in the syllabus of the course / text book that this
question is from.

However, some of the details of dr_hinkle's post are incorrect. It is
not the size of the /observed/ cell counts that are important for
establishing the validity of the chi-square p-value, but rather it is
the /expected/ cell counts that should not be too small. For example,
for the <have cancer/city B> cell the expected cell count is 19/1000 *
350 = 6.65.

Cheers,
S.

Hi, saffie-ga:

You make good points.  But counting the subject line, it is stated
twice to be a probability problem.  The terminology of "events" tends
to confirm that setting.

It might be worth pointing out that the "expected" cell counts (to
which the observed counts are compared in the chi-square test) are
exactly values obtained under independence of the "events" (e.g.
probability of living in city A & having cancer is the product of
these separate probabilities).  Understanding the definition of
independence in probability theory is needed to appreciate the
statistical tests of significance formulated by Pearson, Fisher, and
others.

Of course with whole numbers one cannot observe 6.65 people living in
city B and having cancer.  So without recourse to a "black-box" test,
we can agree with Dr. Hinkle that the differences between the observed
values and those "expected" under independence are "highly
non-significant".

regards, mathtalk-ga