03/10/2025

Introduction to Binomial to Normal Approximation

• Conversion process from binomial distribution to normal approximation.

• Acknowledgement of possible underestimation or overestimation.

• Importance of correction factors in adjustments.

Key Considerations

• Ensuring conditions for normal approximation are met.

  • Example: Checking binomial success probabilities to satisfy conditions (e.g., np and nq > 10).

Steps in Conversion Process

• Determine the binomial distribution parameters.

• Compute mean and standard deviation for binomial distribution:

  • Mean (µ) calculation: µ = np

  • Standard deviation (σ) calculation: σ = √(npq)

Example Calculations

• For example, with n = 500 and p = 0.06:

  • Mean: µ = 500 * 0.06 = 30

  • Standard deviation: σ = √(500 * 0.06 * 0.94) = 5.31

Correcting for Continuity

• Correction factor is necessary when calculating probabilities:

  • Adjust x values for continuity adjustment: e.g., P(x < a) becomes P(x < a - 0.5).

Probability Calculation Example

• For finding P(x < 25) with correction:

  • Calculate z-score: z = (24.5 - 30) / 5.31 = -1.04.

  • Reference z-table for probability: P(Z < -1.04) = 0.1492.

  • Conclusion: Approximately 15% probability that fewer than 25 adults have cancer.

Handling Specific Probabilities

• For exact probabilities, compute using continuity correction on both sides:

  • Example for P(x = 28): Compute for P(27.5 < X < 28.5).

• Z-scores:

  • Lower bound: z = (27.5 - 30) / 5.31 = -0.47.

  • Upper bound: z = (28.5 - 30) / 5.31 = -0.28.

• Use z-table to find probabilities for both z-scores and calculate the difference.

Final Thoughts

• Importance of verifying conditions for using normal approximation.

• Regular practice with these types of problems will improve proficiency in statistical conversion methods.