A binomial distribution counts the number of successes in a fixed number of independent trials, such as correct answers, defective items, or heads in coin flips. When the number of trials is large, calculating exact binomial probabilities can become slow and repetitive. The normal approximation replaces the binomial bars with a smooth bell curve that is much easier to work with.
This matters because it lets students estimate binomial probabilities using z-scores and normal tables or calculators.
Key Facts
- For X ~ Bin(n, p), the mean is μ = np.
- For X ~ Bin(n, p), the standard deviation is σ = sqrt(npq), where q = 1 - p.
- Use the normal approximation when np ≥ 10 and nq ≥ 10.
- Approximate X ~ Bin(n, p) with Y ~ N(np, npq).
- Convert values using z = (x - μ) / σ after applying any needed continuity correction.
- Continuity correction: P(X ≤ k) ≈ P(Y < k + 0.5) and P(X ≥ k) ≈ P(Y > k - 0.5).
Vocabulary
- Binomial distribution
- A probability distribution for the number of successes in n independent trials when each trial has the same success probability p.
- Normal approximation
- A method that estimates a binomial probability using a normal distribution with the same mean and standard deviation.
- Continuity correction
- An adjustment of 0.5 added or subtracted when using a continuous normal curve to estimate a discrete binomial probability.
- Success probability
- The value p, which is the probability that one binomial trial results in a success.
- Z-score
- A standardized value that tells how many standard deviations a data value is from the mean.
Common Mistakes to Avoid
- Skipping the condition check. The normal approximation can be inaccurate if np or nq is too small because the binomial distribution may be too skewed.
- Forgetting the continuity correction. A binomial variable uses whole-number counts, so the normal boundary should be shifted by 0.5 to better match the bar area.
- Using σ = npq instead of σ = sqrt(npq). The variance is npq, but the standard deviation used in z = (x - μ) / σ is the square root of the variance.
- Treating p and q as the same value. Since q = 1 - p, using the wrong q gives the wrong spread and changes the final probability.
Practice Questions
- 1 A multiple-choice test has 80 questions, and a student guesses with probability p = 0.25 of getting each correct. Use the normal approximation with continuity correction to estimate P(X ≥ 25).
- 2 A factory produces items with a defect rate of 0.08. In a sample of 200 items, use the normal approximation with continuity correction to estimate P(X ≤ 10 defects).
- 3 Explain why the continuity correction improves the normal approximation when estimating a binomial probability, and describe when the normal approximation should not be used.