Module 6: Sampling Distributions
Supporting Lectures:
EGN3443 Module 6 - Sampling Distributions and Inference
1. Central Limit Theorem (CLT) and Its Implications
Definition
The Central Limit Theorem states that when independent random samples of size n are drawn from any population with a finite mean (μ) and variance (σ²), the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the shape of the original population distribution.
Key Implications
- The sample mean (x̄) is approximately normally distributed
- The mean of the sampling distribution equals the population mean (μ)
- The standard deviation of the sampling distribution (standard error) depends on the population standard deviation and sample size
Example Calculation
If a population has:
- Mean (μ) = 50
- Standard Deviation (σ) = 10
- Sample Size (n) = 36
Sample Mean Distribution:
- Mean of sample means = 50
- Standard Error = σ / √n = 10 / √36 = 10 / 6 = 1.67
2. Distribution of Sample Means
Definition
The sampling distribution of the mean is a theoretical distribution that describes the possible values of the sample mean for all possible samples of a given size drawn from a population.
Characteristics
- Centered at the population mean
- Spreads out less as sample size increases
- Becomes more normally distributed with larger sample sizes
Example
Consider a population of test scores:
- Original Population: Skewed distribution
- Sample Sizes: 10, 30, 50
- Observe how the distribution of sample means becomes more normal as sample size increases
Calculation
Variance of Sample Mean = σ² / n Standard Error = σ / √n
3. Standard Error
Definition
Standard Error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean.
Formula
Standard Error (SE) = σ / √n
- σ = Population Standard Deviation
- n = Sample Size
Practical Significance
- Measures the precision of the sample mean
- Smaller standard error indicates more precise estimate
- Decreases as sample size increases
Example Calculation
Population σ = 15 Sample Sizes:
- n = 25: SE = 15 / √25 = 15 / 5 = 3
- n = 100: SE = 15 / √100 = 15 / 10 = 1.5
4. Confidence Intervals
Definition
A confidence interval is a range of values that likely contains the true population parameter with a specified level of confidence.
Key Components
- Point Estimate (sample statistic)
- Margin of Error
- Confidence Level (typically 90%, 95%, or 99%)
Calculation Formula
Confidence Interval = x̄ ± (z-score) * (Standard Error)
Example
Sample Mean (x̄) = 75 Standard Error = 3 Confidence Level = 95% z-score = 1.96
95% Confidence Interval:
- Lower Bound: 75 - (1.96 * 3) = 69.12
- Upper Bound: 75 + (1.96 * 3) = 80.88
Interpretation
We are 95% confident that the true population mean falls between 69.12 and 80.88.
A simpler explanation is at https://www.statisticshowto.com/probability-and-statistics/confidence-interval/
5. Sample Size Determination
Definition
Sample size determination involves calculating the number of observations needed to achieve a desired level of precision in statistical estimates.
Factors Influencing Sample Size
- Desired Confidence Level
- Expected Variability
- Desired Margin of Error
- Population Size (for finite populations)
Basic Formula
n = (z²* σ² ) / E²
- z = z-score for desired confidence level
- σ = Population Standard Deviation
- E = Desired Margin of Error
Example Calculation
- Desired Confidence Level: 95% (z = 1.96)
- Population Standard Deviation: 10
- Desired Margin of Error: 2
Sample Size: n = (1.96² * 10²) / 2² = (3.84 * 100) / 4 = 384 / 4 = 96 samples needed
Additional Study Resources