Valid sample size?

How does one determine a valid sample size? For a project I surveyed 100 people as far as which cell phone carrier they used in 4 different cities.


--City A--


carrier 1: 8


carrier 2:10


carrier 3: 4


carrier 4: 3


--City B--


Carrier 1: 9


carrier 2: 9


carrier 3:5


carrier 4: 2


--City C--


carrier 1: 10


carrier 2: 10


carrier 3:4


carrier 4:1


--City D---


Carrier 1: 11


carrier 2: 13


carrier 3: 1


carrier 4: 0





what would the equations be to find out if this sample size is valid?

Valid sample size?
Depends.





First, The samples have to be probability samples to make any inference about the population. If the probability samples are not Simple Random Samples the equations become difficult.





Second, the validity of a sample size, the "power" of a statisitical test, depends upon the particular test being used. Thus if your attempt is to compare cities (ANOVA) would have a completely different formula for power than a Chi Square test formula for power.





Just as 95% is commonly used for confidence levels, a power of 80% is commonly used to determine if a sample size is large enought. That is, the sample size is large enough that, if there were an "important" difference, then their would be an 80% chance of finding the difference statistically significant for that sample size. Important difference is simply a researchers opinion. The classic example is male versus femal birth rate. Given a large enough sample size, it turns out that that there is a slightly more than 50% chance of having a girl, but the margin is so small, that it not considered "important".