Overview
This lecture reviews the steps for applying the Central Limit Theorem (CLT), emphasizing checking its conditions, and explains how to interpret the resulting sampling distribution for making statistical inferences.
Central Limit Theorem Conditions
- Always check all three CLT conditions before proceeding.
- Condition 1: Data must be collected randomly (random selection).
- Condition 2: Population must be normal, OR sample size > 25 (sample size of 30 qualifies).
- Condition 3: Population size must be at least 10 times the sample size (at least 300 women for a sample of 30).
Verifying the Normality Condition
- If the population is not normal (e.g., skewed right), sample size > 25 is sufficient for normality.
- Only one part of the "OR" in condition two needs to be satisfied for CLT to apply.
Importance of the Sampling Distribution
- Once CLT conditions hold, the sampling distribution is always normal.
- A normal sampling distribution allows use of normalcdf to calculate probabilities.
- Calculating probabilities is essential for confidence intervals and hypothesis testing.
Center and Spread of the Sampling Distribution
- The mean (center) of the sampling distribution equals the population mean if data are random.
- The spread (standard error) is calculated as population standard deviation divided by the square root of sample size.
- Standard error formula: SE = σ / √n (σ = population standard deviation, n = sample size).
Statistical Inference
- Use the sampling distribution to make predictions about the population mean.
- Recognize that predictions include uncertainty, captured by the standard error (e.g., ±2.37 BPM).
Key Terms & Definitions
- Central Limit Theorem (CLT) — states that sample means from a large enough sample are normally distributed, regardless of the population shape.
- Sampling Distribution — distribution of a statistic (like mean) over many samples.
- Standard Error (SE) — standard deviation of the sampling distribution, SE = σ / √n.
- Statistical Inference — using sample data to make conclusions or predictions about a population.
- Normalcdf — calculator function for finding probabilities under the normal curve.
Action Items / Next Steps
- Review and check CLT conditions for your own data sets.
- Write the standard error formula on your note sheet for reference.
- Practice calculating sampling distribution mean and standard error with provided values.