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 ($\sigma$): 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

3.2.3. Margin of Error and Confidence Interval#

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

• $Z$: the Z-score representing the confidence level.

• $\sigma$: 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