Sample size determination for the estimation of mean

Suppose that sample of n observations is taken from a normally distributed population with mean and known variance . We know that confidence interval for the population mean is given by

where is the sample mean and is the appropriate cutoff point of the standard normal distribution. This confidence interval is centered on the sample mean and extends a distance of L, the margin of error (also called the sample error, the bound, or the interval half width) is given by

Suppose that we predetermine the size of L and want to find the size of the sample that will yield this margin error. From the above expression, the following formula is obtained that determines the required sample size n.

Definition:

Given the confidence level and standard deviation of the population

(or population variance), the sample size that will produce a predetermined margin error L of the confidence interval estimate of is

Remark 1:

If we do not know , we can take a sample and find sample standard deviation. Then we can use S for in the formula.

Remark 2:

n must be rounded to the next higher integer, because a sample size can not be fractional.

Example:

Suppose that we want to estimate the mean family size for all country families at 99 % confidence level. It is known that the standard deviation

for the sizes of all families in the country is 0.45.

How large a sample should we select if we want its estimate to be within 0.02 of the population mean?

Solution:

We want the 99 % confidence interval for the mean family size to be

.

Hence, the margin of errors is to be 0.02, that is

The value of for a 99 % confidence level is 2.58.

The value of is given to be 0.45. Therefore, substituting all values in the formula and simplifying, we obtain

Thus, the required sample size is 3370. If we will take a sample of 3370 families, compute the mean family size for this sample, and then margin of a 99 % confidence interval around this sample, the margin of error of the estimate will be approximately 0.02.

Дата добавления: 2015-08-05