Question 251235: A marketing advisory service wants to estimate the proportion of the population that responds favorably to a new product. It wants the estimate to be correct within 4% at the 95% confidence level. How big of a sample should they take?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you need three pieces of information to determine that, based on the following link.
http://www.custominsight.com/articles/random-sampling.asp
those pieces of information are:
population size
desired accuracy
confidence level.
you have 2 out of the 3.
you are missing the population size.
it matters.
if a small population, then you will need a proportionally larger sample size to get the same accuracy.
as the population size gets larger, the percentage of the population that needs to be samples is relatively smaller.
example:
if the population is 1,000 people, then you need to get 375 people to provide data in order to get 4% accuracy at the 95% confidence level./
if the population is 10,000 people, then you need to get 566.
if the population is 100,000 people, then you need to get 597.
The percentage of the population required for 4% accuracy at 95% confidence level is progressively smaller as the population gets larger.
The above mentioned article doesn't give you the formula to calculate the right percentages, but it does provide a calculator that provides you with the answer.
the following link uses a similar calculator that provides the same answer.
http://www.surveysystem.com/sscalc.htm
the following link shows the formula they use.
http://www.surveysystem.com/sample-size-formula.htm
there is an additional factor called percent. the default value for this is .5 (50%). this would be a worst case scenario where only 50% of the respondents answer yes to a question and 50% of the respondents answer no. if all answered yes, then the sample size could be smaller. the safe bet is to assume the worst case which is why 50% is the default percent.
An example of how the formula works is as follows:
The formula is:
pss = preliminary sample size. there is an adjustment to this value shown below which will result in fss (final sample size).
p = proportion of the sample respondents that picked the desired answer. the worst case is .5 and that is the typical default value used. a value of .5 requires the largest sample size for a desired accuracy level.
c = confidence interval which is the desired accuracy level you want to achieve. if you want your answer to be +/- 4%, you would make c = .04
z is the z-score. for 95% accuracy, a z-score of 1.96 would be required. this is rounded to the nearest one hundredth.
example:
the following statement is extracted from above.
"if the population is 100,000 people, then you need to get 597."
this means that a sample size of 597 would be required for a population of 100,000 at a 95% confidence level with a 4% confidence interval and a default worst case percent of 50%.
the calculator provided that answer.
the formula that the calcultor used is:
pss is the preliminary value for what we want to find (sample size)
z = 1.96 (95% confidence level)
p = .5 (default value for worst case percent)
c = .04 (confidence interval)
the formula becomes:
this becomes pss = 600.25
there is an adjustment to this answer for population size based on the following formula:
fss is the sample size we are looking for.
pss is the preliminary sample size we derived above.
ps is the population size.
in our example, we have the following values for this formula.
fss = what we want to find.
pss = 600.25
ps = 100,000
the formula becomes:
this becomes:
this becomes fss = 596.6744285 which rounds to fss = 597.
we set sample size equal to fss and we get:
sample size = 597 for a population of 100,000 with a desired confidence interval of 4% at a 95% confidence level with a default percent of 50%.
the question you asked can't be answered because you are missing a very important ingredient. that's the sample size.
include that and you should be able to find the required sample size using the information and tools provided in the references.
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