The sample sizes in the previous problems were always given. Now we are going to determine an appropriate sample size.
In the interval estimation of the population mean , we have seen that the confidence interval is either Z/2 if is known or Z/2 if is not known. In this case the deviation from given by Z/2 or Z/2 is known as the maximum allowable error for (1-) 100% confidence interval. If we denote this error by E, then
E = Z/2 if is known and
E = Z/2 if is not known.
From these, solving for n,
in the first case or n = and n should always be rounded up to the next integer or one can also use the following relation in solving for n.
i.e.
For a finite population: - The sample size estimating the mean is obtained form the confidence interval for the population mean given by
Z/2
where the total allowable error E equals
E
n =
Determination of sample size for a finite population
=
From this solving for the sample size n, we get
Example 1: We want to estimate the population mean within 5, with a 99% level of confidence. The population standard deviation is estimated to be 15. How large a sample is required?
Solution: Given
= 15
E = 5
1- = 0.99 = 0.01 /2 = 0.005
n = ?
n =
n = 60
Example 2: Determine the size of the sample for estimating the true weight of a cereal containers for the universe with N = 5000 on the basis of the following information.
The variance of weight = 4 ounces on the basis of past records.
Estimate should be within 0.8 ounces of the true average weight with 99% probability.
Given
N = 5000 – finite population
= 2
E = 0.8
= 0.01
Z/2 = Z0.005 = 2.58
n =
=
n = 41
