CLT Large Sample Check

Overview

The lecture explains how to check if a sample is large enough to satisfy the Central Limit Theorem (CLT) large sample condition by ensuring both the number of observed successes and failures are at least 10.

Central Limit Theorem Large Sample Condition

• The large sample condition requires both the number of successes and failures to be ≥ 10.
• Success is defined by the characteristic being studied (e.g., students who carry calculators).
• Failure is defined as not having the characteristic (e.g., students who do not carry calculators).

Calculating Number of Successes and Failures

• Number of successes = sample size × observed proportion (n × p̂).
• For 400 students with 65% carrying calculators: 400 × 0.65 = 260 successes.
• Number of failures = sample size – number of successes (n – successes).
• For 400 students: 400 – 260 = 140 failures.

Verifying the Condition

• Both successes (260) and failures (140) are greater than 10.
• Therefore, the large sample condition for applying the CLT is satisfied.

Example Application

• For 346 nurses, with 35 males: number of successes = 35 (males).
• Number who are not male = 346 – 35 = 311.
• Both numbers are above 10, so the large sample condition is met.

Key Terms & Definitions

• Central Limit Theorem (CLT) — a statistical theory stating that the sampling distribution of the sample mean approaches a normal distribution as the sample size grows.
• Success — the outcome of interest in a survey (e.g., carrying a calculator).
• Failure — the opposite outcome (e.g., not carrying a calculator).
• Large Sample Condition — both the number of successes and failures must be at least 10 for the CLT to apply.

Action Items / Next Steps

• Try similar calculations for different characteristics to check if the large sample condition is met.