Overview
The lecture reviews checking the conditions for applying the Central Limit Theorem (CLT) to proportions, using a scenario about sampling HIV rates in a population.
Central Limit Theorem Conditions
- First CLT condition: the sample must be random; look for the word "random" to confirm.
- In this example, the sample of 1,000 people is randomly selected, so condition one is satisfied.
- Second CLT condition: the sample size must be large, checked by the number of expected successes and failures.
- "Success" in this context means an individual has HIV.
- The number of successes is calculated by multiplying the sample size by the given proportion (1,000 × 0.04 = 40).
- To meet the "large sample" condition, the number of successes must be greater than 10.
- In this scenario, there are 40 successes, which is greater than 10, so the large sample condition is satisfied.
Failure to Meet a Condition
- If any CLT condition is not met, the theorem cannot be applied to the data.
- If the sample is not large enough (successes or failures ≤ 10), you must "throw the Central Limit Theorem out the window."
- Do not continue to check other conditions if one fails.
Importance of Conditions
- All required conditions must be satisfied to apply the Central Limit Theorem.
- If all are satisfied, you may proceed to use the CLT for calculations.
- If any condition fails, stop and do not apply the CLT.
Key Terms & Definitions
- Random Sample — A sample where every individual has an equal chance of being chosen.
- Success — An outcome of interest (here, a person who has HIV).
- Central Limit Theorem (CLT) — A statistical theory stating sample means (or proportions) are approximately normally distributed if certain conditions are met.
Action Items / Next Steps
- Review homework problems applying the CLT to proportions.
- Practice calculating expected successes and checking CLT conditions.