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Sample Size

3. Sample Size#

3.1. #

3.2. Traditional Sample Size Design#

\[ n=\frac{Z^2*\sigma^2}{e^2} \]

  • Before we dive into it, let’s introduce some key variables and theory we will use

3.2.1. Key Variables:#

  • Population Size (\(N\)): Total number of units in the population.

  • Sample Size (\(n\)): Number of units to be sampled.

  • Confidence Level: Typically 90%, 95%, or 99%.

  • Margin of Error (\(e\)): The maximum error allowed in the sample estimate.

  • Standard Deviation (\(σ\)): Variability of the population. If unknown, it can be estimated.

  • Z-Score (\(Z\)): Corresponds to the chosen confidence level (e.g., 1.96 for 95%).

3.2.2. Central Limit Theorem (CLT)#

  • The CLT states that, for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution. _ Mean of this normal distribution equals to the population mean (\(\mu\)) _ The standard deviation (Standard error of the mean, \(SEM\)) is determined by both the standard deviation of the population (\(\sigma\)) and sample size

\[ SEM=\frac{\sigma}{\sqrt{n}} \]

3.2.3. Margin of Error and Confidence Interval#

  • The margin of error (\(e\)) is half the width of this confidence interval.

\[ e = Z* SEM \to e = Z*\frac{\sigma}{\sqrt{n}} \to n = \frac{Z^2*\sigma^2}{e^2} \]

  • \(Z\): the Z-score representing the confidence level.

  • \(σ\): the population standard deviation.

  • \(e\): the desired margin of error.

3.2.4. Important Assumption#

  1. Known Population Standard Deviation (\(\sigma\))

  2. Independence of Observations

  3. Homogeneity of Variance

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