Statistics is all about deriving useful information about the population from the statistics derived from a representative samples. Population parameters like mean ‘μ’ and standard deviation ‘σ’ are estimated from sample statistics like sample size ‘n’, sample average ‘x’and sample standard deviation ‘s’ up to desired confidence interval. It is pretty obvious that there will be some error in such estimation as sample size ‘n’ is less than population size ‘N’ and also that increasing the sample size ‘n’ will decrease the error until ‘n’ becomes equal to the population size ‘N’, where the error will become zero as the sample becomes the entire population itself. However, increasing the sample size beyond a limit makes hardly any sense as there is cost involved in collecting data from the sample and this cost increases with increasing sample size. Also, if one has to collect data from entire population, then what is the use of statistics? Therefore, what is done is a trade off between the allowed error and sample size i.e. the cost of data collection. This is illustrated in the following example.
Given,
Allowed errore = x∼μ = $10
Population standard deviation = $500
Confidence Interval = 98%
Therefore, z = 2.33
Sample size n = ? (to be calculated)
Sample size ‘n’ is given by the following formula
Therefore, at a confidence interval of 98%; an error of $10 can be ensured only if a sample size of 13573 is taken.
However, now the cost of data collection comes into the picture. As it cost $5 per person in the sample for data collection; therefore, a sample size of 13573 implies the total cost of data collection to be $67865. However, Piggy bank is not willing to spend this much amount, in stead the bank can afford only $10000 for this exercise. This results in a sample size of 2000.
Therefore, now the problem is to find the error.
Here, Allowed error e =x∼μ= ? (to be calculated)
Population standard deviation = $500
Confidence Interval = 98%
Therefore, z = 2.33
Sample size n = 2000
Sample size ‘n’ is given by the following formula
This means that the error now will be $136. This means reducing the sample size has increased the error and this is the trade off in the present circumstances.
To estimate a single population mean, the sample size should be the same as the population size, this means all the customers of the Piggy Bank will have to be included in the sample.