Determining sample size is a very important issue because samples that are too large may waste time, resources and money, while samples that are too small may lead to inaccurate results. In many cases, we can easily determine the minimum sample size needed to estimate a process parameter, such as the population mean .
When sample data is collected and the sample mean is calculated, that sample mean is typically different from the population mean X. This difference between the sample and population means can be thought of as an error. The margin of error E is the maximum difference between the observed sample mean X and the true value of the population mean \[\mu \]
The formula to calculate sample size is given by \[\frac{z\sigma }{E^2}\]
where z= standard normal value of confidence interval
\[\sigma \] It is the standard deviation of the outcome of interest.
E= is the margin of error that the investigator specifies as important from a clinical or practical standpoint.
Assumption
here y= \[\sigma \]
Boundary condition x > 0
Probelm-We would like to start an ISP and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 95 percent sure that the sample mean is within 1 minute of the population mean \[\mu \]. Assume that a previous survey of household usage has shown \[\sigma \]= 6.95 minutes.
solution
A 95% degree confidence corresponds to \[\alpha =0.05\]
Each of the shaded tails in the following figure has an area of \[\alpha /2=0.025\]
The region to the left of 
and to the right of 
= 0 is 0.5 – 0.025, or 0.475. In the table of the standard normal () distribution, an area of 0.475 corresponds to a value of 1.96. The critical value is therefore = 1.96.
The margin of error 
= 1 and the standard deviation 
= 6.95. Using the formula for sample size, we can calculate 
:
putting all these values in the formula n=\[(1.96*6.95)^2\]=\[(13.62)^2\]=185.55
