Overview
This lecture covers when and how to apply the Central Limit Theorem (CLT) to sample means, including required conditions and calculation of sampling distribution characteristics.
Central Limit Theorem Conditions
- Randomness: The sample must be randomly selected.
- Normality: Either the population is normally distributed, or the sample size is large enough (usually n > 25).
- Population information: If population shape is not given, normality cannot be assumed unless supported by sample size.
- If both population normality and large sample size conditions fail, CLT cannot be applied.
- If the population is explicitly normal, CLT applies regardless of sample size.
- Large Population: Population should be at least 10 times the sample size.
Application to Sample Problem
- Sample: 10 randomly selected students; population mean = 73, population standard deviation = 7.8.
- Initial check: Population shape not given, sample size (n=10) too small for CLT by sample size rule.
- With additional info that population is normal, CLT conditions are satisfied even for n=10.
- Population size (over 100 students) satisfies large population condition.
Sampling Distribution Characteristics
- Shape: Normal, due to normal population shape.
- Center: Mean of sampling distribution is the population mean (73 points).
- Spread: Standard error calculated as population standard deviation divided by the square root of sample size.
- Calculation: 7.8 / √10 ≈ 2.47 points.
- Interpretation: The mean score for a sample of 10 students is expected to be 73, with a standard error of 2.47 points.
Key Terms & Definitions
- Central Limit Theorem (CLT) — Describes the shape, center, and spread of sampling distributions as normal under certain conditions.
- Standard Error — The standard deviation of the sampling distribution of the sample mean (σ/√n).
- Sampling Distribution — Distribution of sample means over repeated samples from the same population.
Action Items / Next Steps
- Practice using CLT formulas for center and spread.
- Learn to check and justify all three CLT conditions for sample means.
- Review related problems in Sections 9.1 and 9.2 for exam preparation.