Lecture Notes: Connection Between Binomial and Normal Distribution

Key Topics

• Binomial Distribution
• Normal Distribution
• Sample Size (n)
• Probability of Success (p)
• Continuity Correction

Binomial Distribution

• Probability of Success (p): Probability of a single success, always a decimal.
  • Example: Probability of a basketball player making a free throw.
• Sample Size (n): Number of attempts or trials.
  • Example: Probability of making a certain number of baskets out of a set number of attempts.

Relationship to Normal Distribution

• Symmetry of Distribution:
  • p close to 0.5 yields a symmetrical (bell-shaped) distribution.
  • p far from 0.5 results in a skewed distribution.
• Role of Sample Size (n):
  • As n increases, the binomial distribution becomes more symmetrical and bell-shaped.
  • Large n allows binomial distribution to be approximated by a normal distribution.

Mean and Standard Deviation

• Mean (μ): μ = np
  • Represents the expected number of successes.
• Standard Deviation (σ): σ = √(np(1-p))
  • Represents the spread of the distribution.

Rule of Thumb for Normal Approximation

• Conditions for using normal approximation:
  • np ≥ 10
  • n(1-p) ≥ 10
  • Ensures both number of successes and failures are large enough.

Continuity Correction

• Used when approximating the binomial distribution with a normal distribution.
• Adds 0.5 to discrete variables to account for continuous properties of normal distribution.
• Not typically used in exams or practical applications unless specified.

Practical Examples

• Basketball free throws scenario used to illustrate concepts.
  • Calculation of probabilities using both binomial and normal distributions.
  • Comparison of exact binomial probability to normal approximation.

Important Formulas

• Mean: μ = np
• Standard Deviation: σ = √(np(1-p))
• Conditions for Normal Approximation:
  • np ≥ 10
  • n(1-p) ≥ 10

Summary

• Larger sample sizes allow binomial distributions to be approximated by normal distributions.
• Continuity correction is a theoretical adjustment not typically used in practical applications.
• Understanding both distributions is essential for analyzing probabilities, especially in statistical modeling contexts like sports performance analysis.