Sample Size

Statistics is the study of the process of collecting, organizing, analyzing, summarizing data and drawing inferences from the data so worked on. In Statistics, we come across two types of data – Population data and Sample data. Population data is a large amount of data that includes the whole area of study, which is termed as population. A population consists of all the elements that are studied for the research. On the other hand, sample data is a part of the population. Usually, it is quite clumsy and difficult to compute the whole population. In this case, a representative sample is selected from the population. This sample is termed as sample data. In this article, let us discuss the sample size definition, formulas, examples in detail.

Sample Size Definition

The sample size is defined as the number of observations used for determining the estimations of a given population. The size of the sample has been drawn from the population. Sampling is the process of selection of a subset of individuals from the population to estimate the characteristics of the whole population. The number of entities in a subset of a population is selected for analysis.

Small Sample Size

Sometimes the sample size can be very small. When the sample size is small (n < 30), we use the t distribution in place of the normal distribution. If the population variance is unknown and the sample size is small, then we use the t statistic to test the null hypothesis with both one-tailed and two-tailed, where \(t = \frac{\bar{X}-\mu }{\frac{s}{\sqrt{n}}}\)

Large Sample Size

Generate for more accurate estimates but large sample size might cause difficulties in interpreting the usual tests of significance, and the same problem may arise in case of very small sample size. Thus, neither too large nor too small sample sizes help research projects.

Sample Size Formula

The sample size formula for the infinite population is given by:

\(SS = \frac{Z^{2}P(1-P)}{C^{2}}\)

Where,

SS = Sample Size

Z = Z -Value

P = Percentage of Population

C = Confidence interval

When the sample input or data is obtained, and the sample mean \(\bar{X}\) is calculated, the sample mean obtained is different from the population mean μ. This difference between the population mean and the sample mean can be considered as an error E, which is the maximum difference between the observed sample mean and the true value of the population mean.

\(E = Z_{\frac{\alpha}{2}}(\frac{\sigma }{\sqrt{n}})\)

The above-given formula can be solved for n, which can be used to determine the minimum sample size.

Therefore, the formula to find the minimum sample size is given by

\(n = \left ( \frac{Z_{\frac{\alpha }{2}}\sigma }{E} \right )^{2}\)

Sample Size Example

Question:

Assuming the heights of students in a college campus are normally distributed with a standard deviation = 5 in, find the minimum size required to construct a 95% confidence interval for mean with a maximum error = 0.5 in.

Solution:

Given: E = 0.5 in, σ = 5 and α = 1 – 0.95 = 0.05

Therefore, Zα/2 = Z 0.025 = 1.96

The formula to find the minimum sample size is

\(n = \left ( \frac{Z_{\frac{\alpha }{2}}\sigma }{E} \right )^{2}\)

Now, substitute the given values in the sample size formula, we get

\(n = \left ( \frac{1.96(5) }{0.5} \right )^{2} = 384.16\)

Therefore, rounding this value up to the next integer, the minimum sample space required is 385.

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