Sample size is nothing but the number of observations that constitute the experiment / research. It is generally denoted by *n. *

**Sample size determination** is closely related to estimation. Sometimes, we may need to know how large a sample is necessary in order to make an accurate estimate. The
answer depends on

- The margin of error.
- The degree of confidence.

In considering large sample confidence intervals for the mean, since the error of estimate is given by E = E = Z_{α/2} (σ / √n), we can solve to find
n, the sample size.

Solving, we have

n = [(Z_{α/2} x σ) / E]^{2}

**Q :** What sample size should be selected to estimate the mean age of workers in the large factory to with in ±1 year at a 95 percent confidence level if the
**standard deviation** for the ages is 3.5 years?

**Sol :** We are given that α=0.05, Z_{α/2} = 1.96 {the value is obtained from the normal distribution chart}, and E=1.

Substituting into the formula, we get the sample size as

n = [(Z_{α/2} x σ) / E]2 = [(1.96 x 3.5)/1]2 = 47.0596 ≈ 48

That is, in order to be 95 percent certain that the estimate is with in 1 year of the true mean age, a sample of at least 48 is selected.

Note: For a large enough sample size (n ≥ 30), when the population standard deviation σ is unknown, we can replace σ with s in the above equations.

The equation for the minimum sample size is:

n = 0.5 x [Z_{α/2}/ E]^{2}

**Q :** The researcher wants to determine the sample size for his research study. If a margin of error of ±0.02 is acceptable at 95 percent confidence interval, what
is the minimum sample size that should be taken**?**

**Sol :** We are given α= 0.05 and E= 0.02. Since, α= 0.05, then Z_{α/2} = 1.96. Thus,

n = 0.5 x [Z_{α/2}/ E]^{2 }

= 0.5 x [1.96/ 0.02]^{2 }

=4802

Thus the researcher should sample at least 4,802 in the research study.