A deep understanding of the sample mean as a random variable is essential. This section expands on the properties of the sample mean, its significance in statistical analysis, and how these properties are applied in problem-solving scenarios.

Introduction to Sample Mean

The sample mean, $\bar{X}$, is crucial in statistics for indicating a sample's average value, selected from a larger population.

Definition: Calculated as $\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}$, where $X_i$ are sample observations and $n$ is the sample size.

Nature: As $\bar{X}$ varies with each sample, it's considered a random variable, essential for estimating the population's central tendency.

Properties of Sample Mean

1. Expected Value (Mean): The average of sample means equals the population mean $(\mu)$. It shows the sample mean is a good guess for the population mean.

2. Variance: How spread out the sample means are. Less spread with bigger samples. Formula: $\frac{\sigma^2}{n}$, where $\sigma^2$ is population variance and $(n)$ is sample size.

Examples

Example 1: Student Scores

A sample of 40 students' scores, with a population variance $(\sigma^2)$ of 25. Calculate the expected value and variance of sample mean.

Solution:

1. Expected Value of Sample Mean $(E(\bar{X}))$:

• $E(\bar{X})$ equals the population mean $(\mu)$.
• If sample mean $(\bar{X})$ is 68, then $E(\bar{X}) = 68$.

2. Variance of Sample Mean $(\text{Var}(\bar{X}))$:

• Formula: $\text{Var}(\bar{X}) = \frac{\sigma^2}{n}$.
• Calculation: $\text{Var}(\bar{X}) = \frac{25}{40} = 0.625$.

Conclusion: The variance of the sample mean is 0.625, suggesting that $\bar{X}$ is a reliable estimator of the population mean.

Example 2: Central Limit Theorem (CLT)

For a sample of 50 from a non-normal distribution with mean 75 and variance 36, apply the Central Limit Theorem to determine the expected value and variance of the sample mean.

Solution:

1. Distribution of Sample Mean:

• With $n = 50$ (adequate for CLT), the sample mean $(\bar{X})$approximates a normal distribution.

2. Expected Value and Variance:

• Expected Value $(E(\bar{X}))$: According to CLT, $E(\bar{X}) = \mu = 75$.
• Variance $(\text{Var}(\bar{X}))$: $\text{Var}(\bar{X}) = \frac{\sigma^2}{n} = \frac{36}{50} = 0.72.$

Interpretation:

• Sample Mean's Distribution: Norma with mean = 75.
• Variance = 0.72: This shows the spread of sample means around 75.
• Implication: Normal distribution analysis is applicable to (\bar{X}), despite the original population's non-normal distribution.

Demonstrating the effect of the Central Limit Theorem, with a sample mean distribution becoming normal.