1. What is Continuity Correction?

Definition: Continuity correction is an adjustment used when approximating a discrete probability distribution (e.g., binomial) with a continuous distribution (e.g., normal). It accounts for the fact that discrete variables take integer values, while continuous variables cover a range.

Purpose: Improves the accuracy of normal approximations for binomial probabilities, especially for moderate sample sizes or probabilities near 0 or 1.

2. How Does the Calculator Work?

The calculator approximates binomial probabilities using a normal distribution with mean \(\mu = N \cdot p\) and standard deviation \(\sigma = \sqrt{N \cdot p \cdot (1 - p)}\). Continuity correction adjusts the discrete value \(n\) by ±0.5:

• \(P(x = n) \approx P\left(\frac{n - 0.5 - \mu}{\sigma} < z < \frac{n + 0.5 - \mu}{\sigma}\right)\)
• \(P(x \leq n) \approx P\left(z \leq \frac{n + 0.5 - \mu}{\sigma}\right)\)
• \(P(x < n) \approx P\left(z \leq \frac{n - 0.5 - \mu}{\sigma}\right)\)
• \(P(x \geq n) \approx P\left(z \geq \frac{n - 0.5 - \mu}{\sigma}\right)\)
• \(P(x > n) \approx P\left(z \geq \frac{n + 0.5 - \mu}{\sigma}\right)\)

Steps:

• Enter the number of trials (\(N\)), successes (\(n\)), and probability of success (\(p\)).
• Click "Calculate" to compute z-scores and probabilities.

3. When to Use Continuity Correction?

Use when approximating a binomial distribution with a normal distribution, provided:

• \(N \cdot p \geq 5\) and \(N \cdot (1 - p) \geq 5\) (Central Limit Theorem).
• Sample size is moderate (e.g., \(N > 30\)) or probabilities are not extreme.

4. Using the Calculator

Example: Suppose \(N = 100\), \(n = 50\), \(p = 0.5\):

• \(\mu = 100 \cdot 0.5 = 50\), \(\sigma = \sqrt{100 \cdot 0.5 \cdot 0.5} = 5\).
• \(P(x = 50) \approx P\left(\frac{49.5 - 50}{5} < z < \frac{50.5 - 50}{5}\right) = P(-0.1 < z < 0.1) \approx 0.0797\).
• \(P(x \leq 50) \approx P\left(z \leq \frac{50.5 - 50}{5}\right) = P(z \leq 0.1) \approx 0.5398\).
• \(P(x < 50) \approx P\left(z \leq \frac{49.5 - 50}{5}\right) = P(z \leq -0.1) \approx 0.4602\).
• \(P(x \geq 50) \approx P\left(z \geq \frac{50 - 0.5 - 50}{5}\right) = P(z \geq -0.1) \approx 0.5398\).
• \(P(x > 50) \approx P\left(z \geq \frac{50.5 - 50}{5}\right) = P(z \geq 0.1) \approx 0.4602\).

5. Frequently Asked Questions (FAQ)

Q: Why use continuity correction?
A: It adjusts for the difference between discrete (integer) and continuous distributions, improving normal approximation accuracy.

Q: When is normal approximation valid?
A: When \(N \cdot p \geq 5\) and \(N \cdot (1 - p) \geq 5\), ensuring the binomial distribution is approximately normal.

Q: How accurate are the probabilities?
A: The calculator uses a polynomial approximation for the normal CDF, which is reasonably accurate but less precise than statistical software.

Q: Can I use this for small sample sizes?
A: Normal approximation is less reliable for small \(N\). Use exact binomial calculations for \(N < 30\).

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